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Theorem rnelfmlem 21803
Description: Lemma for rnelfm 21804. (Contributed by Jeff Hankins, 14-Nov-2009.)
Assertion
Ref Expression
rnelfmlem (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → ran (𝑥𝐿 ↦ (𝐹𝑥)) ∈ (fBas‘𝑌))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹   𝑥,𝐿   𝑥,𝑋   𝑥,𝑌

Proof of Theorem rnelfmlem
Dummy variables 𝑟 𝑠 𝑡 𝑢 𝑣 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl3 1086 . . . . . . 7 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → 𝐹:𝑌𝑋)
2 cnvimass 5520 . . . . . . . 8 (𝐹𝑥) ⊆ dom 𝐹
3 fdm 6089 . . . . . . . 8 (𝐹:𝑌𝑋 → dom 𝐹 = 𝑌)
42, 3syl5sseq 3686 . . . . . . 7 (𝐹:𝑌𝑋 → (𝐹𝑥) ⊆ 𝑌)
51, 4syl 17 . . . . . 6 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → (𝐹𝑥) ⊆ 𝑌)
6 simpl1 1084 . . . . . . 7 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → 𝑌𝐴)
7 elpw2g 4857 . . . . . . 7 (𝑌𝐴 → ((𝐹𝑥) ∈ 𝒫 𝑌 ↔ (𝐹𝑥) ⊆ 𝑌))
86, 7syl 17 . . . . . 6 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → ((𝐹𝑥) ∈ 𝒫 𝑌 ↔ (𝐹𝑥) ⊆ 𝑌))
95, 8mpbird 247 . . . . 5 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → (𝐹𝑥) ∈ 𝒫 𝑌)
109adantr 480 . . . 4 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) → (𝐹𝑥) ∈ 𝒫 𝑌)
11 eqid 2651 . . . 4 (𝑥𝐿 ↦ (𝐹𝑥)) = (𝑥𝐿 ↦ (𝐹𝑥))
1210, 11fmptd 6425 . . 3 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → (𝑥𝐿 ↦ (𝐹𝑥)):𝐿⟶𝒫 𝑌)
13 frn 6091 . . 3 ((𝑥𝐿 ↦ (𝐹𝑥)):𝐿⟶𝒫 𝑌 → ran (𝑥𝐿 ↦ (𝐹𝑥)) ⊆ 𝒫 𝑌)
1412, 13syl 17 . 2 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → ran (𝑥𝐿 ↦ (𝐹𝑥)) ⊆ 𝒫 𝑌)
15 filtop 21706 . . . . . . . 8 (𝐿 ∈ (Fil‘𝑋) → 𝑋𝐿)
16153ad2ant2 1103 . . . . . . 7 ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → 𝑋𝐿)
1716adantr 480 . . . . . 6 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → 𝑋𝐿)
18 fimacnv 6387 . . . . . . . . 9 (𝐹:𝑌𝑋 → (𝐹𝑋) = 𝑌)
1918eqcomd 2657 . . . . . . . 8 (𝐹:𝑌𝑋𝑌 = (𝐹𝑋))
20193ad2ant3 1104 . . . . . . 7 ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → 𝑌 = (𝐹𝑋))
2120adantr 480 . . . . . 6 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → 𝑌 = (𝐹𝑋))
22 imaeq2 5497 . . . . . . . 8 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
2322eqeq2d 2661 . . . . . . 7 (𝑥 = 𝑋 → (𝑌 = (𝐹𝑥) ↔ 𝑌 = (𝐹𝑋)))
2423rspcev 3340 . . . . . 6 ((𝑋𝐿𝑌 = (𝐹𝑋)) → ∃𝑥𝐿 𝑌 = (𝐹𝑥))
2517, 21, 24syl2anc 694 . . . . 5 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → ∃𝑥𝐿 𝑌 = (𝐹𝑥))
2611elrnmpt 5404 . . . . . . 7 (𝑌𝐴 → (𝑌 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐿 𝑌 = (𝐹𝑥)))
27263ad2ant1 1102 . . . . . 6 ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → (𝑌 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐿 𝑌 = (𝐹𝑥)))
2827adantr 480 . . . . 5 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → (𝑌 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐿 𝑌 = (𝐹𝑥)))
2925, 28mpbird 247 . . . 4 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → 𝑌 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)))
30 ne0i 3954 . . . 4 (𝑌 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) → ran (𝑥𝐿 ↦ (𝐹𝑥)) ≠ ∅)
3129, 30syl 17 . . 3 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → ran (𝑥𝐿 ↦ (𝐹𝑥)) ≠ ∅)
32 0nelfil 21700 . . . . . . 7 (𝐿 ∈ (Fil‘𝑋) → ¬ ∅ ∈ 𝐿)
33323ad2ant2 1103 . . . . . 6 ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → ¬ ∅ ∈ 𝐿)
3433adantr 480 . . . . 5 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → ¬ ∅ ∈ 𝐿)
35 0ex 4823 . . . . . . 7 ∅ ∈ V
3611elrnmpt 5404 . . . . . . 7 (∅ ∈ V → (∅ ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐿 ∅ = (𝐹𝑥)))
3735, 36ax-mp 5 . . . . . 6 (∅ ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐿 ∅ = (𝐹𝑥))
38 ffn 6083 . . . . . . . . . . . . . . . . . 18 (𝐹:𝑌𝑋𝐹 Fn 𝑌)
39 fvelrnb 6282 . . . . . . . . . . . . . . . . . 18 (𝐹 Fn 𝑌 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑧𝑌 (𝐹𝑧) = 𝑦))
4038, 39syl 17 . . . . . . . . . . . . . . . . 17 (𝐹:𝑌𝑋 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑧𝑌 (𝐹𝑧) = 𝑦))
41403ad2ant3 1104 . . . . . . . . . . . . . . . 16 ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → (𝑦 ∈ ran 𝐹 ↔ ∃𝑧𝑌 (𝐹𝑧) = 𝑦))
4241ad2antrr 762 . . . . . . . . . . . . . . 15 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ (𝑥𝐿𝑦𝑥)) → (𝑦 ∈ ran 𝐹 ↔ ∃𝑧𝑌 (𝐹𝑧) = 𝑦))
43 eleq1 2718 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹𝑧) = 𝑦 → ((𝐹𝑧) ∈ 𝑥𝑦𝑥))
4443biimparc 503 . . . . . . . . . . . . . . . . . . . 20 ((𝑦𝑥 ∧ (𝐹𝑧) = 𝑦) → (𝐹𝑧) ∈ 𝑥)
4544ad2ant2l 797 . . . . . . . . . . . . . . . . . . 19 (((𝑥𝐿𝑦𝑥) ∧ (𝑧𝑌 ∧ (𝐹𝑧) = 𝑦)) → (𝐹𝑧) ∈ 𝑥)
4645adantll 750 . . . . . . . . . . . . . . . . . 18 (((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ (𝑥𝐿𝑦𝑥)) ∧ (𝑧𝑌 ∧ (𝐹𝑧) = 𝑦)) → (𝐹𝑧) ∈ 𝑥)
47 ffun 6086 . . . . . . . . . . . . . . . . . . . . 21 (𝐹:𝑌𝑋 → Fun 𝐹)
48473ad2ant3 1104 . . . . . . . . . . . . . . . . . . . 20 ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → Fun 𝐹)
4948ad3antrrr 766 . . . . . . . . . . . . . . . . . . 19 (((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ (𝑥𝐿𝑦𝑥)) ∧ (𝑧𝑌 ∧ (𝐹𝑧) = 𝑦)) → Fun 𝐹)
503eleq2d 2716 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐹:𝑌𝑋 → (𝑧 ∈ dom 𝐹𝑧𝑌))
5150biimpar 501 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐹:𝑌𝑋𝑧𝑌) → 𝑧 ∈ dom 𝐹)
52513ad2antl3 1245 . . . . . . . . . . . . . . . . . . . . 21 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ 𝑧𝑌) → 𝑧 ∈ dom 𝐹)
5352adantlr 751 . . . . . . . . . . . . . . . . . . . 20 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑧𝑌) → 𝑧 ∈ dom 𝐹)
5453ad2ant2r 798 . . . . . . . . . . . . . . . . . . 19 (((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ (𝑥𝐿𝑦𝑥)) ∧ (𝑧𝑌 ∧ (𝐹𝑧) = 𝑦)) → 𝑧 ∈ dom 𝐹)
55 fvimacnv 6372 . . . . . . . . . . . . . . . . . . 19 ((Fun 𝐹𝑧 ∈ dom 𝐹) → ((𝐹𝑧) ∈ 𝑥𝑧 ∈ (𝐹𝑥)))
5649, 54, 55syl2anc 694 . . . . . . . . . . . . . . . . . 18 (((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ (𝑥𝐿𝑦𝑥)) ∧ (𝑧𝑌 ∧ (𝐹𝑧) = 𝑦)) → ((𝐹𝑧) ∈ 𝑥𝑧 ∈ (𝐹𝑥)))
5746, 56mpbid 222 . . . . . . . . . . . . . . . . 17 (((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ (𝑥𝐿𝑦𝑥)) ∧ (𝑧𝑌 ∧ (𝐹𝑧) = 𝑦)) → 𝑧 ∈ (𝐹𝑥))
58 n0i 3953 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ (𝐹𝑥) → ¬ (𝐹𝑥) = ∅)
59 eqcom 2658 . . . . . . . . . . . . . . . . . 18 ((𝐹𝑥) = ∅ ↔ ∅ = (𝐹𝑥))
6058, 59sylnib 317 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ (𝐹𝑥) → ¬ ∅ = (𝐹𝑥))
6157, 60syl 17 . . . . . . . . . . . . . . . 16 (((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ (𝑥𝐿𝑦𝑥)) ∧ (𝑧𝑌 ∧ (𝐹𝑧) = 𝑦)) → ¬ ∅ = (𝐹𝑥))
6261rexlimdvaa 3061 . . . . . . . . . . . . . . 15 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ (𝑥𝐿𝑦𝑥)) → (∃𝑧𝑌 (𝐹𝑧) = 𝑦 → ¬ ∅ = (𝐹𝑥)))
6342, 62sylbid 230 . . . . . . . . . . . . . 14 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ (𝑥𝐿𝑦𝑥)) → (𝑦 ∈ ran 𝐹 → ¬ ∅ = (𝐹𝑥)))
6463con2d 129 . . . . . . . . . . . . 13 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ (𝑥𝐿𝑦𝑥)) → (∅ = (𝐹𝑥) → ¬ 𝑦 ∈ ran 𝐹))
6564expr 642 . . . . . . . . . . . 12 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) → (𝑦𝑥 → (∅ = (𝐹𝑥) → ¬ 𝑦 ∈ ran 𝐹)))
6665com23 86 . . . . . . . . . . 11 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) → (∅ = (𝐹𝑥) → (𝑦𝑥 → ¬ 𝑦 ∈ ran 𝐹)))
6766impr 648 . . . . . . . . . 10 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ (𝑥𝐿 ∧ ∅ = (𝐹𝑥))) → (𝑦𝑥 → ¬ 𝑦 ∈ ran 𝐹))
6867alrimiv 1895 . . . . . . . . 9 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ (𝑥𝐿 ∧ ∅ = (𝐹𝑥))) → ∀𝑦(𝑦𝑥 → ¬ 𝑦 ∈ ran 𝐹))
69 imnan 437 . . . . . . . . . . . 12 ((𝑦𝑥 → ¬ 𝑦 ∈ ran 𝐹) ↔ ¬ (𝑦𝑥𝑦 ∈ ran 𝐹))
70 elin 3829 . . . . . . . . . . . 12 (𝑦 ∈ (𝑥 ∩ ran 𝐹) ↔ (𝑦𝑥𝑦 ∈ ran 𝐹))
7169, 70xchbinxr 324 . . . . . . . . . . 11 ((𝑦𝑥 → ¬ 𝑦 ∈ ran 𝐹) ↔ ¬ 𝑦 ∈ (𝑥 ∩ ran 𝐹))
7271albii 1787 . . . . . . . . . 10 (∀𝑦(𝑦𝑥 → ¬ 𝑦 ∈ ran 𝐹) ↔ ∀𝑦 ¬ 𝑦 ∈ (𝑥 ∩ ran 𝐹))
73 eq0 3962 . . . . . . . . . 10 ((𝑥 ∩ ran 𝐹) = ∅ ↔ ∀𝑦 ¬ 𝑦 ∈ (𝑥 ∩ ran 𝐹))
74 eqcom 2658 . . . . . . . . . 10 ((𝑥 ∩ ran 𝐹) = ∅ ↔ ∅ = (𝑥 ∩ ran 𝐹))
7572, 73, 743bitr2i 288 . . . . . . . . 9 (∀𝑦(𝑦𝑥 → ¬ 𝑦 ∈ ran 𝐹) ↔ ∅ = (𝑥 ∩ ran 𝐹))
7668, 75sylib 208 . . . . . . . 8 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ (𝑥𝐿 ∧ ∅ = (𝐹𝑥))) → ∅ = (𝑥 ∩ ran 𝐹))
77 simpll2 1121 . . . . . . . . 9 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ (𝑥𝐿 ∧ ∅ = (𝐹𝑥))) → 𝐿 ∈ (Fil‘𝑋))
78 simprl 809 . . . . . . . . 9 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ (𝑥𝐿 ∧ ∅ = (𝐹𝑥))) → 𝑥𝐿)
79 simplr 807 . . . . . . . . 9 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ (𝑥𝐿 ∧ ∅ = (𝐹𝑥))) → ran 𝐹𝐿)
80 filin 21705 . . . . . . . . 9 ((𝐿 ∈ (Fil‘𝑋) ∧ 𝑥𝐿 ∧ ran 𝐹𝐿) → (𝑥 ∩ ran 𝐹) ∈ 𝐿)
8177, 78, 79, 80syl3anc 1366 . . . . . . . 8 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ (𝑥𝐿 ∧ ∅ = (𝐹𝑥))) → (𝑥 ∩ ran 𝐹) ∈ 𝐿)
8276, 81eqeltrd 2730 . . . . . . 7 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ (𝑥𝐿 ∧ ∅ = (𝐹𝑥))) → ∅ ∈ 𝐿)
8382rexlimdvaa 3061 . . . . . 6 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → (∃𝑥𝐿 ∅ = (𝐹𝑥) → ∅ ∈ 𝐿))
8437, 83syl5bi 232 . . . . 5 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → (∅ ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) → ∅ ∈ 𝐿))
8534, 84mtod 189 . . . 4 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → ¬ ∅ ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)))
86 df-nel 2927 . . . 4 (∅ ∉ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ¬ ∅ ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)))
8785, 86sylibr 224 . . 3 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → ∅ ∉ ran (𝑥𝐿 ↦ (𝐹𝑥)))
88 vex 3234 . . . . . . . . 9 𝑟 ∈ V
8911elrnmpt 5404 . . . . . . . . 9 (𝑟 ∈ V → (𝑟 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐿 𝑟 = (𝐹𝑥)))
9088, 89ax-mp 5 . . . . . . . 8 (𝑟 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐿 𝑟 = (𝐹𝑥))
91 imaeq2 5497 . . . . . . . . . 10 (𝑥 = 𝑢 → (𝐹𝑥) = (𝐹𝑢))
9291eqeq2d 2661 . . . . . . . . 9 (𝑥 = 𝑢 → (𝑟 = (𝐹𝑥) ↔ 𝑟 = (𝐹𝑢)))
9392cbvrexv 3202 . . . . . . . 8 (∃𝑥𝐿 𝑟 = (𝐹𝑥) ↔ ∃𝑢𝐿 𝑟 = (𝐹𝑢))
9490, 93bitri 264 . . . . . . 7 (𝑟 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑢𝐿 𝑟 = (𝐹𝑢))
95 vex 3234 . . . . . . . . 9 𝑠 ∈ V
9611elrnmpt 5404 . . . . . . . . 9 (𝑠 ∈ V → (𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐿 𝑠 = (𝐹𝑥)))
9795, 96ax-mp 5 . . . . . . . 8 (𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐿 𝑠 = (𝐹𝑥))
98 imaeq2 5497 . . . . . . . . . 10 (𝑥 = 𝑣 → (𝐹𝑥) = (𝐹𝑣))
9998eqeq2d 2661 . . . . . . . . 9 (𝑥 = 𝑣 → (𝑠 = (𝐹𝑥) ↔ 𝑠 = (𝐹𝑣)))
10099cbvrexv 3202 . . . . . . . 8 (∃𝑥𝐿 𝑠 = (𝐹𝑥) ↔ ∃𝑣𝐿 𝑠 = (𝐹𝑣))
10197, 100bitri 264 . . . . . . 7 (𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑣𝐿 𝑠 = (𝐹𝑣))
10294, 101anbi12i 733 . . . . . 6 ((𝑟 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ∧ 𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))) ↔ (∃𝑢𝐿 𝑟 = (𝐹𝑢) ∧ ∃𝑣𝐿 𝑠 = (𝐹𝑣)))
103 reeanv 3136 . . . . . 6 (∃𝑢𝐿𝑣𝐿 (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣)) ↔ (∃𝑢𝐿 𝑟 = (𝐹𝑢) ∧ ∃𝑣𝐿 𝑠 = (𝐹𝑣)))
104102, 103bitr4i 267 . . . . 5 ((𝑟 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ∧ 𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))) ↔ ∃𝑢𝐿𝑣𝐿 (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣)))
105 filin 21705 . . . . . . . . . . . . . 14 ((𝐿 ∈ (Fil‘𝑋) ∧ 𝑢𝐿𝑣𝐿) → (𝑢𝑣) ∈ 𝐿)
1061053expb 1285 . . . . . . . . . . . . 13 ((𝐿 ∈ (Fil‘𝑋) ∧ (𝑢𝐿𝑣𝐿)) → (𝑢𝑣) ∈ 𝐿)
107106adantlr 751 . . . . . . . . . . . 12 (((𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ (𝑢𝐿𝑣𝐿)) → (𝑢𝑣) ∈ 𝐿)
108 eqidd 2652 . . . . . . . . . . . 12 (((𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ (𝑢𝐿𝑣𝐿)) → (𝐹 “ (𝑢𝑣)) = (𝐹 “ (𝑢𝑣)))
109 imaeq2 5497 . . . . . . . . . . . . . 14 (𝑥 = (𝑢𝑣) → (𝐹𝑥) = (𝐹 “ (𝑢𝑣)))
110109eqeq2d 2661 . . . . . . . . . . . . 13 (𝑥 = (𝑢𝑣) → ((𝐹 “ (𝑢𝑣)) = (𝐹𝑥) ↔ (𝐹 “ (𝑢𝑣)) = (𝐹 “ (𝑢𝑣))))
111110rspcev 3340 . . . . . . . . . . . 12 (((𝑢𝑣) ∈ 𝐿 ∧ (𝐹 “ (𝑢𝑣)) = (𝐹 “ (𝑢𝑣))) → ∃𝑥𝐿 (𝐹 “ (𝑢𝑣)) = (𝐹𝑥))
112107, 108, 111syl2anc 694 . . . . . . . . . . 11 (((𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ (𝑢𝐿𝑣𝐿)) → ∃𝑥𝐿 (𝐹 “ (𝑢𝑣)) = (𝐹𝑥))
1131123adantl1 1237 . . . . . . . . . 10 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ (𝑢𝐿𝑣𝐿)) → ∃𝑥𝐿 (𝐹 “ (𝑢𝑣)) = (𝐹𝑥))
114113ad2ant2r 798 . . . . . . . . 9 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ ((𝑢𝐿𝑣𝐿) ∧ (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣)))) → ∃𝑥𝐿 (𝐹 “ (𝑢𝑣)) = (𝐹𝑥))
115 simpll1 1120 . . . . . . . . . . 11 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ ((𝑢𝐿𝑣𝐿) ∧ (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣)))) → 𝑌𝐴)
116 cnvimass 5520 . . . . . . . . . . . . . 14 (𝐹 “ (𝑢𝑣)) ⊆ dom 𝐹
117116, 3syl5sseq 3686 . . . . . . . . . . . . 13 (𝐹:𝑌𝑋 → (𝐹 “ (𝑢𝑣)) ⊆ 𝑌)
1181173ad2ant3 1104 . . . . . . . . . . . 12 ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → (𝐹 “ (𝑢𝑣)) ⊆ 𝑌)
119118ad2antrr 762 . . . . . . . . . . 11 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ ((𝑢𝐿𝑣𝐿) ∧ (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣)))) → (𝐹 “ (𝑢𝑣)) ⊆ 𝑌)
120115, 119ssexd 4838 . . . . . . . . . 10 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ ((𝑢𝐿𝑣𝐿) ∧ (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣)))) → (𝐹 “ (𝑢𝑣)) ∈ V)
12111elrnmpt 5404 . . . . . . . . . 10 ((𝐹 “ (𝑢𝑣)) ∈ V → ((𝐹 “ (𝑢𝑣)) ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐿 (𝐹 “ (𝑢𝑣)) = (𝐹𝑥)))
122120, 121syl 17 . . . . . . . . 9 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ ((𝑢𝐿𝑣𝐿) ∧ (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣)))) → ((𝐹 “ (𝑢𝑣)) ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐿 (𝐹 “ (𝑢𝑣)) = (𝐹𝑥)))
123114, 122mpbird 247 . . . . . . . 8 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ ((𝑢𝐿𝑣𝐿) ∧ (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣)))) → (𝐹 “ (𝑢𝑣)) ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)))
124 simprrl 821 . . . . . . . . . . 11 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ ((𝑢𝐿𝑣𝐿) ∧ (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣)))) → 𝑟 = (𝐹𝑢))
125 simprrr 822 . . . . . . . . . . 11 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ ((𝑢𝐿𝑣𝐿) ∧ (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣)))) → 𝑠 = (𝐹𝑣))
126124, 125ineq12d 3848 . . . . . . . . . 10 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ ((𝑢𝐿𝑣𝐿) ∧ (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣)))) → (𝑟𝑠) = ((𝐹𝑢) ∩ (𝐹𝑣)))
127 funcnvcnv 5994 . . . . . . . . . . . . 13 (Fun 𝐹 → Fun 𝐹)
128 imain 6012 . . . . . . . . . . . . 13 (Fun 𝐹 → (𝐹 “ (𝑢𝑣)) = ((𝐹𝑢) ∩ (𝐹𝑣)))
12947, 127, 1283syl 18 . . . . . . . . . . . 12 (𝐹:𝑌𝑋 → (𝐹 “ (𝑢𝑣)) = ((𝐹𝑢) ∩ (𝐹𝑣)))
1301293ad2ant3 1104 . . . . . . . . . . 11 ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → (𝐹 “ (𝑢𝑣)) = ((𝐹𝑢) ∩ (𝐹𝑣)))
131130ad2antrr 762 . . . . . . . . . 10 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ ((𝑢𝐿𝑣𝐿) ∧ (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣)))) → (𝐹 “ (𝑢𝑣)) = ((𝐹𝑢) ∩ (𝐹𝑣)))
132126, 131eqtr4d 2688 . . . . . . . . 9 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ ((𝑢𝐿𝑣𝐿) ∧ (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣)))) → (𝑟𝑠) = (𝐹 “ (𝑢𝑣)))
133 eqimss2 3691 . . . . . . . . 9 ((𝑟𝑠) = (𝐹 “ (𝑢𝑣)) → (𝐹 “ (𝑢𝑣)) ⊆ (𝑟𝑠))
134132, 133syl 17 . . . . . . . 8 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ ((𝑢𝐿𝑣𝐿) ∧ (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣)))) → (𝐹 “ (𝑢𝑣)) ⊆ (𝑟𝑠))
135 sseq1 3659 . . . . . . . . 9 (𝑡 = (𝐹 “ (𝑢𝑣)) → (𝑡 ⊆ (𝑟𝑠) ↔ (𝐹 “ (𝑢𝑣)) ⊆ (𝑟𝑠)))
136135rspcev 3340 . . . . . . . 8 (((𝐹 “ (𝑢𝑣)) ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ∧ (𝐹 “ (𝑢𝑣)) ⊆ (𝑟𝑠)) → ∃𝑡 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))𝑡 ⊆ (𝑟𝑠))
137123, 134, 136syl2anc 694 . . . . . . 7 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ ((𝑢𝐿𝑣𝐿) ∧ (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣)))) → ∃𝑡 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))𝑡 ⊆ (𝑟𝑠))
138137exp32 630 . . . . . 6 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → ((𝑢𝐿𝑣𝐿) → ((𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣)) → ∃𝑡 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))𝑡 ⊆ (𝑟𝑠))))
139138rexlimdvv 3066 . . . . 5 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → (∃𝑢𝐿𝑣𝐿 (𝑟 = (𝐹𝑢) ∧ 𝑠 = (𝐹𝑣)) → ∃𝑡 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))𝑡 ⊆ (𝑟𝑠)))
140104, 139syl5bi 232 . . . 4 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → ((𝑟 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ∧ 𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))) → ∃𝑡 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))𝑡 ⊆ (𝑟𝑠)))
141140ralrimivv 2999 . . 3 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → ∀𝑟 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))∀𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))∃𝑡 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))𝑡 ⊆ (𝑟𝑠))
14231, 87, 1413jca 1261 . 2 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → (ran (𝑥𝐿 ↦ (𝐹𝑥)) ≠ ∅ ∧ ∅ ∉ ran (𝑥𝐿 ↦ (𝐹𝑥)) ∧ ∀𝑟 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))∀𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))∃𝑡 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))𝑡 ⊆ (𝑟𝑠)))
143 isfbas2 21686 . . 3 (𝑌𝐴 → (ran (𝑥𝐿 ↦ (𝐹𝑥)) ∈ (fBas‘𝑌) ↔ (ran (𝑥𝐿 ↦ (𝐹𝑥)) ⊆ 𝒫 𝑌 ∧ (ran (𝑥𝐿 ↦ (𝐹𝑥)) ≠ ∅ ∧ ∅ ∉ ran (𝑥𝐿 ↦ (𝐹𝑥)) ∧ ∀𝑟 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))∀𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))∃𝑡 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))𝑡 ⊆ (𝑟𝑠)))))
1446, 143syl 17 . 2 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → (ran (𝑥𝐿 ↦ (𝐹𝑥)) ∈ (fBas‘𝑌) ↔ (ran (𝑥𝐿 ↦ (𝐹𝑥)) ⊆ 𝒫 𝑌 ∧ (ran (𝑥𝐿 ↦ (𝐹𝑥)) ≠ ∅ ∧ ∅ ∉ ran (𝑥𝐿 ↦ (𝐹𝑥)) ∧ ∀𝑟 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))∀𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))∃𝑡 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))𝑡 ⊆ (𝑟𝑠)))))
14514, 142, 144mpbir2and 977 1 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → ran (𝑥𝐿 ↦ (𝐹𝑥)) ∈ (fBas‘𝑌))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1054  wal 1521   = wceq 1523  wcel 2030  wne 2823  wnel 2926  wral 2941  wrex 2942  Vcvv 3231  cin 3606  wss 3607  c0 3948  𝒫 cpw 4191  cmpt 4762  ccnv 5142  dom cdm 5143  ran crn 5144  cima 5146  Fun wfun 5920   Fn wfn 5921  wf 5922  cfv 5926  fBascfbas 19782  Filcfil 21696
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
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  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-nel 2927  df-ral 2946  df-rex 2947  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-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  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-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-fbas 19791  df-fil 21697
This theorem is referenced by:  rnelfm  21804  fmfnfm  21809
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