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Theorem locfinreflem 30035
Description: A locally finite refinement of an open cover induces a locally finite open cover with the original index set. This is fact 2 of http://at.yorku.ca/p/a/c/a/02.pdf, it is expressed by exposing a function 𝑓 from the original cover 𝑈, which is taken as the index set. The solution is constructed by building unions, so the same method can be used to prove a similar theorem about closed covers. (Contributed by Thierry Arnoux, 29-Jan-2020.)
Hypotheses
Ref Expression
locfinref.x 𝑋 = 𝐽
locfinref.1 (𝜑𝑈𝐽)
locfinref.2 (𝜑𝑋 = 𝑈)
locfinref.3 (𝜑𝑉𝐽)
locfinref.4 (𝜑𝑉Ref𝑈)
locfinref.5 (𝜑𝑉 ∈ (LocFin‘𝐽))
Assertion
Ref Expression
locfinreflem (𝜑 → ∃𝑓((Fun 𝑓 ∧ dom 𝑓𝑈 ∧ ran 𝑓𝐽) ∧ (ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽))))
Distinct variable groups:   𝑓,𝐽   𝑈,𝑓   𝑓,𝑉   𝜑,𝑓
Allowed substitution hint:   𝑋(𝑓)

Proof of Theorem locfinreflem
Dummy variables 𝑔 𝑗 𝑘 𝑢 𝑣 𝑤 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 locfinref.4 . . . 4 (𝜑𝑉Ref𝑈)
2 locfinref.5 . . . . 5 (𝜑𝑉 ∈ (LocFin‘𝐽))
3 reff 30034 . . . . 5 (𝑉 ∈ (LocFin‘𝐽) → (𝑉Ref𝑈 ↔ ( 𝑈 𝑉 ∧ ∃𝑔(𝑔:𝑉𝑈 ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)))))
42, 3syl 17 . . . 4 (𝜑 → (𝑉Ref𝑈 ↔ ( 𝑈 𝑉 ∧ ∃𝑔(𝑔:𝑉𝑈 ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)))))
51, 4mpbid 222 . . 3 (𝜑 → ( 𝑈 𝑉 ∧ ∃𝑔(𝑔:𝑉𝑈 ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣))))
65simprd 478 . 2 (𝜑 → ∃𝑔(𝑔:𝑉𝑈 ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)))
7 funmpt 5964 . . . . . 6 Fun (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))
87a1i 11 . . . . 5 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → Fun (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})))
9 eqid 2651 . . . . . . 7 (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) = (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))
109dmmptss 5669 . . . . . 6 dom (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ⊆ ran 𝑔
11 frn 6091 . . . . . . 7 (𝑔:𝑉𝑈 → ran 𝑔𝑈)
1211ad2antlr 763 . . . . . 6 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → ran 𝑔𝑈)
1310, 12syl5ss 3647 . . . . 5 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → dom (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ⊆ 𝑈)
14 locfintop 21372 . . . . . . . . . 10 (𝑉 ∈ (LocFin‘𝐽) → 𝐽 ∈ Top)
152, 14syl 17 . . . . . . . . 9 (𝜑𝐽 ∈ Top)
1615ad3antrrr 766 . . . . . . . 8 ((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑢 ∈ ran 𝑔) → 𝐽 ∈ Top)
17 cnvimass 5520 . . . . . . . . . 10 (𝑔 “ {𝑢}) ⊆ dom 𝑔
18 fdm 6089 . . . . . . . . . . 11 (𝑔:𝑉𝑈 → dom 𝑔 = 𝑉)
1918ad3antlr 767 . . . . . . . . . 10 ((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑢 ∈ ran 𝑔) → dom 𝑔 = 𝑉)
2017, 19syl5sseq 3686 . . . . . . . . 9 ((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑢 ∈ ran 𝑔) → (𝑔 “ {𝑢}) ⊆ 𝑉)
21 locfinref.3 . . . . . . . . . 10 (𝜑𝑉𝐽)
2221ad3antrrr 766 . . . . . . . . 9 ((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑢 ∈ ran 𝑔) → 𝑉𝐽)
2320, 22sstrd 3646 . . . . . . . 8 ((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑢 ∈ ran 𝑔) → (𝑔 “ {𝑢}) ⊆ 𝐽)
24 uniopn 20750 . . . . . . . 8 ((𝐽 ∈ Top ∧ (𝑔 “ {𝑢}) ⊆ 𝐽) → (𝑔 “ {𝑢}) ∈ 𝐽)
2516, 23, 24syl2anc 694 . . . . . . 7 ((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑢 ∈ ran 𝑔) → (𝑔 “ {𝑢}) ∈ 𝐽)
2625ralrimiva 2995 . . . . . 6 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → ∀𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}) ∈ 𝐽)
279rnmptss 6432 . . . . . 6 (∀𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}) ∈ 𝐽 → ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ⊆ 𝐽)
2826, 27syl 17 . . . . 5 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ⊆ 𝐽)
29 eqid 2651 . . . . . . . . . 10 𝑉 = 𝑉
30 eqid 2651 . . . . . . . . . 10 𝑈 = 𝑈
3129, 30refbas 21361 . . . . . . . . 9 (𝑉Ref𝑈 𝑈 = 𝑉)
321, 31syl 17 . . . . . . . 8 (𝜑 𝑈 = 𝑉)
3332ad2antrr 762 . . . . . . 7 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → 𝑈 = 𝑉)
34 nfv 1883 . . . . . . . . . . . . 13 𝑣(𝜑𝑔:𝑉𝑈)
35 nfra1 2970 . . . . . . . . . . . . 13 𝑣𝑣𝑉 𝑣 ⊆ (𝑔𝑣)
3634, 35nfan 1868 . . . . . . . . . . . 12 𝑣((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣))
37 nfre1 3034 . . . . . . . . . . . 12 𝑣𝑣𝑉 𝑥𝑣
3836, 37nfan 1868 . . . . . . . . . . 11 𝑣(((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ ∃𝑣𝑉 𝑥𝑣)
39 ffn 6083 . . . . . . . . . . . . . . 15 (𝑔:𝑉𝑈𝑔 Fn 𝑉)
4039ad4antlr 771 . . . . . . . . . . . . . 14 (((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑣𝑉) ∧ 𝑥𝑣) → 𝑔 Fn 𝑉)
41 simplr 807 . . . . . . . . . . . . . 14 (((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑣𝑉) ∧ 𝑥𝑣) → 𝑣𝑉)
42 fnfvelrn 6396 . . . . . . . . . . . . . 14 ((𝑔 Fn 𝑉𝑣𝑉) → (𝑔𝑣) ∈ ran 𝑔)
4340, 41, 42syl2anc 694 . . . . . . . . . . . . 13 (((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑣𝑉) ∧ 𝑥𝑣) → (𝑔𝑣) ∈ ran 𝑔)
44 ssid 3657 . . . . . . . . . . . . . . 15 𝑣𝑣
4539ad3antlr 767 . . . . . . . . . . . . . . . 16 ((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑣𝑉) → 𝑔 Fn 𝑉)
46 eqid 2651 . . . . . . . . . . . . . . . . 17 (𝑔𝑣) = (𝑔𝑣)
47 fniniseg 6378 . . . . . . . . . . . . . . . . . 18 (𝑔 Fn 𝑉 → (𝑣 ∈ (𝑔 “ {(𝑔𝑣)}) ↔ (𝑣𝑉 ∧ (𝑔𝑣) = (𝑔𝑣))))
4847biimpar 501 . . . . . . . . . . . . . . . . 17 ((𝑔 Fn 𝑉 ∧ (𝑣𝑉 ∧ (𝑔𝑣) = (𝑔𝑣))) → 𝑣 ∈ (𝑔 “ {(𝑔𝑣)}))
4946, 48mpanr2 720 . . . . . . . . . . . . . . . 16 ((𝑔 Fn 𝑉𝑣𝑉) → 𝑣 ∈ (𝑔 “ {(𝑔𝑣)}))
5045, 49sylancom 702 . . . . . . . . . . . . . . 15 ((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑣𝑉) → 𝑣 ∈ (𝑔 “ {(𝑔𝑣)}))
51 ssuni 4491 . . . . . . . . . . . . . . 15 ((𝑣𝑣𝑣 ∈ (𝑔 “ {(𝑔𝑣)})) → 𝑣 (𝑔 “ {(𝑔𝑣)}))
5244, 50, 51sylancr 696 . . . . . . . . . . . . . 14 ((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑣𝑉) → 𝑣 (𝑔 “ {(𝑔𝑣)}))
5352sselda 3636 . . . . . . . . . . . . 13 (((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑣𝑉) ∧ 𝑥𝑣) → 𝑥 (𝑔 “ {(𝑔𝑣)}))
54 sneq 4220 . . . . . . . . . . . . . . . . 17 (𝑢 = (𝑔𝑣) → {𝑢} = {(𝑔𝑣)})
5554imaeq2d 5501 . . . . . . . . . . . . . . . 16 (𝑢 = (𝑔𝑣) → (𝑔 “ {𝑢}) = (𝑔 “ {(𝑔𝑣)}))
5655unieqd 4478 . . . . . . . . . . . . . . 15 (𝑢 = (𝑔𝑣) → (𝑔 “ {𝑢}) = (𝑔 “ {(𝑔𝑣)}))
5756eleq2d 2716 . . . . . . . . . . . . . 14 (𝑢 = (𝑔𝑣) → (𝑥 (𝑔 “ {𝑢}) ↔ 𝑥 (𝑔 “ {(𝑔𝑣)})))
5857rspcev 3340 . . . . . . . . . . . . 13 (((𝑔𝑣) ∈ ran 𝑔𝑥 (𝑔 “ {(𝑔𝑣)})) → ∃𝑢 ∈ ran 𝑔 𝑥 (𝑔 “ {𝑢}))
5943, 53, 58syl2anc 694 . . . . . . . . . . . 12 (((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑣𝑉) ∧ 𝑥𝑣) → ∃𝑢 ∈ ran 𝑔 𝑥 (𝑔 “ {𝑢}))
6059adantllr 755 . . . . . . . . . . 11 ((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ ∃𝑣𝑉 𝑥𝑣) ∧ 𝑣𝑉) ∧ 𝑥𝑣) → ∃𝑢 ∈ ran 𝑔 𝑥 (𝑔 “ {𝑢}))
61 simpr 476 . . . . . . . . . . 11 ((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ ∃𝑣𝑉 𝑥𝑣) → ∃𝑣𝑉 𝑥𝑣)
6238, 60, 61r19.29af 3105 . . . . . . . . . 10 ((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ ∃𝑣𝑉 𝑥𝑣) → ∃𝑢 ∈ ran 𝑔 𝑥 (𝑔 “ {𝑢}))
63 nfv 1883 . . . . . . . . . . . . . 14 𝑣 𝑢 ∈ ran 𝑔
6436, 63nfan 1868 . . . . . . . . . . . . 13 𝑣(((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑢 ∈ ran 𝑔)
65 nfv 1883 . . . . . . . . . . . . 13 𝑣 𝑥 (𝑔 “ {𝑢})
6664, 65nfan 1868 . . . . . . . . . . . 12 𝑣((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑢 ∈ ran 𝑔) ∧ 𝑥 (𝑔 “ {𝑢}))
6720ad3antrrr 766 . . . . . . . . . . . . 13 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑢 ∈ ran 𝑔) ∧ 𝑥 (𝑔 “ {𝑢})) ∧ 𝑣 ∈ (𝑔 “ {𝑢})) ∧ 𝑥𝑣) → (𝑔 “ {𝑢}) ⊆ 𝑉)
68 simplr 807 . . . . . . . . . . . . 13 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑢 ∈ ran 𝑔) ∧ 𝑥 (𝑔 “ {𝑢})) ∧ 𝑣 ∈ (𝑔 “ {𝑢})) ∧ 𝑥𝑣) → 𝑣 ∈ (𝑔 “ {𝑢}))
6967, 68sseldd 3637 . . . . . . . . . . . 12 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑢 ∈ ran 𝑔) ∧ 𝑥 (𝑔 “ {𝑢})) ∧ 𝑣 ∈ (𝑔 “ {𝑢})) ∧ 𝑥𝑣) → 𝑣𝑉)
70 simpr 476 . . . . . . . . . . . 12 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑢 ∈ ran 𝑔) ∧ 𝑥 (𝑔 “ {𝑢})) ∧ 𝑣 ∈ (𝑔 “ {𝑢})) ∧ 𝑥𝑣) → 𝑥𝑣)
71 simpr 476 . . . . . . . . . . . . 13 (((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑢 ∈ ran 𝑔) ∧ 𝑥 (𝑔 “ {𝑢})) → 𝑥 (𝑔 “ {𝑢}))
72 eluni2 4472 . . . . . . . . . . . . 13 (𝑥 (𝑔 “ {𝑢}) ↔ ∃𝑣 ∈ (𝑔 “ {𝑢})𝑥𝑣)
7371, 72sylib 208 . . . . . . . . . . . 12 (((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑢 ∈ ran 𝑔) ∧ 𝑥 (𝑔 “ {𝑢})) → ∃𝑣 ∈ (𝑔 “ {𝑢})𝑥𝑣)
7466, 69, 70, 73reximd2a 3042 . . . . . . . . . . 11 (((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑢 ∈ ran 𝑔) ∧ 𝑥 (𝑔 “ {𝑢})) → ∃𝑣𝑉 𝑥𝑣)
7574r19.29an 3106 . . . . . . . . . 10 ((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ ∃𝑢 ∈ ran 𝑔 𝑥 (𝑔 “ {𝑢})) → ∃𝑣𝑉 𝑥𝑣)
7662, 75impbida 895 . . . . . . . . 9 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → (∃𝑣𝑉 𝑥𝑣 ↔ ∃𝑢 ∈ ran 𝑔 𝑥 (𝑔 “ {𝑢})))
77 eluni2 4472 . . . . . . . . 9 (𝑥 𝑉 ↔ ∃𝑣𝑉 𝑥𝑣)
78 eliun 4556 . . . . . . . . 9 (𝑥 𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}) ↔ ∃𝑢 ∈ ran 𝑔 𝑥 (𝑔 “ {𝑢}))
7976, 77, 783bitr4g 303 . . . . . . . 8 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → (𝑥 𝑉𝑥 𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})))
8079eqrdv 2649 . . . . . . 7 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → 𝑉 = 𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))
81 dfiun3g 5410 . . . . . . . 8 (∀𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}) ∈ 𝐽 𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}) = ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})))
8226, 81syl 17 . . . . . . 7 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → 𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}) = ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})))
8333, 80, 823eqtrd 2689 . . . . . 6 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → 𝑈 = ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})))
8411ad3antlr 767 . . . . . . . . 9 ((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))) → ran 𝑔𝑈)
85 vex 3234 . . . . . . . . . . 11 𝑤 ∈ V
869elrnmpt 5404 . . . . . . . . . . 11 (𝑤 ∈ V → (𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ↔ ∃𝑢 ∈ ran 𝑔 𝑤 = (𝑔 “ {𝑢})))
8785, 86mp1i 13 . . . . . . . . . 10 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → (𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ↔ ∃𝑢 ∈ ran 𝑔 𝑤 = (𝑔 “ {𝑢})))
8887biimpa 500 . . . . . . . . 9 ((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))) → ∃𝑢 ∈ ran 𝑔 𝑤 = (𝑔 “ {𝑢}))
89 ssrexv 3700 . . . . . . . . 9 (ran 𝑔𝑈 → (∃𝑢 ∈ ran 𝑔 𝑤 = (𝑔 “ {𝑢}) → ∃𝑢𝑈 𝑤 = (𝑔 “ {𝑢})))
9084, 88, 89sylc 65 . . . . . . . 8 ((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))) → ∃𝑢𝑈 𝑤 = (𝑔 “ {𝑢}))
91 nfv 1883 . . . . . . . . . 10 𝑢((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣))
92 nfmpt1 4780 . . . . . . . . . . . 12 𝑢(𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))
9392nfrn 5400 . . . . . . . . . . 11 𝑢ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))
9493nfcri 2787 . . . . . . . . . 10 𝑢 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))
9591, 94nfan 1868 . . . . . . . . 9 𝑢(((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})))
96 simpr 476 . . . . . . . . . . 11 ((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))) ∧ 𝑢𝑈) ∧ 𝑤 = (𝑔 “ {𝑢})) → 𝑤 = (𝑔 “ {𝑢}))
97 nfv 1883 . . . . . . . . . . . . . . . 16 𝑣 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))
9836, 97nfan 1868 . . . . . . . . . . . . . . 15 𝑣(((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})))
99 nfv 1883 . . . . . . . . . . . . . . 15 𝑣 𝑢𝑈
10098, 99nfan 1868 . . . . . . . . . . . . . 14 𝑣((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))) ∧ 𝑢𝑈)
101 nfv 1883 . . . . . . . . . . . . . 14 𝑣 𝑤 = (𝑔 “ {𝑢})
102100, 101nfan 1868 . . . . . . . . . . . . 13 𝑣(((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))) ∧ 𝑢𝑈) ∧ 𝑤 = (𝑔 “ {𝑢}))
103 simp-5r 826 . . . . . . . . . . . . . . . 16 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))) ∧ 𝑢𝑈) ∧ 𝑤 = (𝑔 “ {𝑢})) ∧ 𝑣 ∈ (𝑔 “ {𝑢})) → ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣))
10439ad5antlr 775 . . . . . . . . . . . . . . . . . . 19 ((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))) ∧ 𝑢𝑈) ∧ 𝑤 = (𝑔 “ {𝑢})) → 𝑔 Fn 𝑉)
105 fniniseg 6378 . . . . . . . . . . . . . . . . . . 19 (𝑔 Fn 𝑉 → (𝑣 ∈ (𝑔 “ {𝑢}) ↔ (𝑣𝑉 ∧ (𝑔𝑣) = 𝑢)))
106104, 105syl 17 . . . . . . . . . . . . . . . . . 18 ((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))) ∧ 𝑢𝑈) ∧ 𝑤 = (𝑔 “ {𝑢})) → (𝑣 ∈ (𝑔 “ {𝑢}) ↔ (𝑣𝑉 ∧ (𝑔𝑣) = 𝑢)))
107106biimpa 500 . . . . . . . . . . . . . . . . 17 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))) ∧ 𝑢𝑈) ∧ 𝑤 = (𝑔 “ {𝑢})) ∧ 𝑣 ∈ (𝑔 “ {𝑢})) → (𝑣𝑉 ∧ (𝑔𝑣) = 𝑢))
108107simpld 474 . . . . . . . . . . . . . . . 16 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))) ∧ 𝑢𝑈) ∧ 𝑤 = (𝑔 “ {𝑢})) ∧ 𝑣 ∈ (𝑔 “ {𝑢})) → 𝑣𝑉)
109 rspa 2959 . . . . . . . . . . . . . . . 16 ((∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣) ∧ 𝑣𝑉) → 𝑣 ⊆ (𝑔𝑣))
110103, 108, 109syl2anc 694 . . . . . . . . . . . . . . 15 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))) ∧ 𝑢𝑈) ∧ 𝑤 = (𝑔 “ {𝑢})) ∧ 𝑣 ∈ (𝑔 “ {𝑢})) → 𝑣 ⊆ (𝑔𝑣))
111107simprd 478 . . . . . . . . . . . . . . 15 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))) ∧ 𝑢𝑈) ∧ 𝑤 = (𝑔 “ {𝑢})) ∧ 𝑣 ∈ (𝑔 “ {𝑢})) → (𝑔𝑣) = 𝑢)
112110, 111sseqtrd 3674 . . . . . . . . . . . . . 14 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))) ∧ 𝑢𝑈) ∧ 𝑤 = (𝑔 “ {𝑢})) ∧ 𝑣 ∈ (𝑔 “ {𝑢})) → 𝑣𝑢)
113112ex 449 . . . . . . . . . . . . 13 ((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))) ∧ 𝑢𝑈) ∧ 𝑤 = (𝑔 “ {𝑢})) → (𝑣 ∈ (𝑔 “ {𝑢}) → 𝑣𝑢))
114102, 113ralrimi 2986 . . . . . . . . . . . 12 ((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))) ∧ 𝑢𝑈) ∧ 𝑤 = (𝑔 “ {𝑢})) → ∀𝑣 ∈ (𝑔 “ {𝑢})𝑣𝑢)
115 unissb 4501 . . . . . . . . . . . 12 ( (𝑔 “ {𝑢}) ⊆ 𝑢 ↔ ∀𝑣 ∈ (𝑔 “ {𝑢})𝑣𝑢)
116114, 115sylibr 224 . . . . . . . . . . 11 ((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))) ∧ 𝑢𝑈) ∧ 𝑤 = (𝑔 “ {𝑢})) → (𝑔 “ {𝑢}) ⊆ 𝑢)
11796, 116eqsstrd 3672 . . . . . . . . . 10 ((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))) ∧ 𝑢𝑈) ∧ 𝑤 = (𝑔 “ {𝑢})) → 𝑤𝑢)
118117exp31 629 . . . . . . . . 9 ((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))) → (𝑢𝑈 → (𝑤 = (𝑔 “ {𝑢}) → 𝑤𝑢)))
11995, 118reximdai 3041 . . . . . . . 8 ((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))) → (∃𝑢𝑈 𝑤 = (𝑔 “ {𝑢}) → ∃𝑢𝑈 𝑤𝑢))
12090, 119mpd 15 . . . . . . 7 ((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))) → ∃𝑢𝑈 𝑤𝑢)
121120ralrimiva 2995 . . . . . 6 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → ∀𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))∃𝑢𝑈 𝑤𝑢)
122 vex 3234 . . . . . . . . . 10 𝑔 ∈ V
123122rnex 7142 . . . . . . . . 9 ran 𝑔 ∈ V
124123mptex 6527 . . . . . . . 8 (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∈ V
125 rnexg 7140 . . . . . . . 8 ((𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∈ V → ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∈ V)
126124, 125mp1i 13 . . . . . . 7 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∈ V)
127 eqid 2651 . . . . . . . 8 ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) = ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))
128127, 30isref 21360 . . . . . . 7 (ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∈ V → (ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))Ref𝑈 ↔ ( 𝑈 = ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∧ ∀𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))∃𝑢𝑈 𝑤𝑢)))
129126, 128syl 17 . . . . . 6 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → (ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))Ref𝑈 ↔ ( 𝑈 = ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∧ ∀𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))∃𝑢𝑈 𝑤𝑢)))
13083, 121, 129mpbir2and 977 . . . . 5 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))Ref𝑈)
13115ad2antrr 762 . . . . . 6 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → 𝐽 ∈ Top)
132 locfinref.2 . . . . . . . 8 (𝜑𝑋 = 𝑈)
133132ad2antrr 762 . . . . . . 7 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → 𝑋 = 𝑈)
134133, 83eqtrd 2685 . . . . . 6 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → 𝑋 = ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})))
135 nfv 1883 . . . . . . . . 9 𝑣 𝑥𝑋
13636, 135nfan 1868 . . . . . . . 8 𝑣(((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋)
137 simplr 807 . . . . . . . 8 ((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ (𝑥𝑣 ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin)) → 𝑣𝐽)
138 ffun 6086 . . . . . . . . . . . . . 14 (𝑔:𝑉𝑈 → Fun 𝑔)
139138ad6antlr 779 . . . . . . . . . . . . 13 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) → Fun 𝑔)
140 imafi 8300 . . . . . . . . . . . . 13 ((Fun 𝑔 ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) → (𝑔 “ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅}) ∈ Fin)
141139, 140sylancom 702 . . . . . . . . . . . 12 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) → (𝑔 “ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅}) ∈ Fin)
142 simp3 1083 . . . . . . . . . . . . . . . . . . 19 ((((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) ∧ 𝑘 ∈ ran 𝑔𝑤 = ((𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))‘𝑘)) → 𝑤 = ((𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))‘𝑘))
143 sneq 4220 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑢 = 𝑘 → {𝑢} = {𝑘})
144143imaeq2d 5501 . . . . . . . . . . . . . . . . . . . . . 22 (𝑢 = 𝑘 → (𝑔 “ {𝑢}) = (𝑔 “ {𝑘}))
145144unieqd 4478 . . . . . . . . . . . . . . . . . . . . 21 (𝑢 = 𝑘 (𝑔 “ {𝑢}) = (𝑔 “ {𝑘}))
146122cnvex 7155 . . . . . . . . . . . . . . . . . . . . . . 23 𝑔 ∈ V
147 imaexg 7145 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑔 ∈ V → (𝑔 “ {𝑘}) ∈ V)
148146, 147ax-mp 5 . . . . . . . . . . . . . . . . . . . . . 22 (𝑔 “ {𝑘}) ∈ V
149148uniex 6995 . . . . . . . . . . . . . . . . . . . . 21 (𝑔 “ {𝑘}) ∈ V
150145, 9, 149fvmpt 6321 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ ran 𝑔 → ((𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))‘𝑘) = (𝑔 “ {𝑘}))
1511503ad2ant2 1103 . . . . . . . . . . . . . . . . . . 19 ((((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) ∧ 𝑘 ∈ ran 𝑔𝑤 = ((𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))‘𝑘)) → ((𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))‘𝑘) = (𝑔 “ {𝑘}))
152142, 151eqtrd 2685 . . . . . . . . . . . . . . . . . 18 ((((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) ∧ 𝑘 ∈ ran 𝑔𝑤 = ((𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))‘𝑘)) → 𝑤 = (𝑔 “ {𝑘}))
153152ineq1d 3846 . . . . . . . . . . . . . . . . 17 ((((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) ∧ 𝑘 ∈ ran 𝑔𝑤 = ((𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))‘𝑘)) → (𝑤𝑣) = ( (𝑔 “ {𝑘}) ∩ 𝑣))
154153neeq1d 2882 . . . . . . . . . . . . . . . 16 ((((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) ∧ 𝑘 ∈ ran 𝑔𝑤 = ((𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))‘𝑘)) → ((𝑤𝑣) ≠ ∅ ↔ ( (𝑔 “ {𝑘}) ∩ 𝑣) ≠ ∅))
155123a1i 11 . . . . . . . . . . . . . . . 16 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) → ran 𝑔 ∈ V)
156 imaexg 7145 . . . . . . . . . . . . . . . . . . . . 21 (𝑔 ∈ V → (𝑔 “ {𝑢}) ∈ V)
157146, 156ax-mp 5 . . . . . . . . . . . . . . . . . . . 20 (𝑔 “ {𝑢}) ∈ V
158157uniex 6995 . . . . . . . . . . . . . . . . . . 19 (𝑔 “ {𝑢}) ∈ V
159158, 9fnmpti 6060 . . . . . . . . . . . . . . . . . 18 (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) Fn ran 𝑔
160 dffn4 6159 . . . . . . . . . . . . . . . . . 18 ((𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) Fn ran 𝑔 ↔ (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})):ran 𝑔onto→ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})))
161159, 160mpbi 220 . . . . . . . . . . . . . . . . 17 (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})):ran 𝑔onto→ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))
162161a1i 11 . . . . . . . . . . . . . . . 16 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) → (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})):ran 𝑔onto→ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})))
163154, 155, 162rabfodom 29470 . . . . . . . . . . . . . . 15 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) → {𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∣ (𝑤𝑣) ≠ ∅} ≼ {𝑘 ∈ ran 𝑔 ∣ ( (𝑔 “ {𝑘}) ∩ 𝑣) ≠ ∅})
164 sneq 4220 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝑢 → {𝑘} = {𝑢})
165164imaeq2d 5501 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑢 → (𝑔 “ {𝑘}) = (𝑔 “ {𝑢}))
166165unieqd 4478 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑢 (𝑔 “ {𝑘}) = (𝑔 “ {𝑢}))
167166ineq1d 3846 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑢 → ( (𝑔 “ {𝑘}) ∩ 𝑣) = ( (𝑔 “ {𝑢}) ∩ 𝑣))
168167neeq1d 2882 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑢 → (( (𝑔 “ {𝑘}) ∩ 𝑣) ≠ ∅ ↔ ( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅))
169168cbvrabv 3230 . . . . . . . . . . . . . . 15 {𝑘 ∈ ran 𝑔 ∣ ( (𝑔 “ {𝑘}) ∩ 𝑣) ≠ ∅} = {𝑢 ∈ ran 𝑔 ∣ ( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅}
170163, 169syl6breq 4726 . . . . . . . . . . . . . 14 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) → {𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∣ (𝑤𝑣) ≠ ∅} ≼ {𝑢 ∈ ran 𝑔 ∣ ( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅})
171123rabex 4845 . . . . . . . . . . . . . . 15 {𝑢 ∈ ran 𝑔 ∣ ∃𝑘 ∈ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} (𝑔𝑘) = 𝑢} ∈ V
172 nfv 1883 . . . . . . . . . . . . . . . . . . . . 21 𝑗(((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣)
173 nfrab1 3152 . . . . . . . . . . . . . . . . . . . . . 22 𝑗{𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅}
174173nfel1 2808 . . . . . . . . . . . . . . . . . . . . 21 𝑗{𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin
175172, 174nfan 1868 . . . . . . . . . . . . . . . . . . . 20 𝑗((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin)
176 nfv 1883 . . . . . . . . . . . . . . . . . . . 20 𝑗 𝑢 ∈ ran 𝑔
177175, 176nfan 1868 . . . . . . . . . . . . . . . . . . 19 𝑗(((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) ∧ 𝑢 ∈ ran 𝑔)
178 nfv 1883 . . . . . . . . . . . . . . . . . . 19 𝑗( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅
179177, 178nfan 1868 . . . . . . . . . . . . . . . . . 18 𝑗((((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) ∧ 𝑢 ∈ ran 𝑔) ∧ ( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅)
180 nfv 1883 . . . . . . . . . . . . . . . . . . 19 𝑗(𝑔𝑘) = 𝑢
181173, 180nfrex 3036 . . . . . . . . . . . . . . . . . 18 𝑗𝑘 ∈ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} (𝑔𝑘) = 𝑢
18239ad5antlr 775 . . . . . . . . . . . . . . . . . . . . . . 23 ((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) → 𝑔 Fn 𝑉)
183182ad5antr 773 . . . . . . . . . . . . . . . . . . . . . 22 (((((((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) ∧ 𝑢 ∈ ran 𝑔) ∧ ( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅) ∧ 𝑗 ∈ (𝑔 “ {𝑢})) ∧ (𝑗𝑣) ≠ ∅) → 𝑔 Fn 𝑉)
184 simplr 807 . . . . . . . . . . . . . . . . . . . . . 22 (((((((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) ∧ 𝑢 ∈ ran 𝑔) ∧ ( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅) ∧ 𝑗 ∈ (𝑔 “ {𝑢})) ∧ (𝑗𝑣) ≠ ∅) → 𝑗 ∈ (𝑔 “ {𝑢}))
185 fniniseg 6378 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑔 Fn 𝑉 → (𝑗 ∈ (𝑔 “ {𝑢}) ↔ (𝑗𝑉 ∧ (𝑔𝑗) = 𝑢)))
186185biimpa 500 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑔 Fn 𝑉𝑗 ∈ (𝑔 “ {𝑢})) → (𝑗𝑉 ∧ (𝑔𝑗) = 𝑢))
187183, 184, 186syl2anc 694 . . . . . . . . . . . . . . . . . . . . 21 (((((((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) ∧ 𝑢 ∈ ran 𝑔) ∧ ( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅) ∧ 𝑗 ∈ (𝑔 “ {𝑢})) ∧ (𝑗𝑣) ≠ ∅) → (𝑗𝑉 ∧ (𝑔𝑗) = 𝑢))
188187simpld 474 . . . . . . . . . . . . . . . . . . . 20 (((((((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) ∧ 𝑢 ∈ ran 𝑔) ∧ ( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅) ∧ 𝑗 ∈ (𝑔 “ {𝑢})) ∧ (𝑗𝑣) ≠ ∅) → 𝑗𝑉)
189 simpr 476 . . . . . . . . . . . . . . . . . . . 20 (((((((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) ∧ 𝑢 ∈ ran 𝑔) ∧ ( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅) ∧ 𝑗 ∈ (𝑔 “ {𝑢})) ∧ (𝑗𝑣) ≠ ∅) → (𝑗𝑣) ≠ ∅)
190 rabid 3145 . . . . . . . . . . . . . . . . . . . 20 (𝑗 ∈ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ↔ (𝑗𝑉 ∧ (𝑗𝑣) ≠ ∅))
191188, 189, 190sylanbrc 699 . . . . . . . . . . . . . . . . . . 19 (((((((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) ∧ 𝑢 ∈ ran 𝑔) ∧ ( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅) ∧ 𝑗 ∈ (𝑔 “ {𝑢})) ∧ (𝑗𝑣) ≠ ∅) → 𝑗 ∈ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅})
192187simprd 478 . . . . . . . . . . . . . . . . . . 19 (((((((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) ∧ 𝑢 ∈ ran 𝑔) ∧ ( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅) ∧ 𝑗 ∈ (𝑔 “ {𝑢})) ∧ (𝑗𝑣) ≠ ∅) → (𝑔𝑗) = 𝑢)
193 fveq2 6229 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 𝑗 → (𝑔𝑘) = (𝑔𝑗))
194193eqeq1d 2653 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝑗 → ((𝑔𝑘) = 𝑢 ↔ (𝑔𝑗) = 𝑢))
195194rspcev 3340 . . . . . . . . . . . . . . . . . . 19 ((𝑗 ∈ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∧ (𝑔𝑗) = 𝑢) → ∃𝑘 ∈ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} (𝑔𝑘) = 𝑢)
196191, 192, 195syl2anc 694 . . . . . . . . . . . . . . . . . 18 (((((((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) ∧ 𝑢 ∈ ran 𝑔) ∧ ( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅) ∧ 𝑗 ∈ (𝑔 “ {𝑢})) ∧ (𝑗𝑣) ≠ ∅) → ∃𝑘 ∈ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} (𝑔𝑘) = 𝑢)
197 uniinn0 29492 . . . . . . . . . . . . . . . . . . . 20 (( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅ ↔ ∃𝑗 ∈ (𝑔 “ {𝑢})(𝑗𝑣) ≠ ∅)
198197biimpi 206 . . . . . . . . . . . . . . . . . . 19 (( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅ → ∃𝑗 ∈ (𝑔 “ {𝑢})(𝑗𝑣) ≠ ∅)
199198adantl 481 . . . . . . . . . . . . . . . . . 18 (((((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) ∧ 𝑢 ∈ ran 𝑔) ∧ ( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅) → ∃𝑗 ∈ (𝑔 “ {𝑢})(𝑗𝑣) ≠ ∅)
200179, 181, 196, 199r19.29af2 3104 . . . . . . . . . . . . . . . . 17 (((((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) ∧ 𝑢 ∈ ran 𝑔) ∧ ( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅) → ∃𝑘 ∈ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} (𝑔𝑘) = 𝑢)
201200ex 449 . . . . . . . . . . . . . . . 16 ((((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) ∧ 𝑢 ∈ ran 𝑔) → (( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅ → ∃𝑘 ∈ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} (𝑔𝑘) = 𝑢))
202201ss2rabdv 3716 . . . . . . . . . . . . . . 15 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) → {𝑢 ∈ ran 𝑔 ∣ ( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅} ⊆ {𝑢 ∈ ran 𝑔 ∣ ∃𝑘 ∈ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} (𝑔𝑘) = 𝑢})
203 ssdomg 8043 . . . . . . . . . . . . . . 15 ({𝑢 ∈ ran 𝑔 ∣ ∃𝑘 ∈ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} (𝑔𝑘) = 𝑢} ∈ V → ({𝑢 ∈ ran 𝑔 ∣ ( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅} ⊆ {𝑢 ∈ ran 𝑔 ∣ ∃𝑘 ∈ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} (𝑔𝑘) = 𝑢} → {𝑢 ∈ ran 𝑔 ∣ ( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅} ≼ {𝑢 ∈ ran 𝑔 ∣ ∃𝑘 ∈ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} (𝑔𝑘) = 𝑢}))
204171, 202, 203mpsyl 68 . . . . . . . . . . . . . 14 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) → {𝑢 ∈ ran 𝑔 ∣ ( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅} ≼ {𝑢 ∈ ran 𝑔 ∣ ∃𝑘 ∈ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} (𝑔𝑘) = 𝑢})
205 domtr 8050 . . . . . . . . . . . . . 14 (({𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∣ (𝑤𝑣) ≠ ∅} ≼ {𝑢 ∈ ran 𝑔 ∣ ( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅} ∧ {𝑢 ∈ ran 𝑔 ∣ ( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅} ≼ {𝑢 ∈ ran 𝑔 ∣ ∃𝑘 ∈ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} (𝑔𝑘) = 𝑢}) → {𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∣ (𝑤𝑣) ≠ ∅} ≼ {𝑢 ∈ ran 𝑔 ∣ ∃𝑘 ∈ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} (𝑔𝑘) = 𝑢})
206170, 204, 205syl2anc 694 . . . . . . . . . . . . 13 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) → {𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∣ (𝑤𝑣) ≠ ∅} ≼ {𝑢 ∈ ran 𝑔 ∣ ∃𝑘 ∈ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} (𝑔𝑘) = 𝑢})
207182adantr 480 . . . . . . . . . . . . . 14 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) → 𝑔 Fn 𝑉)
208 dffn3 6092 . . . . . . . . . . . . . . 15 (𝑔 Fn 𝑉𝑔:𝑉⟶ran 𝑔)
209208biimpi 206 . . . . . . . . . . . . . 14 (𝑔 Fn 𝑉𝑔:𝑉⟶ran 𝑔)
210 ssrab2 3720 . . . . . . . . . . . . . . 15 {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ⊆ 𝑉
211 fimarab 29573 . . . . . . . . . . . . . . 15 ((𝑔:𝑉⟶ran 𝑔 ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ⊆ 𝑉) → (𝑔 “ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅}) = {𝑢 ∈ ran 𝑔 ∣ ∃𝑘 ∈ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} (𝑔𝑘) = 𝑢})
212210, 211mpan2 707 . . . . . . . . . . . . . 14 (𝑔:𝑉⟶ran 𝑔 → (𝑔 “ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅}) = {𝑢 ∈ ran 𝑔 ∣ ∃𝑘 ∈ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} (𝑔𝑘) = 𝑢})
213207, 209, 2123syl 18 . . . . . . . . . . . . 13 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) → (𝑔 “ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅}) = {𝑢 ∈ ran 𝑔 ∣ ∃𝑘 ∈ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} (𝑔𝑘) = 𝑢})
214206, 213breqtrrd 4713 . . . . . . . . . . . 12 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) → {𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∣ (𝑤𝑣) ≠ ∅} ≼ (𝑔 “ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅}))
215 domfi 8222 . . . . . . . . . . . 12 (((𝑔 “ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅}) ∈ Fin ∧ {𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∣ (𝑤𝑣) ≠ ∅} ≼ (𝑔 “ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅})) → {𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∣ (𝑤𝑣) ≠ ∅} ∈ Fin)
216141, 214, 215syl2anc 694 . . . . . . . . . . 11 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) → {𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∣ (𝑤𝑣) ≠ ∅} ∈ Fin)
217216ex 449 . . . . . . . . . 10 ((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) → ({𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin → {𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∣ (𝑤𝑣) ≠ ∅} ∈ Fin))
218217imdistanda 729 . . . . . . . . 9 (((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) → ((𝑥𝑣 ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) → (𝑥𝑣 ∧ {𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∣ (𝑤𝑣) ≠ ∅} ∈ Fin)))
219218imp 444 . . . . . . . 8 ((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ (𝑥𝑣 ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin)) → (𝑥𝑣 ∧ {𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∣ (𝑤𝑣) ≠ ∅} ∈ Fin))
220 simplll 813 . . . . . . . . 9 ((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) → 𝜑)
221 locfinref.x . . . . . . . . . . . . 13 𝑋 = 𝐽
222221, 29islocfin 21368 . . . . . . . . . . . 12 (𝑉 ∈ (LocFin‘𝐽) ↔ (𝐽 ∈ Top ∧ 𝑋 = 𝑉 ∧ ∀𝑥𝑋𝑣𝐽 (𝑥𝑣 ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin)))
2232, 222sylib 208 . . . . . . . . . . 11 (𝜑 → (𝐽 ∈ Top ∧ 𝑋 = 𝑉 ∧ ∀𝑥𝑋𝑣𝐽 (𝑥𝑣 ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin)))
224223simp3d 1095 . . . . . . . . . 10 (𝜑 → ∀𝑥𝑋𝑣𝐽 (𝑥𝑣 ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin))
225224r19.21bi 2961 . . . . . . . . 9 ((𝜑𝑥𝑋) → ∃𝑣𝐽 (𝑥𝑣 ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin))
226220, 225sylancom 702 . . . . . . . 8 ((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) → ∃𝑣𝐽 (𝑥𝑣 ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin))
227136, 137, 219, 226reximd2a 3042 . . . . . . 7 ((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) → ∃𝑣𝐽 (𝑥𝑣 ∧ {𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∣ (𝑤𝑣) ≠ ∅} ∈ Fin))
228227ralrimiva 2995 . . . . . 6 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → ∀𝑥𝑋𝑣𝐽 (𝑥𝑣 ∧ {𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∣ (𝑤𝑣) ≠ ∅} ∈ Fin))
229221, 127islocfin 21368 . . . . . 6 (ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∈ (LocFin‘𝐽) ↔ (𝐽 ∈ Top ∧ 𝑋 = ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∧ ∀𝑥𝑋𝑣𝐽 (𝑥𝑣 ∧ {𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∣ (𝑤𝑣) ≠ ∅} ∈ Fin)))
230131, 134, 228, 229syl3anbrc 1265 . . . . 5 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∈ (LocFin‘𝐽))
231 funeq 5946 . . . . . . . 8 (𝑓 = (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) → (Fun 𝑓 ↔ Fun (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))))
232 dmeq 5356 . . . . . . . . 9 (𝑓 = (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) → dom 𝑓 = dom (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})))
233232sseq1d 3665 . . . . . . . 8 (𝑓 = (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) → (dom 𝑓𝑈 ↔ dom (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ⊆ 𝑈))
234 rneq 5383 . . . . . . . . 9 (𝑓 = (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) → ran 𝑓 = ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})))
235234sseq1d 3665 . . . . . . . 8 (𝑓 = (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) → (ran 𝑓𝐽 ↔ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ⊆ 𝐽))
236231, 233, 2353anbi123d 1439 . . . . . . 7 (𝑓 = (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) → ((Fun 𝑓 ∧ dom 𝑓𝑈 ∧ ran 𝑓𝐽) ↔ (Fun (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∧ dom (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ⊆ 𝑈 ∧ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ⊆ 𝐽)))
237234breq1d 4695 . . . . . . . 8 (𝑓 = (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) → (ran 𝑓Ref𝑈 ↔ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))Ref𝑈))
238234eleq1d 2715 . . . . . . . 8 (𝑓 = (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) → (ran 𝑓 ∈ (LocFin‘𝐽) ↔ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∈ (LocFin‘𝐽)))
239237, 238anbi12d 747 . . . . . . 7 (𝑓 = (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) → ((ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽)) ↔ (ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))Ref𝑈 ∧ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∈ (LocFin‘𝐽))))
240236, 239anbi12d 747 . . . . . 6 (𝑓 = (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) → (((Fun 𝑓 ∧ dom 𝑓𝑈 ∧ ran 𝑓𝐽) ∧ (ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽))) ↔ ((Fun (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∧ dom (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ⊆ 𝑈 ∧ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ⊆ 𝐽) ∧ (ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))Ref𝑈 ∧ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∈ (LocFin‘𝐽)))))
241124, 240spcev 3331 . . . . 5 (((Fun (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∧ dom (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ⊆ 𝑈 ∧ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ⊆ 𝐽) ∧ (ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))Ref𝑈 ∧ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∈ (LocFin‘𝐽))) → ∃𝑓((Fun 𝑓 ∧ dom 𝑓𝑈 ∧ ran 𝑓𝐽) ∧ (ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽))))
2428, 13, 28, 130, 230, 241syl32anc 1374 . . . 4 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → ∃𝑓((Fun 𝑓 ∧ dom 𝑓𝑈 ∧ ran 𝑓𝐽) ∧ (ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽))))
243242expl 647 . . 3 (𝜑 → ((𝑔:𝑉𝑈 ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → ∃𝑓((Fun 𝑓 ∧ dom 𝑓𝑈 ∧ ran 𝑓𝐽) ∧ (ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽)))))
244243exlimdv 1901 . 2 (𝜑 → (∃𝑔(𝑔:𝑉𝑈 ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → ∃𝑓((Fun 𝑓 ∧ dom 𝑓𝑈 ∧ ran 𝑓𝐽) ∧ (ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽)))))
2456, 244mpd 15 1 (𝜑 → ∃𝑓((Fun 𝑓 ∧ dom 𝑓𝑈 ∧ ran 𝑓𝐽) ∧ (ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wex 1744  wcel 2030  wne 2823  wral 2941  wrex 2942  {crab 2945  Vcvv 3231  cin 3606  wss 3607  c0 3948  {csn 4210   cuni 4468   ciun 4552   class class class wbr 4685  cmpt 4762  ccnv 5142  dom cdm 5143  ran crn 5144  cima 5146  Fun wfun 5920   Fn wfn 5921  wf 5922  ontowfo 5924  cfv 5926  cdom 7995  Fincfn 7997  Topctop 20746  Refcref 21353  LocFinclocfin 21355
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-reg 8538  ax-inf2 8576  ax-ac2 9323
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-er 7787  df-en 7998  df-dom 7999  df-fin 8001  df-r1 8665  df-rank 8666  df-card 8803  df-ac 8977  df-top 20747  df-ref 21356  df-locfin 21358
This theorem is referenced by:  locfinref  30036
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