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Theorem rnelfm 23304
Description: A condition for a filter to be an image filter for a given function. (Contributed by Jeff Hankins, 14-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
Assertion
Ref Expression
rnelfm ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → (𝐿 ∈ ran (𝑋 FilMap 𝐹) ↔ ran 𝐹𝐿))

Proof of Theorem rnelfm
Dummy variables 𝑏 𝑠 𝑡 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 filtop 23206 . . . . . . 7 (𝐿 ∈ (Fil‘𝑋) → 𝑋𝐿)
213ad2ant2 1134 . . . . . 6 ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → 𝑋𝐿)
3 simp1 1136 . . . . . 6 ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → 𝑌𝐴)
4 simp3 1138 . . . . . 6 ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → 𝐹:𝑌𝑋)
5 fmf 23296 . . . . . 6 ((𝑋𝐿𝑌𝐴𝐹:𝑌𝑋) → (𝑋 FilMap 𝐹):(fBas‘𝑌)⟶(Fil‘𝑋))
62, 3, 4, 5syl3anc 1371 . . . . 5 ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → (𝑋 FilMap 𝐹):(fBas‘𝑌)⟶(Fil‘𝑋))
76ffnd 6669 . . . 4 ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → (𝑋 FilMap 𝐹) Fn (fBas‘𝑌))
8 fvelrnb 6903 . . . 4 ((𝑋 FilMap 𝐹) Fn (fBas‘𝑌) → (𝐿 ∈ ran (𝑋 FilMap 𝐹) ↔ ∃𝑏 ∈ (fBas‘𝑌)((𝑋 FilMap 𝐹)‘𝑏) = 𝐿))
97, 8syl 17 . . 3 ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → (𝐿 ∈ ran (𝑋 FilMap 𝐹) ↔ ∃𝑏 ∈ (fBas‘𝑌)((𝑋 FilMap 𝐹)‘𝑏) = 𝐿))
10 ffn 6668 . . . . . . . . . . . 12 (𝐹:𝑌𝑋𝐹 Fn 𝑌)
11 dffn4 6762 . . . . . . . . . . . 12 (𝐹 Fn 𝑌𝐹:𝑌onto→ran 𝐹)
1210, 11sylib 217 . . . . . . . . . . 11 (𝐹:𝑌𝑋𝐹:𝑌onto→ran 𝐹)
13 foima 6761 . . . . . . . . . . 11 (𝐹:𝑌onto→ran 𝐹 → (𝐹𝑌) = ran 𝐹)
1412, 13syl 17 . . . . . . . . . 10 (𝐹:𝑌𝑋 → (𝐹𝑌) = ran 𝐹)
1514ad2antlr 725 . . . . . . . . 9 (((𝑋𝐿𝐹:𝑌𝑋) ∧ 𝑏 ∈ (fBas‘𝑌)) → (𝐹𝑌) = ran 𝐹)
16 simpll 765 . . . . . . . . . 10 (((𝑋𝐿𝐹:𝑌𝑋) ∧ 𝑏 ∈ (fBas‘𝑌)) → 𝑋𝐿)
17 simpr 485 . . . . . . . . . 10 (((𝑋𝐿𝐹:𝑌𝑋) ∧ 𝑏 ∈ (fBas‘𝑌)) → 𝑏 ∈ (fBas‘𝑌))
18 simplr 767 . . . . . . . . . 10 (((𝑋𝐿𝐹:𝑌𝑋) ∧ 𝑏 ∈ (fBas‘𝑌)) → 𝐹:𝑌𝑋)
19 fgcl 23229 . . . . . . . . . . . 12 (𝑏 ∈ (fBas‘𝑌) → (𝑌filGen𝑏) ∈ (Fil‘𝑌))
20 filtop 23206 . . . . . . . . . . . 12 ((𝑌filGen𝑏) ∈ (Fil‘𝑌) → 𝑌 ∈ (𝑌filGen𝑏))
2119, 20syl 17 . . . . . . . . . . 11 (𝑏 ∈ (fBas‘𝑌) → 𝑌 ∈ (𝑌filGen𝑏))
2221adantl 482 . . . . . . . . . 10 (((𝑋𝐿𝐹:𝑌𝑋) ∧ 𝑏 ∈ (fBas‘𝑌)) → 𝑌 ∈ (𝑌filGen𝑏))
23 eqid 2736 . . . . . . . . . . 11 (𝑌filGen𝑏) = (𝑌filGen𝑏)
2423imaelfm 23302 . . . . . . . . . 10 (((𝑋𝐿𝑏 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑌 ∈ (𝑌filGen𝑏)) → (𝐹𝑌) ∈ ((𝑋 FilMap 𝐹)‘𝑏))
2516, 17, 18, 22, 24syl31anc 1373 . . . . . . . . 9 (((𝑋𝐿𝐹:𝑌𝑋) ∧ 𝑏 ∈ (fBas‘𝑌)) → (𝐹𝑌) ∈ ((𝑋 FilMap 𝐹)‘𝑏))
2615, 25eqeltrrd 2839 . . . . . . . 8 (((𝑋𝐿𝐹:𝑌𝑋) ∧ 𝑏 ∈ (fBas‘𝑌)) → ran 𝐹 ∈ ((𝑋 FilMap 𝐹)‘𝑏))
27 eleq2 2826 . . . . . . . 8 (((𝑋 FilMap 𝐹)‘𝑏) = 𝐿 → (ran 𝐹 ∈ ((𝑋 FilMap 𝐹)‘𝑏) ↔ ran 𝐹𝐿))
2826, 27syl5ibcom 244 . . . . . . 7 (((𝑋𝐿𝐹:𝑌𝑋) ∧ 𝑏 ∈ (fBas‘𝑌)) → (((𝑋 FilMap 𝐹)‘𝑏) = 𝐿 → ran 𝐹𝐿))
2928ex 413 . . . . . 6 ((𝑋𝐿𝐹:𝑌𝑋) → (𝑏 ∈ (fBas‘𝑌) → (((𝑋 FilMap 𝐹)‘𝑏) = 𝐿 → ran 𝐹𝐿)))
301, 29sylan 580 . . . . 5 ((𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → (𝑏 ∈ (fBas‘𝑌) → (((𝑋 FilMap 𝐹)‘𝑏) = 𝐿 → ran 𝐹𝐿)))
31303adant1 1130 . . . 4 ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → (𝑏 ∈ (fBas‘𝑌) → (((𝑋 FilMap 𝐹)‘𝑏) = 𝐿 → ran 𝐹𝐿)))
3231rexlimdv 3150 . . 3 ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → (∃𝑏 ∈ (fBas‘𝑌)((𝑋 FilMap 𝐹)‘𝑏) = 𝐿 → ran 𝐹𝐿))
339, 32sylbid 239 . 2 ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → (𝐿 ∈ ran (𝑋 FilMap 𝐹) → ran 𝐹𝐿))
34 simpl2 1192 . . . . . . . . 9 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → 𝐿 ∈ (Fil‘𝑋))
35 filelss 23203 . . . . . . . . . 10 ((𝐿 ∈ (Fil‘𝑋) ∧ 𝑡𝐿) → 𝑡𝑋)
3635ex 413 . . . . . . . . 9 (𝐿 ∈ (Fil‘𝑋) → (𝑡𝐿𝑡𝑋))
3734, 36syl 17 . . . . . . . 8 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → (𝑡𝐿𝑡𝑋))
38 simpr 485 . . . . . . . . . . . 12 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑡𝐿) → 𝑡𝐿)
39 eqidd 2737 . . . . . . . . . . . 12 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑡𝐿) → (𝐹𝑡) = (𝐹𝑡))
40 imaeq2 6009 . . . . . . . . . . . . 13 (𝑥 = 𝑡 → (𝐹𝑥) = (𝐹𝑡))
4140rspceeqv 3595 . . . . . . . . . . . 12 ((𝑡𝐿 ∧ (𝐹𝑡) = (𝐹𝑡)) → ∃𝑥𝐿 (𝐹𝑡) = (𝐹𝑥))
4238, 39, 41syl2anc 584 . . . . . . . . . . 11 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑡𝐿) → ∃𝑥𝐿 (𝐹𝑡) = (𝐹𝑥))
43 simpl1 1191 . . . . . . . . . . . . . 14 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → 𝑌𝐴)
44 cnvimass 6033 . . . . . . . . . . . . . . . . 17 (𝐹𝑡) ⊆ dom 𝐹
45 fdm 6677 . . . . . . . . . . . . . . . . 17 (𝐹:𝑌𝑋 → dom 𝐹 = 𝑌)
4644, 45sseqtrid 3996 . . . . . . . . . . . . . . . 16 (𝐹:𝑌𝑋 → (𝐹𝑡) ⊆ 𝑌)
47463ad2ant3 1135 . . . . . . . . . . . . . . 15 ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → (𝐹𝑡) ⊆ 𝑌)
4847adantr 481 . . . . . . . . . . . . . 14 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → (𝐹𝑡) ⊆ 𝑌)
4943, 48ssexd 5281 . . . . . . . . . . . . 13 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → (𝐹𝑡) ∈ V)
50 eqid 2736 . . . . . . . . . . . . . 14 (𝑥𝐿 ↦ (𝐹𝑥)) = (𝑥𝐿 ↦ (𝐹𝑥))
5150elrnmpt 5911 . . . . . . . . . . . . 13 ((𝐹𝑡) ∈ V → ((𝐹𝑡) ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐿 (𝐹𝑡) = (𝐹𝑥)))
5249, 51syl 17 . . . . . . . . . . . 12 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → ((𝐹𝑡) ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐿 (𝐹𝑡) = (𝐹𝑥)))
5352adantr 481 . . . . . . . . . . 11 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑡𝐿) → ((𝐹𝑡) ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐿 (𝐹𝑡) = (𝐹𝑥)))
5442, 53mpbird 256 . . . . . . . . . 10 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑡𝐿) → (𝐹𝑡) ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)))
55 ssid 3966 . . . . . . . . . . 11 (𝐹𝑡) ⊆ (𝐹𝑡)
56 ffun 6671 . . . . . . . . . . . . . 14 (𝐹:𝑌𝑋 → Fun 𝐹)
57563ad2ant3 1135 . . . . . . . . . . . . 13 ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → Fun 𝐹)
5857ad2antrr 724 . . . . . . . . . . . 12 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑡𝐿) → Fun 𝐹)
59 funimass3 7004 . . . . . . . . . . . 12 ((Fun 𝐹 ∧ (𝐹𝑡) ⊆ dom 𝐹) → ((𝐹 “ (𝐹𝑡)) ⊆ 𝑡 ↔ (𝐹𝑡) ⊆ (𝐹𝑡)))
6058, 44, 59sylancl 586 . . . . . . . . . . 11 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑡𝐿) → ((𝐹 “ (𝐹𝑡)) ⊆ 𝑡 ↔ (𝐹𝑡) ⊆ (𝐹𝑡)))
6155, 60mpbiri 257 . . . . . . . . . 10 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑡𝐿) → (𝐹 “ (𝐹𝑡)) ⊆ 𝑡)
62 imaeq2 6009 . . . . . . . . . . . 12 (𝑠 = (𝐹𝑡) → (𝐹𝑠) = (𝐹 “ (𝐹𝑡)))
6362sseq1d 3975 . . . . . . . . . . 11 (𝑠 = (𝐹𝑡) → ((𝐹𝑠) ⊆ 𝑡 ↔ (𝐹 “ (𝐹𝑡)) ⊆ 𝑡))
6463rspcev 3581 . . . . . . . . . 10 (((𝐹𝑡) ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ∧ (𝐹 “ (𝐹𝑡)) ⊆ 𝑡) → ∃𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))(𝐹𝑠) ⊆ 𝑡)
6554, 61, 64syl2anc 584 . . . . . . . . 9 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑡𝐿) → ∃𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))(𝐹𝑠) ⊆ 𝑡)
6665ex 413 . . . . . . . 8 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → (𝑡𝐿 → ∃𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))(𝐹𝑠) ⊆ 𝑡))
6737, 66jcad 513 . . . . . . 7 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → (𝑡𝐿 → (𝑡𝑋 ∧ ∃𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))(𝐹𝑠) ⊆ 𝑡)))
6834adantr 481 . . . . . . . . . . 11 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ ((𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ∧ (𝐹𝑠) ⊆ 𝑡) ∧ 𝑡𝑋)) → 𝐿 ∈ (Fil‘𝑋))
6950elrnmpt 5911 . . . . . . . . . . . . . 14 (𝑠 ∈ V → (𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐿 𝑠 = (𝐹𝑥)))
7069elv 3451 . . . . . . . . . . . . 13 (𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐿 𝑠 = (𝐹𝑥))
71 ssid 3966 . . . . . . . . . . . . . . . . . . . 20 (𝐹𝑥) ⊆ (𝐹𝑥)
7257ad3antrrr 728 . . . . . . . . . . . . . . . . . . . . 21 (((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → Fun 𝐹)
73 cnvimass 6033 . . . . . . . . . . . . . . . . . . . . 21 (𝐹𝑥) ⊆ dom 𝐹
74 funimass3 7004 . . . . . . . . . . . . . . . . . . . . 21 ((Fun 𝐹 ∧ (𝐹𝑥) ⊆ dom 𝐹) → ((𝐹 “ (𝐹𝑥)) ⊆ 𝑥 ↔ (𝐹𝑥) ⊆ (𝐹𝑥)))
7572, 73, 74sylancl 586 . . . . . . . . . . . . . . . . . . . 20 (((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → ((𝐹 “ (𝐹𝑥)) ⊆ 𝑥 ↔ (𝐹𝑥) ⊆ (𝐹𝑥)))
7671, 75mpbiri 257 . . . . . . . . . . . . . . . . . . 19 (((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → (𝐹 “ (𝐹𝑥)) ⊆ 𝑥)
77 imassrn 6024 . . . . . . . . . . . . . . . . . . 19 (𝐹 “ (𝐹𝑥)) ⊆ ran 𝐹
78 ssin 4190 . . . . . . . . . . . . . . . . . . 19 (((𝐹 “ (𝐹𝑥)) ⊆ 𝑥 ∧ (𝐹 “ (𝐹𝑥)) ⊆ ran 𝐹) ↔ (𝐹 “ (𝐹𝑥)) ⊆ (𝑥 ∩ ran 𝐹))
7976, 77, 78sylanblc 589 . . . . . . . . . . . . . . . . . 18 (((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → (𝐹 “ (𝐹𝑥)) ⊆ (𝑥 ∩ ran 𝐹))
80 elin 3926 . . . . . . . . . . . . . . . . . . . 20 (𝑧 ∈ (𝑥 ∩ ran 𝐹) ↔ (𝑧𝑥𝑧 ∈ ran 𝐹))
81 fvelrnb 6903 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐹 Fn 𝑌 → (𝑧 ∈ ran 𝐹 ↔ ∃𝑦𝑌 (𝐹𝑦) = 𝑧))
8210, 81syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐹:𝑌𝑋 → (𝑧 ∈ ran 𝐹 ↔ ∃𝑦𝑌 (𝐹𝑦) = 𝑧))
83823ad2ant3 1135 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → (𝑧 ∈ ran 𝐹 ↔ ∃𝑦𝑌 (𝐹𝑦) = 𝑧))
8483ad3antrrr 728 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → (𝑧 ∈ ran 𝐹 ↔ ∃𝑦𝑌 (𝐹𝑦) = 𝑧))
8572ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) ∧ 𝑦𝑌) ∧ (𝐹𝑦) ∈ 𝑥) → Fun 𝐹)
8685, 73jctir 521 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) ∧ 𝑦𝑌) ∧ (𝐹𝑦) ∈ 𝑥) → (Fun 𝐹 ∧ (𝐹𝑥) ⊆ dom 𝐹))
8757ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) → Fun 𝐹)
8887ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) ∧ 𝑦𝑌) → Fun 𝐹)
89453ad2ant3 1135 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → dom 𝐹 = 𝑌)
9089ad3antrrr 728 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → dom 𝐹 = 𝑌)
9190eleq2d 2823 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → (𝑦 ∈ dom 𝐹𝑦𝑌))
9291biimpar 478 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) ∧ 𝑦𝑌) → 𝑦 ∈ dom 𝐹)
93 fvimacnv 7003 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((Fun 𝐹𝑦 ∈ dom 𝐹) → ((𝐹𝑦) ∈ 𝑥𝑦 ∈ (𝐹𝑥)))
9488, 92, 93syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) ∧ 𝑦𝑌) → ((𝐹𝑦) ∈ 𝑥𝑦 ∈ (𝐹𝑥)))
9594biimpa 477 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) ∧ 𝑦𝑌) ∧ (𝐹𝑦) ∈ 𝑥) → 𝑦 ∈ (𝐹𝑥))
96 funfvima2 7181 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((Fun 𝐹 ∧ (𝐹𝑥) ⊆ dom 𝐹) → (𝑦 ∈ (𝐹𝑥) → (𝐹𝑦) ∈ (𝐹 “ (𝐹𝑥))))
9786, 95, 96sylc 65 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) ∧ 𝑦𝑌) ∧ (𝐹𝑦) ∈ 𝑥) → (𝐹𝑦) ∈ (𝐹 “ (𝐹𝑥)))
9897ex 413 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) ∧ 𝑦𝑌) → ((𝐹𝑦) ∈ 𝑥 → (𝐹𝑦) ∈ (𝐹 “ (𝐹𝑥))))
99 eleq1 2825 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐹𝑦) = 𝑧 → ((𝐹𝑦) ∈ 𝑥𝑧𝑥))
100 eleq1 2825 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐹𝑦) = 𝑧 → ((𝐹𝑦) ∈ (𝐹 “ (𝐹𝑥)) ↔ 𝑧 ∈ (𝐹 “ (𝐹𝑥))))
10199, 100imbi12d 344 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐹𝑦) = 𝑧 → (((𝐹𝑦) ∈ 𝑥 → (𝐹𝑦) ∈ (𝐹 “ (𝐹𝑥))) ↔ (𝑧𝑥𝑧 ∈ (𝐹 “ (𝐹𝑥)))))
10298, 101syl5ibcom 244 . . . . . . . . . . . . . . . . . . . . . . 23 ((((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) ∧ 𝑦𝑌) → ((𝐹𝑦) = 𝑧 → (𝑧𝑥𝑧 ∈ (𝐹 “ (𝐹𝑥)))))
103102rexlimdva 3152 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → (∃𝑦𝑌 (𝐹𝑦) = 𝑧 → (𝑧𝑥𝑧 ∈ (𝐹 “ (𝐹𝑥)))))
10484, 103sylbid 239 . . . . . . . . . . . . . . . . . . . . 21 (((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → (𝑧 ∈ ran 𝐹 → (𝑧𝑥𝑧 ∈ (𝐹 “ (𝐹𝑥)))))
105104impcomd 412 . . . . . . . . . . . . . . . . . . . 20 (((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → ((𝑧𝑥𝑧 ∈ ran 𝐹) → 𝑧 ∈ (𝐹 “ (𝐹𝑥))))
10680, 105biimtrid 241 . . . . . . . . . . . . . . . . . . 19 (((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → (𝑧 ∈ (𝑥 ∩ ran 𝐹) → 𝑧 ∈ (𝐹 “ (𝐹𝑥))))
107106ssrdv 3950 . . . . . . . . . . . . . . . . . 18 (((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → (𝑥 ∩ ran 𝐹) ⊆ (𝐹 “ (𝐹𝑥)))
10879, 107eqssd 3961 . . . . . . . . . . . . . . . . 17 (((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → (𝐹 “ (𝐹𝑥)) = (𝑥 ∩ ran 𝐹))
109 filin 23205 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐿 ∈ (Fil‘𝑋) ∧ 𝑥𝐿 ∧ ran 𝐹𝐿) → (𝑥 ∩ ran 𝐹) ∈ 𝐿)
1101093exp 1119 . . . . . . . . . . . . . . . . . . . . 21 (𝐿 ∈ (Fil‘𝑋) → (𝑥𝐿 → (ran 𝐹𝐿 → (𝑥 ∩ ran 𝐹) ∈ 𝐿)))
111110com23 86 . . . . . . . . . . . . . . . . . . . 20 (𝐿 ∈ (Fil‘𝑋) → (ran 𝐹𝐿 → (𝑥𝐿 → (𝑥 ∩ ran 𝐹) ∈ 𝐿)))
1121113ad2ant2 1134 . . . . . . . . . . . . . . . . . . 19 ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → (ran 𝐹𝐿 → (𝑥𝐿 → (𝑥 ∩ ran 𝐹) ∈ 𝐿)))
113112imp31 418 . . . . . . . . . . . . . . . . . 18 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) → (𝑥 ∩ ran 𝐹) ∈ 𝐿)
114113adantr 481 . . . . . . . . . . . . . . . . 17 (((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → (𝑥 ∩ ran 𝐹) ∈ 𝐿)
115108, 114eqeltrd 2838 . . . . . . . . . . . . . . . 16 (((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → (𝐹 “ (𝐹𝑥)) ∈ 𝐿)
116115exp32 421 . . . . . . . . . . . . . . 15 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) → ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡 → (𝑡𝑋 → (𝐹 “ (𝐹𝑥)) ∈ 𝐿)))
117 imaeq2 6009 . . . . . . . . . . . . . . . . 17 (𝑠 = (𝐹𝑥) → (𝐹𝑠) = (𝐹 “ (𝐹𝑥)))
118117sseq1d 3975 . . . . . . . . . . . . . . . 16 (𝑠 = (𝐹𝑥) → ((𝐹𝑠) ⊆ 𝑡 ↔ (𝐹 “ (𝐹𝑥)) ⊆ 𝑡))
119117eleq1d 2822 . . . . . . . . . . . . . . . . 17 (𝑠 = (𝐹𝑥) → ((𝐹𝑠) ∈ 𝐿 ↔ (𝐹 “ (𝐹𝑥)) ∈ 𝐿))
120119imbi2d 340 . . . . . . . . . . . . . . . 16 (𝑠 = (𝐹𝑥) → ((𝑡𝑋 → (𝐹𝑠) ∈ 𝐿) ↔ (𝑡𝑋 → (𝐹 “ (𝐹𝑥)) ∈ 𝐿)))
121118, 120imbi12d 344 . . . . . . . . . . . . . . 15 (𝑠 = (𝐹𝑥) → (((𝐹𝑠) ⊆ 𝑡 → (𝑡𝑋 → (𝐹𝑠) ∈ 𝐿)) ↔ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡 → (𝑡𝑋 → (𝐹 “ (𝐹𝑥)) ∈ 𝐿))))
122116, 121syl5ibrcom 246 . . . . . . . . . . . . . 14 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) → (𝑠 = (𝐹𝑥) → ((𝐹𝑠) ⊆ 𝑡 → (𝑡𝑋 → (𝐹𝑠) ∈ 𝐿))))
123122rexlimdva 3152 . . . . . . . . . . . . 13 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → (∃𝑥𝐿 𝑠 = (𝐹𝑥) → ((𝐹𝑠) ⊆ 𝑡 → (𝑡𝑋 → (𝐹𝑠) ∈ 𝐿))))
12470, 123biimtrid 241 . . . . . . . . . . . 12 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → (𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) → ((𝐹𝑠) ⊆ 𝑡 → (𝑡𝑋 → (𝐹𝑠) ∈ 𝐿))))
125124imp44 429 . . . . . . . . . . 11 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ ((𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ∧ (𝐹𝑠) ⊆ 𝑡) ∧ 𝑡𝑋)) → (𝐹𝑠) ∈ 𝐿)
126 simprr 771 . . . . . . . . . . 11 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ ((𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ∧ (𝐹𝑠) ⊆ 𝑡) ∧ 𝑡𝑋)) → 𝑡𝑋)
127 simprlr 778 . . . . . . . . . . 11 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ ((𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ∧ (𝐹𝑠) ⊆ 𝑡) ∧ 𝑡𝑋)) → (𝐹𝑠) ⊆ 𝑡)
128 filss 23204 . . . . . . . . . . 11 ((𝐿 ∈ (Fil‘𝑋) ∧ ((𝐹𝑠) ∈ 𝐿𝑡𝑋 ∧ (𝐹𝑠) ⊆ 𝑡)) → 𝑡𝐿)
12968, 125, 126, 127, 128syl13anc 1372 . . . . . . . . . 10 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ ((𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ∧ (𝐹𝑠) ⊆ 𝑡) ∧ 𝑡𝑋)) → 𝑡𝐿)
130129exp44 438 . . . . . . . . 9 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → (𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) → ((𝐹𝑠) ⊆ 𝑡 → (𝑡𝑋𝑡𝐿))))
131130rexlimdv 3150 . . . . . . . 8 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → (∃𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))(𝐹𝑠) ⊆ 𝑡 → (𝑡𝑋𝑡𝐿)))
132131impcomd 412 . . . . . . 7 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → ((𝑡𝑋 ∧ ∃𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))(𝐹𝑠) ⊆ 𝑡) → 𝑡𝐿))
13367, 132impbid 211 . . . . . 6 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → (𝑡𝐿 ↔ (𝑡𝑋 ∧ ∃𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))(𝐹𝑠) ⊆ 𝑡)))
1342adantr 481 . . . . . . 7 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → 𝑋𝐿)
135 rnelfmlem 23303 . . . . . . 7 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → ran (𝑥𝐿 ↦ (𝐹𝑥)) ∈ (fBas‘𝑌))
136 simpl3 1193 . . . . . . 7 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → 𝐹:𝑌𝑋)
137 elfm 23298 . . . . . . 7 ((𝑋𝐿 ∧ ran (𝑥𝐿 ↦ (𝐹𝑥)) ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝑡 ∈ ((𝑋 FilMap 𝐹)‘ran (𝑥𝐿 ↦ (𝐹𝑥))) ↔ (𝑡𝑋 ∧ ∃𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))(𝐹𝑠) ⊆ 𝑡)))
138134, 135, 136, 137syl3anc 1371 . . . . . 6 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → (𝑡 ∈ ((𝑋 FilMap 𝐹)‘ran (𝑥𝐿 ↦ (𝐹𝑥))) ↔ (𝑡𝑋 ∧ ∃𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))(𝐹𝑠) ⊆ 𝑡)))
139133, 138bitr4d 281 . . . . 5 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → (𝑡𝐿𝑡 ∈ ((𝑋 FilMap 𝐹)‘ran (𝑥𝐿 ↦ (𝐹𝑥)))))
140139eqrdv 2734 . . . 4 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → 𝐿 = ((𝑋 FilMap 𝐹)‘ran (𝑥𝐿 ↦ (𝐹𝑥))))
1417adantr 481 . . . . 5 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → (𝑋 FilMap 𝐹) Fn (fBas‘𝑌))
142 fnfvelrn 7031 . . . . 5 (((𝑋 FilMap 𝐹) Fn (fBas‘𝑌) ∧ ran (𝑥𝐿 ↦ (𝐹𝑥)) ∈ (fBas‘𝑌)) → ((𝑋 FilMap 𝐹)‘ran (𝑥𝐿 ↦ (𝐹𝑥))) ∈ ran (𝑋 FilMap 𝐹))
143141, 135, 142syl2anc 584 . . . 4 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → ((𝑋 FilMap 𝐹)‘ran (𝑥𝐿 ↦ (𝐹𝑥))) ∈ ran (𝑋 FilMap 𝐹))
144140, 143eqeltrd 2838 . . 3 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → 𝐿 ∈ ran (𝑋 FilMap 𝐹))
145144ex 413 . 2 ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → (ran 𝐹𝐿𝐿 ∈ ran (𝑋 FilMap 𝐹)))
14633, 145impbid 211 1 ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → (𝐿 ∈ ran (𝑋 FilMap 𝐹) ↔ ran 𝐹𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wrex 3073  Vcvv 3445  cin 3909  wss 3910  cmpt 5188  ccnv 5632  dom cdm 5633  ran crn 5634  cima 5636  Fun wfun 6490   Fn wfn 6491  wf 6492  ontowfo 6494  cfv 6496  (class class class)co 7357  fBascfbas 20784  filGencfg 20785  Filcfil 23196   FilMap cfm 23284
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-fbas 20793  df-fg 20794  df-fil 23197  df-fm 23289
This theorem is referenced by: (None)
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