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Theorem comppfsc 21383
Description: A space where every open cover has a point-finite subcover is compact. This is significant in part because it shows half of the proposition that if only half the generalization in the definition of metacompactness (and consequently paracompactness) is performed, one does not obtain any more spaces. (Contributed by Jeff Hankins, 21-Jan-2010.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
Hypothesis
Ref Expression
comppfsc.1 𝑋 = 𝐽
Assertion
Ref Expression
comppfsc (𝐽 ∈ Top → (𝐽 ∈ Comp ↔ ∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑))))
Distinct variable groups:   𝑐,𝑑,𝐽   𝑋,𝑐,𝑑

Proof of Theorem comppfsc
Dummy variables 𝑎 𝑏 𝑓 𝑝 𝑞 𝑠 𝑥 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elpwi 4201 . . . 4 (𝑐 ∈ 𝒫 𝐽𝑐𝐽)
2 comppfsc.1 . . . . . . 7 𝑋 = 𝐽
32cmpcov 21240 . . . . . 6 ((𝐽 ∈ Comp ∧ 𝑐𝐽𝑋 = 𝑐) → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑)
4 elfpw 8309 . . . . . . . 8 (𝑑 ∈ (𝒫 𝑐 ∩ Fin) ↔ (𝑑𝑐𝑑 ∈ Fin))
5 finptfin 21369 . . . . . . . . . . 11 (𝑑 ∈ Fin → 𝑑 ∈ PtFin)
65anim1i 591 . . . . . . . . . 10 ((𝑑 ∈ Fin ∧ (𝑑𝑐𝑋 = 𝑑)) → (𝑑 ∈ PtFin ∧ (𝑑𝑐𝑋 = 𝑑)))
76anassrs 681 . . . . . . . . 9 (((𝑑 ∈ Fin ∧ 𝑑𝑐) ∧ 𝑋 = 𝑑) → (𝑑 ∈ PtFin ∧ (𝑑𝑐𝑋 = 𝑑)))
87ancom1s 864 . . . . . . . 8 (((𝑑𝑐𝑑 ∈ Fin) ∧ 𝑋 = 𝑑) → (𝑑 ∈ PtFin ∧ (𝑑𝑐𝑋 = 𝑑)))
94, 8sylanb 488 . . . . . . 7 ((𝑑 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑋 = 𝑑) → (𝑑 ∈ PtFin ∧ (𝑑𝑐𝑋 = 𝑑)))
109reximi2 3039 . . . . . 6 (∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑))
113, 10syl 17 . . . . 5 ((𝐽 ∈ Comp ∧ 𝑐𝐽𝑋 = 𝑐) → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑))
12113exp 1283 . . . 4 (𝐽 ∈ Comp → (𝑐𝐽 → (𝑋 = 𝑐 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑))))
131, 12syl5 34 . . 3 (𝐽 ∈ Comp → (𝑐 ∈ 𝒫 𝐽 → (𝑋 = 𝑐 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑))))
1413ralrimiv 2994 . 2 (𝐽 ∈ Comp → ∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑)))
15 elpwi 4201 . . . . . . 7 (𝑎 ∈ 𝒫 𝐽𝑎𝐽)
16 0elpw 4864 . . . . . . . . . . . 12 ∅ ∈ 𝒫 𝑎
17 0fin 8229 . . . . . . . . . . . 12 ∅ ∈ Fin
18 elin 3829 . . . . . . . . . . . 12 (∅ ∈ (𝒫 𝑎 ∩ Fin) ↔ (∅ ∈ 𝒫 𝑎 ∧ ∅ ∈ Fin))
1916, 17, 18mpbir2an 975 . . . . . . . . . . 11 ∅ ∈ (𝒫 𝑎 ∩ Fin)
20 unieq 4476 . . . . . . . . . . . . . 14 (𝑏 = ∅ → 𝑏 = ∅)
21 uni0 4497 . . . . . . . . . . . . . 14 ∅ = ∅
2220, 21syl6eq 2701 . . . . . . . . . . . . 13 (𝑏 = ∅ → 𝑏 = ∅)
2322eqeq2d 2661 . . . . . . . . . . . 12 (𝑏 = ∅ → (𝑋 = 𝑏𝑋 = ∅))
2423rspcev 3340 . . . . . . . . . . 11 ((∅ ∈ (𝒫 𝑎 ∩ Fin) ∧ 𝑋 = ∅) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏)
2519, 24mpan 706 . . . . . . . . . 10 (𝑋 = ∅ → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏)
2625a1d 25 . . . . . . . . 9 (𝑋 = ∅ → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑)) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏))
2726a1i 11 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) → (𝑋 = ∅ → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑)) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏)))
28 n0 3964 . . . . . . . . 9 (𝑋 ≠ ∅ ↔ ∃𝑥 𝑥𝑋)
29 simp2 1082 . . . . . . . . . . . . . 14 ((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) → 𝑋 = 𝑎)
3029eleq2d 2716 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) → (𝑥𝑋𝑥 𝑎))
3130biimpd 219 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) → (𝑥𝑋𝑥 𝑎))
32 eluni2 4472 . . . . . . . . . . . 12 (𝑥 𝑎 ↔ ∃𝑠𝑎 𝑥𝑠)
3331, 32syl6ib 241 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) → (𝑥𝑋 → ∃𝑠𝑎 𝑥𝑠))
34 simpl3 1086 . . . . . . . . . . . . . . . . . . . . 21 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → 𝑎𝐽)
35 simprl 809 . . . . . . . . . . . . . . . . . . . . 21 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → 𝑠𝑎)
3634, 35sseldd 3637 . . . . . . . . . . . . . . . . . . . 20 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → 𝑠𝐽)
37 elssuni 4499 . . . . . . . . . . . . . . . . . . . . 21 (𝑠𝐽𝑠 𝐽)
3837, 2syl6sseqr 3685 . . . . . . . . . . . . . . . . . . . 20 (𝑠𝐽𝑠𝑋)
3936, 38syl 17 . . . . . . . . . . . . . . . . . . 19 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → 𝑠𝑋)
4039ralrimivw 2996 . . . . . . . . . . . . . . . . . 18 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → ∀𝑝𝑎 𝑠𝑋)
41 iunss 4593 . . . . . . . . . . . . . . . . . 18 ( 𝑝𝑎 𝑠𝑋 ↔ ∀𝑝𝑎 𝑠𝑋)
4240, 41sylibr 224 . . . . . . . . . . . . . . . . 17 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → 𝑝𝑎 𝑠𝑋)
43 ssequn1 3816 . . . . . . . . . . . . . . . . 17 ( 𝑝𝑎 𝑠𝑋 ↔ ( 𝑝𝑎 𝑠𝑋) = 𝑋)
4442, 43sylib 208 . . . . . . . . . . . . . . . 16 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → ( 𝑝𝑎 𝑠𝑋) = 𝑋)
45 simpl2 1085 . . . . . . . . . . . . . . . . . 18 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → 𝑋 = 𝑎)
46 uniiun 4605 . . . . . . . . . . . . . . . . . 18 𝑎 = 𝑝𝑎 𝑝
4745, 46syl6eq 2701 . . . . . . . . . . . . . . . . 17 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → 𝑋 = 𝑝𝑎 𝑝)
4847uneq2d 3800 . . . . . . . . . . . . . . . 16 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → ( 𝑝𝑎 𝑠𝑋) = ( 𝑝𝑎 𝑠 𝑝𝑎 𝑝))
4944, 48eqtr3d 2687 . . . . . . . . . . . . . . 15 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → 𝑋 = ( 𝑝𝑎 𝑠 𝑝𝑎 𝑝))
50 iunun 4636 . . . . . . . . . . . . . . . 16 𝑝𝑎 (𝑠𝑝) = ( 𝑝𝑎 𝑠 𝑝𝑎 𝑝)
51 vex 3234 . . . . . . . . . . . . . . . . . 18 𝑠 ∈ V
52 vex 3234 . . . . . . . . . . . . . . . . . 18 𝑝 ∈ V
5351, 52unex 6998 . . . . . . . . . . . . . . . . 17 (𝑠𝑝) ∈ V
5453dfiun3 5412 . . . . . . . . . . . . . . . 16 𝑝𝑎 (𝑠𝑝) = ran (𝑝𝑎 ↦ (𝑠𝑝))
5550, 54eqtr3i 2675 . . . . . . . . . . . . . . 15 ( 𝑝𝑎 𝑠 𝑝𝑎 𝑝) = ran (𝑝𝑎 ↦ (𝑠𝑝))
5649, 55syl6eq 2701 . . . . . . . . . . . . . 14 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → 𝑋 = ran (𝑝𝑎 ↦ (𝑠𝑝)))
57 simpll1 1120 . . . . . . . . . . . . . . . . . . 19 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ 𝑝𝑎) → 𝐽 ∈ Top)
5836adantr 480 . . . . . . . . . . . . . . . . . . 19 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ 𝑝𝑎) → 𝑠𝐽)
5934sselda 3636 . . . . . . . . . . . . . . . . . . 19 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ 𝑝𝑎) → 𝑝𝐽)
60 unopn 20756 . . . . . . . . . . . . . . . . . . 19 ((𝐽 ∈ Top ∧ 𝑠𝐽𝑝𝐽) → (𝑠𝑝) ∈ 𝐽)
6157, 58, 59, 60syl3anc 1366 . . . . . . . . . . . . . . . . . 18 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ 𝑝𝑎) → (𝑠𝑝) ∈ 𝐽)
62 eqid 2651 . . . . . . . . . . . . . . . . . 18 (𝑝𝑎 ↦ (𝑠𝑝)) = (𝑝𝑎 ↦ (𝑠𝑝))
6361, 62fmptd 6425 . . . . . . . . . . . . . . . . 17 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → (𝑝𝑎 ↦ (𝑠𝑝)):𝑎𝐽)
64 frn 6091 . . . . . . . . . . . . . . . . 17 ((𝑝𝑎 ↦ (𝑠𝑝)):𝑎𝐽 → ran (𝑝𝑎 ↦ (𝑠𝑝)) ⊆ 𝐽)
6563, 64syl 17 . . . . . . . . . . . . . . . 16 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → ran (𝑝𝑎 ↦ (𝑠𝑝)) ⊆ 𝐽)
66 elpw2g 4857 . . . . . . . . . . . . . . . . . 18 (𝐽 ∈ Top → (ran (𝑝𝑎 ↦ (𝑠𝑝)) ∈ 𝒫 𝐽 ↔ ran (𝑝𝑎 ↦ (𝑠𝑝)) ⊆ 𝐽))
67663ad2ant1 1102 . . . . . . . . . . . . . . . . 17 ((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) → (ran (𝑝𝑎 ↦ (𝑠𝑝)) ∈ 𝒫 𝐽 ↔ ran (𝑝𝑎 ↦ (𝑠𝑝)) ⊆ 𝐽))
6867adantr 480 . . . . . . . . . . . . . . . 16 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → (ran (𝑝𝑎 ↦ (𝑠𝑝)) ∈ 𝒫 𝐽 ↔ ran (𝑝𝑎 ↦ (𝑠𝑝)) ⊆ 𝐽))
6965, 68mpbird 247 . . . . . . . . . . . . . . 15 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → ran (𝑝𝑎 ↦ (𝑠𝑝)) ∈ 𝒫 𝐽)
70 unieq 4476 . . . . . . . . . . . . . . . . . 18 (𝑐 = ran (𝑝𝑎 ↦ (𝑠𝑝)) → 𝑐 = ran (𝑝𝑎 ↦ (𝑠𝑝)))
7170eqeq2d 2661 . . . . . . . . . . . . . . . . 17 (𝑐 = ran (𝑝𝑎 ↦ (𝑠𝑝)) → (𝑋 = 𝑐𝑋 = ran (𝑝𝑎 ↦ (𝑠𝑝))))
72 sseq2 3660 . . . . . . . . . . . . . . . . . . 19 (𝑐 = ran (𝑝𝑎 ↦ (𝑠𝑝)) → (𝑑𝑐𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝))))
7372anbi1d 741 . . . . . . . . . . . . . . . . . 18 (𝑐 = ran (𝑝𝑎 ↦ (𝑠𝑝)) → ((𝑑𝑐𝑋 = 𝑑) ↔ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)))
7473rexbidv 3081 . . . . . . . . . . . . . . . . 17 (𝑐 = ran (𝑝𝑎 ↦ (𝑠𝑝)) → (∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑) ↔ ∃𝑑 ∈ PtFin (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)))
7571, 74imbi12d 333 . . . . . . . . . . . . . . . 16 (𝑐 = ran (𝑝𝑎 ↦ (𝑠𝑝)) → ((𝑋 = 𝑐 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑)) ↔ (𝑋 = ran (𝑝𝑎 ↦ (𝑠𝑝)) → ∃𝑑 ∈ PtFin (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑))))
7675rspcv 3336 . . . . . . . . . . . . . . 15 (ran (𝑝𝑎 ↦ (𝑠𝑝)) ∈ 𝒫 𝐽 → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑)) → (𝑋 = ran (𝑝𝑎 ↦ (𝑠𝑝)) → ∃𝑑 ∈ PtFin (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑))))
7769, 76syl 17 . . . . . . . . . . . . . 14 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑)) → (𝑋 = ran (𝑝𝑎 ↦ (𝑠𝑝)) → ∃𝑑 ∈ PtFin (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑))))
7856, 77mpid 44 . . . . . . . . . . . . 13 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑)) → ∃𝑑 ∈ PtFin (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)))
79 simprr 811 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → 𝑥𝑠)
80 ssel2 3631 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑎𝐽𝑠𝑎) → 𝑠𝐽)
81803ad2antl3 1245 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ 𝑠𝑎) → 𝑠𝐽)
8281adantrr 753 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → 𝑠𝐽)
83 elunii 4473 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥𝑠𝑠𝐽) → 𝑥 𝐽)
8479, 82, 83syl2anc 694 . . . . . . . . . . . . . . . . . . . . 21 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → 𝑥 𝐽)
8584, 2syl6eleqr 2741 . . . . . . . . . . . . . . . . . . . 20 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → 𝑥𝑋)
8685adantr 480 . . . . . . . . . . . . . . . . . . 19 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) → 𝑥𝑋)
87 simprr 811 . . . . . . . . . . . . . . . . . . 19 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) → 𝑋 = 𝑑)
8886, 87eleqtrd 2732 . . . . . . . . . . . . . . . . . 18 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) → 𝑥 𝑑)
89 eqid 2651 . . . . . . . . . . . . . . . . . . . 20 𝑑 = 𝑑
9089ptfinfin 21370 . . . . . . . . . . . . . . . . . . 19 ((𝑑 ∈ PtFin ∧ 𝑥 𝑑) → {𝑧𝑑𝑥𝑧} ∈ Fin)
9190expcom 450 . . . . . . . . . . . . . . . . . 18 (𝑥 𝑑 → (𝑑 ∈ PtFin → {𝑧𝑑𝑥𝑧} ∈ Fin))
9288, 91syl 17 . . . . . . . . . . . . . . . . 17 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) → (𝑑 ∈ PtFin → {𝑧𝑑𝑥𝑧} ∈ Fin))
93 simprl 809 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) → 𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)))
94 elun1 3813 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥𝑠𝑥 ∈ (𝑠𝑝))
9594ad2antll 765 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → 𝑥 ∈ (𝑠𝑝))
9695ralrimivw 2996 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → ∀𝑝𝑎 𝑥 ∈ (𝑠𝑝))
9753rgenw 2953 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑝𝑎 (𝑠𝑝) ∈ V
98 eleq2 2719 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑧 = (𝑠𝑝) → (𝑥𝑧𝑥 ∈ (𝑠𝑝)))
9962, 98ralrnmpt 6408 . . . . . . . . . . . . . . . . . . . . . . . 24 (∀𝑝𝑎 (𝑠𝑝) ∈ V → (∀𝑧 ∈ ran (𝑝𝑎 ↦ (𝑠𝑝))𝑥𝑧 ↔ ∀𝑝𝑎 𝑥 ∈ (𝑠𝑝)))
10097, 99ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . 23 (∀𝑧 ∈ ran (𝑝𝑎 ↦ (𝑠𝑝))𝑥𝑧 ↔ ∀𝑝𝑎 𝑥 ∈ (𝑠𝑝))
10196, 100sylibr 224 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → ∀𝑧 ∈ ran (𝑝𝑎 ↦ (𝑠𝑝))𝑥𝑧)
102101adantr 480 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) → ∀𝑧 ∈ ran (𝑝𝑎 ↦ (𝑠𝑝))𝑥𝑧)
103 ssralv 3699 . . . . . . . . . . . . . . . . . . . . 21 (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) → (∀𝑧 ∈ ran (𝑝𝑎 ↦ (𝑠𝑝))𝑥𝑧 → ∀𝑧𝑑 𝑥𝑧))
10493, 102, 103sylc 65 . . . . . . . . . . . . . . . . . . . 20 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) → ∀𝑧𝑑 𝑥𝑧)
105 rabid2 3148 . . . . . . . . . . . . . . . . . . . 20 (𝑑 = {𝑧𝑑𝑥𝑧} ↔ ∀𝑧𝑑 𝑥𝑧)
106104, 105sylibr 224 . . . . . . . . . . . . . . . . . . 19 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) → 𝑑 = {𝑧𝑑𝑥𝑧})
107106eleq1d 2715 . . . . . . . . . . . . . . . . . 18 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) → (𝑑 ∈ Fin ↔ {𝑧𝑑𝑥𝑧} ∈ Fin))
108107biimprd 238 . . . . . . . . . . . . . . . . 17 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) → ({𝑧𝑑𝑥𝑧} ∈ Fin → 𝑑 ∈ Fin))
10962rnmpt 5403 . . . . . . . . . . . . . . . . . . . . 21 ran (𝑝𝑎 ↦ (𝑠𝑝)) = {𝑞 ∣ ∃𝑝𝑎 𝑞 = (𝑠𝑝)}
11093, 109syl6sseq 3684 . . . . . . . . . . . . . . . . . . . 20 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) → 𝑑 ⊆ {𝑞 ∣ ∃𝑝𝑎 𝑞 = (𝑠𝑝)})
111 ssabral 3706 . . . . . . . . . . . . . . . . . . . 20 (𝑑 ⊆ {𝑞 ∣ ∃𝑝𝑎 𝑞 = (𝑠𝑝)} ↔ ∀𝑞𝑑𝑝𝑎 𝑞 = (𝑠𝑝))
112110, 111sylib 208 . . . . . . . . . . . . . . . . . . 19 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) → ∀𝑞𝑑𝑝𝑎 𝑞 = (𝑠𝑝))
113 uneq2 3794 . . . . . . . . . . . . . . . . . . . . . 22 (𝑝 = (𝑓𝑞) → (𝑠𝑝) = (𝑠 ∪ (𝑓𝑞)))
114113eqeq2d 2661 . . . . . . . . . . . . . . . . . . . . 21 (𝑝 = (𝑓𝑞) → (𝑞 = (𝑠𝑝) ↔ 𝑞 = (𝑠 ∪ (𝑓𝑞))))
115114ac6sfi 8245 . . . . . . . . . . . . . . . . . . . 20 ((𝑑 ∈ Fin ∧ ∀𝑞𝑑𝑝𝑎 𝑞 = (𝑠𝑝)) → ∃𝑓(𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))
116115expcom 450 . . . . . . . . . . . . . . . . . . 19 (∀𝑞𝑑𝑝𝑎 𝑞 = (𝑠𝑝) → (𝑑 ∈ Fin → ∃𝑓(𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞)))))
117112, 116syl 17 . . . . . . . . . . . . . . . . . 18 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) → (𝑑 ∈ Fin → ∃𝑓(𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞)))))
118 frn 6091 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑓:𝑑𝑎 → ran 𝑓𝑎)
119118adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))) → ran 𝑓𝑎)
120119ad2antll 765 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → ran 𝑓𝑎)
12135ad2antrr 762 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → 𝑠𝑎)
122121snssd 4372 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → {𝑠} ⊆ 𝑎)
123120, 122unssd 3822 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → (ran 𝑓 ∪ {𝑠}) ⊆ 𝑎)
124 simprl 809 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → 𝑑 ∈ Fin)
125 simprrl 821 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → 𝑓:𝑑𝑎)
126 ffn 6083 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑓:𝑑𝑎𝑓 Fn 𝑑)
127125, 126syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → 𝑓 Fn 𝑑)
128 dffn4 6159 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑓 Fn 𝑑𝑓:𝑑onto→ran 𝑓)
129127, 128sylib 208 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → 𝑓:𝑑onto→ran 𝑓)
130 fofi 8293 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑑 ∈ Fin ∧ 𝑓:𝑑onto→ran 𝑓) → ran 𝑓 ∈ Fin)
131124, 129, 130syl2anc 694 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → ran 𝑓 ∈ Fin)
132 snfi 8079 . . . . . . . . . . . . . . . . . . . . . . . 24 {𝑠} ∈ Fin
133 unfi 8268 . . . . . . . . . . . . . . . . . . . . . . . 24 ((ran 𝑓 ∈ Fin ∧ {𝑠} ∈ Fin) → (ran 𝑓 ∪ {𝑠}) ∈ Fin)
134131, 132, 133sylancl 695 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → (ran 𝑓 ∪ {𝑠}) ∈ Fin)
135 elfpw 8309 . . . . . . . . . . . . . . . . . . . . . . 23 ((ran 𝑓 ∪ {𝑠}) ∈ (𝒫 𝑎 ∩ Fin) ↔ ((ran 𝑓 ∪ {𝑠}) ⊆ 𝑎 ∧ (ran 𝑓 ∪ {𝑠}) ∈ Fin))
136123, 134, 135sylanbrc 699 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → (ran 𝑓 ∪ {𝑠}) ∈ (𝒫 𝑎 ∩ Fin))
137 simplrr 818 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → 𝑋 = 𝑑)
138 uniiun 4605 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑑 = 𝑞𝑑 𝑞
139 simprrr 822 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞)))
140 iuneq2 4569 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞)) → 𝑞𝑑 𝑞 = 𝑞𝑑 (𝑠 ∪ (𝑓𝑞)))
141139, 140syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → 𝑞𝑑 𝑞 = 𝑞𝑑 (𝑠 ∪ (𝑓𝑞)))
142138, 141syl5eq 2697 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → 𝑑 = 𝑞𝑑 (𝑠 ∪ (𝑓𝑞)))
143137, 142eqtrd 2685 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → 𝑋 = 𝑞𝑑 (𝑠 ∪ (𝑓𝑞)))
144 ssun2 3810 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 {𝑠} ⊆ (ran 𝑓 ∪ {𝑠})
145 vsnid 4242 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 𝑠 ∈ {𝑠}
146144, 145sselii 3633 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 𝑠 ∈ (ran 𝑓 ∪ {𝑠})
147 elssuni 4499 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑠 ∈ (ran 𝑓 ∪ {𝑠}) → 𝑠 (ran 𝑓 ∪ {𝑠}))
148146, 147ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑠 (ran 𝑓 ∪ {𝑠})
149 fvssunirn 6255 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑓𝑞) ⊆ ran 𝑓
150 ssun1 3809 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ran 𝑓 ⊆ (ran 𝑓 ∪ {𝑠})
151150unissi 4493 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ran 𝑓 (ran 𝑓 ∪ {𝑠})
152149, 151sstri 3645 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑓𝑞) ⊆ (ran 𝑓 ∪ {𝑠})
153148, 152unssi 3821 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑠 ∪ (𝑓𝑞)) ⊆ (ran 𝑓 ∪ {𝑠})
154153rgenw 2953 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑞𝑑 (𝑠 ∪ (𝑓𝑞)) ⊆ (ran 𝑓 ∪ {𝑠})
155 iunss 4593 . . . . . . . . . . . . . . . . . . . . . . . . 25 ( 𝑞𝑑 (𝑠 ∪ (𝑓𝑞)) ⊆ (ran 𝑓 ∪ {𝑠}) ↔ ∀𝑞𝑑 (𝑠 ∪ (𝑓𝑞)) ⊆ (ran 𝑓 ∪ {𝑠}))
156154, 155mpbir 221 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑞𝑑 (𝑠 ∪ (𝑓𝑞)) ⊆ (ran 𝑓 ∪ {𝑠})
157143, 156syl6eqss 3688 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → 𝑋 (ran 𝑓 ∪ {𝑠}))
15834ad2antrr 762 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → 𝑎𝐽)
159120, 158sstrd 3646 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → ran 𝑓𝐽)
16036ad2antrr 762 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → 𝑠𝐽)
161160snssd 4372 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → {𝑠} ⊆ 𝐽)
162159, 161unssd 3822 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → (ran 𝑓 ∪ {𝑠}) ⊆ 𝐽)
163 uniss 4490 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((ran 𝑓 ∪ {𝑠}) ⊆ 𝐽 (ran 𝑓 ∪ {𝑠}) ⊆ 𝐽)
164163, 2syl6sseqr 3685 . . . . . . . . . . . . . . . . . . . . . . . 24 ((ran 𝑓 ∪ {𝑠}) ⊆ 𝐽 (ran 𝑓 ∪ {𝑠}) ⊆ 𝑋)
165162, 164syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → (ran 𝑓 ∪ {𝑠}) ⊆ 𝑋)
166157, 165eqssd 3653 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → 𝑋 = (ran 𝑓 ∪ {𝑠}))
167 unieq 4476 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑏 = (ran 𝑓 ∪ {𝑠}) → 𝑏 = (ran 𝑓 ∪ {𝑠}))
168167eqeq2d 2661 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑏 = (ran 𝑓 ∪ {𝑠}) → (𝑋 = 𝑏𝑋 = (ran 𝑓 ∪ {𝑠})))
169168rspcev 3340 . . . . . . . . . . . . . . . . . . . . . 22 (((ran 𝑓 ∪ {𝑠}) ∈ (𝒫 𝑎 ∩ Fin) ∧ 𝑋 = (ran 𝑓 ∪ {𝑠})) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏)
170136, 166, 169syl2anc 694 . . . . . . . . . . . . . . . . . . . . 21 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏)
171170expr 642 . . . . . . . . . . . . . . . . . . . 20 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ 𝑑 ∈ Fin) → ((𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏))
172171exlimdv 1901 . . . . . . . . . . . . . . . . . . 19 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ 𝑑 ∈ Fin) → (∃𝑓(𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏))
173172ex 449 . . . . . . . . . . . . . . . . . 18 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) → (𝑑 ∈ Fin → (∃𝑓(𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏)))
174117, 173mpdd 43 . . . . . . . . . . . . . . . . 17 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) → (𝑑 ∈ Fin → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏))
17592, 108, 1743syld 60 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) → (𝑑 ∈ PtFin → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏))
176175ex 449 . . . . . . . . . . . . . . 15 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → ((𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑) → (𝑑 ∈ PtFin → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏)))
177176com23 86 . . . . . . . . . . . . . 14 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → (𝑑 ∈ PtFin → ((𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏)))
178177rexlimdv 3059 . . . . . . . . . . . . 13 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → (∃𝑑 ∈ PtFin (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏))
17978, 178syld 47 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑)) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏))
180179rexlimdvaa 3061 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) → (∃𝑠𝑎 𝑥𝑠 → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑)) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏)))
18133, 180syld 47 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) → (𝑥𝑋 → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑)) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏)))
182181exlimdv 1901 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) → (∃𝑥 𝑥𝑋 → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑)) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏)))
18328, 182syl5bi 232 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) → (𝑋 ≠ ∅ → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑)) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏)))
18427, 183pm2.61dne 2909 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑)) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏))
18515, 184syl3an3 1401 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎 ∈ 𝒫 𝐽) → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑)) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏))
1861853exp 1283 . . . . 5 (𝐽 ∈ Top → (𝑋 = 𝑎 → (𝑎 ∈ 𝒫 𝐽 → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑)) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏))))
187186com24 95 . . . 4 (𝐽 ∈ Top → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑)) → (𝑎 ∈ 𝒫 𝐽 → (𝑋 = 𝑎 → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏))))
188187ralrimdv 2997 . . 3 (𝐽 ∈ Top → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑)) → ∀𝑎 ∈ 𝒫 𝐽(𝑋 = 𝑎 → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏)))
1892iscmp 21239 . . . 4 (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑎 ∈ 𝒫 𝐽(𝑋 = 𝑎 → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏)))
190189baibr 965 . . 3 (𝐽 ∈ Top → (∀𝑎 ∈ 𝒫 𝐽(𝑋 = 𝑎 → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏) ↔ 𝐽 ∈ Comp))
191188, 190sylibd 229 . 2 (𝐽 ∈ Top → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑)) → 𝐽 ∈ Comp))
19214, 191impbid2 216 1 (𝐽 ∈ Top → (𝐽 ∈ Comp ↔ ∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wex 1744  wcel 2030  {cab 2637  wne 2823  wral 2941  wrex 2942  {crab 2945  Vcvv 3231  cun 3605  cin 3606  wss 3607  c0 3948  𝒫 cpw 4191  {csn 4210   cuni 4468   ciun 4552  cmpt 4762  ran crn 5144   Fn wfn 5921  wf 5922  ontowfo 5924  cfv 5926  Fincfn 7997  Topctop 20746  Compccmp 21237  PtFincptfin 21354
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-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
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-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-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-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-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-fin 8001  df-top 20747  df-cmp 21238  df-ptfin 21357
This theorem is referenced by: (None)
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