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Theorem rnelfm 23977
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 23879 . . . . . . 7 (𝐿 ∈ (Fil‘𝑋) → 𝑋𝐿)
213ad2ant2 1133 . . . . . 6 ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → 𝑋𝐿)
3 simp1 1135 . . . . . 6 ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → 𝑌𝐴)
4 simp3 1137 . . . . . 6 ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → 𝐹:𝑌𝑋)
5 fmf 23969 . . . . . 6 ((𝑋𝐿𝑌𝐴𝐹:𝑌𝑋) → (𝑋 FilMap 𝐹):(fBas‘𝑌)⟶(Fil‘𝑋))
62, 3, 4, 5syl3anc 1370 . . . . 5 ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → (𝑋 FilMap 𝐹):(fBas‘𝑌)⟶(Fil‘𝑋))
76ffnd 6738 . . . 4 ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → (𝑋 FilMap 𝐹) Fn (fBas‘𝑌))
8 fvelrnb 6969 . . . 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 6737 . . . . . . . . . . . 12 (𝐹:𝑌𝑋𝐹 Fn 𝑌)
11 dffn4 6827 . . . . . . . . . . . 12 (𝐹 Fn 𝑌𝐹:𝑌onto→ran 𝐹)
1210, 11sylib 218 . . . . . . . . . . 11 (𝐹:𝑌𝑋𝐹:𝑌onto→ran 𝐹)
13 foima 6826 . . . . . . . . . . 11 (𝐹:𝑌onto→ran 𝐹 → (𝐹𝑌) = ran 𝐹)
1412, 13syl 17 . . . . . . . . . 10 (𝐹:𝑌𝑋 → (𝐹𝑌) = ran 𝐹)
1514ad2antlr 727 . . . . . . . . 9 (((𝑋𝐿𝐹:𝑌𝑋) ∧ 𝑏 ∈ (fBas‘𝑌)) → (𝐹𝑌) = ran 𝐹)
16 simpll 767 . . . . . . . . . 10 (((𝑋𝐿𝐹:𝑌𝑋) ∧ 𝑏 ∈ (fBas‘𝑌)) → 𝑋𝐿)
17 simpr 484 . . . . . . . . . 10 (((𝑋𝐿𝐹:𝑌𝑋) ∧ 𝑏 ∈ (fBas‘𝑌)) → 𝑏 ∈ (fBas‘𝑌))
18 simplr 769 . . . . . . . . . 10 (((𝑋𝐿𝐹:𝑌𝑋) ∧ 𝑏 ∈ (fBas‘𝑌)) → 𝐹:𝑌𝑋)
19 fgcl 23902 . . . . . . . . . . . 12 (𝑏 ∈ (fBas‘𝑌) → (𝑌filGen𝑏) ∈ (Fil‘𝑌))
20 filtop 23879 . . . . . . . . . . . 12 ((𝑌filGen𝑏) ∈ (Fil‘𝑌) → 𝑌 ∈ (𝑌filGen𝑏))
2119, 20syl 17 . . . . . . . . . . 11 (𝑏 ∈ (fBas‘𝑌) → 𝑌 ∈ (𝑌filGen𝑏))
2221adantl 481 . . . . . . . . . 10 (((𝑋𝐿𝐹:𝑌𝑋) ∧ 𝑏 ∈ (fBas‘𝑌)) → 𝑌 ∈ (𝑌filGen𝑏))
23 eqid 2735 . . . . . . . . . . 11 (𝑌filGen𝑏) = (𝑌filGen𝑏)
2423imaelfm 23975 . . . . . . . . . 10 (((𝑋𝐿𝑏 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑌 ∈ (𝑌filGen𝑏)) → (𝐹𝑌) ∈ ((𝑋 FilMap 𝐹)‘𝑏))
2516, 17, 18, 22, 24syl31anc 1372 . . . . . . . . 9 (((𝑋𝐿𝐹:𝑌𝑋) ∧ 𝑏 ∈ (fBas‘𝑌)) → (𝐹𝑌) ∈ ((𝑋 FilMap 𝐹)‘𝑏))
2615, 25eqeltrrd 2840 . . . . . . . 8 (((𝑋𝐿𝐹:𝑌𝑋) ∧ 𝑏 ∈ (fBas‘𝑌)) → ran 𝐹 ∈ ((𝑋 FilMap 𝐹)‘𝑏))
27 eleq2 2828 . . . . . . . 8 (((𝑋 FilMap 𝐹)‘𝑏) = 𝐿 → (ran 𝐹 ∈ ((𝑋 FilMap 𝐹)‘𝑏) ↔ ran 𝐹𝐿))
2826, 27syl5ibcom 245 . . . . . . 7 (((𝑋𝐿𝐹:𝑌𝑋) ∧ 𝑏 ∈ (fBas‘𝑌)) → (((𝑋 FilMap 𝐹)‘𝑏) = 𝐿 → ran 𝐹𝐿))
2928ex 412 . . . . . 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 1129 . . . 4 ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → (𝑏 ∈ (fBas‘𝑌) → (((𝑋 FilMap 𝐹)‘𝑏) = 𝐿 → ran 𝐹𝐿)))
3231rexlimdv 3151 . . 3 ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → (∃𝑏 ∈ (fBas‘𝑌)((𝑋 FilMap 𝐹)‘𝑏) = 𝐿 → ran 𝐹𝐿))
339, 32sylbid 240 . 2 ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → (𝐿 ∈ ran (𝑋 FilMap 𝐹) → ran 𝐹𝐿))
34 simpl2 1191 . . . . . . . . 9 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → 𝐿 ∈ (Fil‘𝑋))
35 filelss 23876 . . . . . . . . . 10 ((𝐿 ∈ (Fil‘𝑋) ∧ 𝑡𝐿) → 𝑡𝑋)
3635ex 412 . . . . . . . . 9 (𝐿 ∈ (Fil‘𝑋) → (𝑡𝐿𝑡𝑋))
3734, 36syl 17 . . . . . . . 8 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → (𝑡𝐿𝑡𝑋))
38 simpr 484 . . . . . . . . . . . 12 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑡𝐿) → 𝑡𝐿)
39 eqidd 2736 . . . . . . . . . . . 12 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑡𝐿) → (𝐹𝑡) = (𝐹𝑡))
40 imaeq2 6076 . . . . . . . . . . . . 13 (𝑥 = 𝑡 → (𝐹𝑥) = (𝐹𝑡))
4140rspceeqv 3645 . . . . . . . . . . . 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 1190 . . . . . . . . . . . . . 14 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → 𝑌𝐴)
44 cnvimass 6102 . . . . . . . . . . . . . . . . 17 (𝐹𝑡) ⊆ dom 𝐹
45 fdm 6746 . . . . . . . . . . . . . . . . 17 (𝐹:𝑌𝑋 → dom 𝐹 = 𝑌)
4644, 45sseqtrid 4048 . . . . . . . . . . . . . . . 16 (𝐹:𝑌𝑋 → (𝐹𝑡) ⊆ 𝑌)
47463ad2ant3 1134 . . . . . . . . . . . . . . 15 ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → (𝐹𝑡) ⊆ 𝑌)
4847adantr 480 . . . . . . . . . . . . . 14 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → (𝐹𝑡) ⊆ 𝑌)
4943, 48ssexd 5330 . . . . . . . . . . . . 13 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → (𝐹𝑡) ∈ V)
50 eqid 2735 . . . . . . . . . . . . . 14 (𝑥𝐿 ↦ (𝐹𝑥)) = (𝑥𝐿 ↦ (𝐹𝑥))
5150elrnmpt 5972 . . . . . . . . . . . . 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 480 . . . . . . . . . . 11 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑡𝐿) → ((𝐹𝑡) ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐿 (𝐹𝑡) = (𝐹𝑥)))
5442, 53mpbird 257 . . . . . . . . . 10 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑡𝐿) → (𝐹𝑡) ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)))
55 ssid 4018 . . . . . . . . . . 11 (𝐹𝑡) ⊆ (𝐹𝑡)
56 ffun 6740 . . . . . . . . . . . . . 14 (𝐹:𝑌𝑋 → Fun 𝐹)
57563ad2ant3 1134 . . . . . . . . . . . . 13 ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → Fun 𝐹)
5857ad2antrr 726 . . . . . . . . . . . 12 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑡𝐿) → Fun 𝐹)
59 funimass3 7074 . . . . . . . . . . . 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 258 . . . . . . . . . 10 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑡𝐿) → (𝐹 “ (𝐹𝑡)) ⊆ 𝑡)
62 imaeq2 6076 . . . . . . . . . . . 12 (𝑠 = (𝐹𝑡) → (𝐹𝑠) = (𝐹 “ (𝐹𝑡)))
6362sseq1d 4027 . . . . . . . . . . 11 (𝑠 = (𝐹𝑡) → ((𝐹𝑠) ⊆ 𝑡 ↔ (𝐹 “ (𝐹𝑡)) ⊆ 𝑡))
6463rspcev 3622 . . . . . . . . . 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 412 . . . . . . . 8 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → (𝑡𝐿 → ∃𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))(𝐹𝑠) ⊆ 𝑡))
6737, 66jcad 512 . . . . . . 7 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → (𝑡𝐿 → (𝑡𝑋 ∧ ∃𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))(𝐹𝑠) ⊆ 𝑡)))
6834adantr 480 . . . . . . . . . . 11 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ ((𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ∧ (𝐹𝑠) ⊆ 𝑡) ∧ 𝑡𝑋)) → 𝐿 ∈ (Fil‘𝑋))
6950elrnmpt 5972 . . . . . . . . . . . . . 14 (𝑠 ∈ V → (𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐿 𝑠 = (𝐹𝑥)))
7069elv 3483 . . . . . . . . . . . . 13 (𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐿 𝑠 = (𝐹𝑥))
71 ssid 4018 . . . . . . . . . . . . . . . . . . . 20 (𝐹𝑥) ⊆ (𝐹𝑥)
7257ad3antrrr 730 . . . . . . . . . . . . . . . . . . . . 21 (((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → Fun 𝐹)
73 cnvimass 6102 . . . . . . . . . . . . . . . . . . . . 21 (𝐹𝑥) ⊆ dom 𝐹
74 funimass3 7074 . . . . . . . . . . . . . . . . . . . . 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 258 . . . . . . . . . . . . . . . . . . 19 (((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → (𝐹 “ (𝐹𝑥)) ⊆ 𝑥)
77 imassrn 6091 . . . . . . . . . . . . . . . . . . 19 (𝐹 “ (𝐹𝑥)) ⊆ ran 𝐹
78 ssin 4247 . . . . . . . . . . . . . . . . . . 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 3979 . . . . . . . . . . . . . . . . . . . 20 (𝑧 ∈ (𝑥 ∩ ran 𝐹) ↔ (𝑧𝑥𝑧 ∈ ran 𝐹))
81 fvelrnb 6969 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐹 Fn 𝑌 → (𝑧 ∈ ran 𝐹 ↔ ∃𝑦𝑌 (𝐹𝑦) = 𝑧))
8210, 81syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐹:𝑌𝑋 → (𝑧 ∈ ran 𝐹 ↔ ∃𝑦𝑌 (𝐹𝑦) = 𝑧))
83823ad2ant3 1134 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → (𝑧 ∈ ran 𝐹 ↔ ∃𝑦𝑌 (𝐹𝑦) = 𝑧))
8483ad3antrrr 730 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → (𝑧 ∈ ran 𝐹 ↔ ∃𝑦𝑌 (𝐹𝑦) = 𝑧))
8572ad2antrr 726 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) ∧ 𝑦𝑌) ∧ (𝐹𝑦) ∈ 𝑥) → Fun 𝐹)
8685, 73jctir 520 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) ∧ 𝑦𝑌) ∧ (𝐹𝑦) ∈ 𝑥) → (Fun 𝐹 ∧ (𝐹𝑥) ⊆ dom 𝐹))
8757ad2antrr 726 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) → Fun 𝐹)
8887ad2antrr 726 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) ∧ 𝑦𝑌) → Fun 𝐹)
89453ad2ant3 1134 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → dom 𝐹 = 𝑌)
9089ad3antrrr 730 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → dom 𝐹 = 𝑌)
9190eleq2d 2825 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → (𝑦 ∈ dom 𝐹𝑦𝑌))
9291biimpar 477 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) ∧ 𝑦𝑌) → 𝑦 ∈ dom 𝐹)
93 fvimacnv 7073 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 476 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) ∧ 𝑦𝑌) ∧ (𝐹𝑦) ∈ 𝑥) → 𝑦 ∈ (𝐹𝑥))
96 funfvima2 7251 . . . . . . . . . . . . . . . . . . . . . . . . . 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 412 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) ∧ 𝑦𝑌) → ((𝐹𝑦) ∈ 𝑥 → (𝐹𝑦) ∈ (𝐹 “ (𝐹𝑥))))
99 eleq1 2827 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐹𝑦) = 𝑧 → ((𝐹𝑦) ∈ 𝑥𝑧𝑥))
100 eleq1 2827 . . . . . . . . . . . . . . . . . . . . . . . . 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 245 . . . . . . . . . . . . . . . . . . . . . . 23 ((((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) ∧ 𝑦𝑌) → ((𝐹𝑦) = 𝑧 → (𝑧𝑥𝑧 ∈ (𝐹 “ (𝐹𝑥)))))
103102rexlimdva 3153 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → (∃𝑦𝑌 (𝐹𝑦) = 𝑧 → (𝑧𝑥𝑧 ∈ (𝐹 “ (𝐹𝑥)))))
10484, 103sylbid 240 . . . . . . . . . . . . . . . . . . . . 21 (((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → (𝑧 ∈ ran 𝐹 → (𝑧𝑥𝑧 ∈ (𝐹 “ (𝐹𝑥)))))
105104impcomd 411 . . . . . . . . . . . . . . . . . . . 20 (((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → ((𝑧𝑥𝑧 ∈ ran 𝐹) → 𝑧 ∈ (𝐹 “ (𝐹𝑥))))
10680, 105biimtrid 242 . . . . . . . . . . . . . . . . . . 19 (((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → (𝑧 ∈ (𝑥 ∩ ran 𝐹) → 𝑧 ∈ (𝐹 “ (𝐹𝑥))))
107106ssrdv 4001 . . . . . . . . . . . . . . . . . 18 (((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → (𝑥 ∩ ran 𝐹) ⊆ (𝐹 “ (𝐹𝑥)))
10879, 107eqssd 4013 . . . . . . . . . . . . . . . . 17 (((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → (𝐹 “ (𝐹𝑥)) = (𝑥 ∩ ran 𝐹))
109 filin 23878 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐿 ∈ (Fil‘𝑋) ∧ 𝑥𝐿 ∧ ran 𝐹𝐿) → (𝑥 ∩ ran 𝐹) ∈ 𝐿)
1101093exp 1118 . . . . . . . . . . . . . . . . . . . . 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 1133 . . . . . . . . . . . . . . . . . . 19 ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → (ran 𝐹𝐿 → (𝑥𝐿 → (𝑥 ∩ ran 𝐹) ∈ 𝐿)))
113112imp31 417 . . . . . . . . . . . . . . . . . 18 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) → (𝑥 ∩ ran 𝐹) ∈ 𝐿)
114113adantr 480 . . . . . . . . . . . . . . . . 17 (((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → (𝑥 ∩ ran 𝐹) ∈ 𝐿)
115108, 114eqeltrd 2839 . . . . . . . . . . . . . . . 16 (((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → (𝐹 “ (𝐹𝑥)) ∈ 𝐿)
116115exp32 420 . . . . . . . . . . . . . . 15 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) → ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡 → (𝑡𝑋 → (𝐹 “ (𝐹𝑥)) ∈ 𝐿)))
117 imaeq2 6076 . . . . . . . . . . . . . . . . 17 (𝑠 = (𝐹𝑥) → (𝐹𝑠) = (𝐹 “ (𝐹𝑥)))
118117sseq1d 4027 . . . . . . . . . . . . . . . 16 (𝑠 = (𝐹𝑥) → ((𝐹𝑠) ⊆ 𝑡 ↔ (𝐹 “ (𝐹𝑥)) ⊆ 𝑡))
119117eleq1d 2824 . . . . . . . . . . . . . . . . 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 247 . . . . . . . . . . . . . 14 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) → (𝑠 = (𝐹𝑥) → ((𝐹𝑠) ⊆ 𝑡 → (𝑡𝑋 → (𝐹𝑠) ∈ 𝐿))))
123122rexlimdva 3153 . . . . . . . . . . . . 13 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → (∃𝑥𝐿 𝑠 = (𝐹𝑥) → ((𝐹𝑠) ⊆ 𝑡 → (𝑡𝑋 → (𝐹𝑠) ∈ 𝐿))))
12470, 123biimtrid 242 . . . . . . . . . . . 12 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → (𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) → ((𝐹𝑠) ⊆ 𝑡 → (𝑡𝑋 → (𝐹𝑠) ∈ 𝐿))))
125124imp44 428 . . . . . . . . . . 11 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ ((𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ∧ (𝐹𝑠) ⊆ 𝑡) ∧ 𝑡𝑋)) → (𝐹𝑠) ∈ 𝐿)
126 simprr 773 . . . . . . . . . . 11 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ ((𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ∧ (𝐹𝑠) ⊆ 𝑡) ∧ 𝑡𝑋)) → 𝑡𝑋)
127 simprlr 780 . . . . . . . . . . 11 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ ((𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ∧ (𝐹𝑠) ⊆ 𝑡) ∧ 𝑡𝑋)) → (𝐹𝑠) ⊆ 𝑡)
128 filss 23877 . . . . . . . . . . 11 ((𝐿 ∈ (Fil‘𝑋) ∧ ((𝐹𝑠) ∈ 𝐿𝑡𝑋 ∧ (𝐹𝑠) ⊆ 𝑡)) → 𝑡𝐿)
12968, 125, 126, 127, 128syl13anc 1371 . . . . . . . . . 10 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ ((𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ∧ (𝐹𝑠) ⊆ 𝑡) ∧ 𝑡𝑋)) → 𝑡𝐿)
130129exp44 437 . . . . . . . . 9 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → (𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) → ((𝐹𝑠) ⊆ 𝑡 → (𝑡𝑋𝑡𝐿))))
131130rexlimdv 3151 . . . . . . . 8 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → (∃𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))(𝐹𝑠) ⊆ 𝑡 → (𝑡𝑋𝑡𝐿)))
132131impcomd 411 . . . . . . 7 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → ((𝑡𝑋 ∧ ∃𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))(𝐹𝑠) ⊆ 𝑡) → 𝑡𝐿))
13367, 132impbid 212 . . . . . 6 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → (𝑡𝐿 ↔ (𝑡𝑋 ∧ ∃𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))(𝐹𝑠) ⊆ 𝑡)))
1342adantr 480 . . . . . . 7 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → 𝑋𝐿)
135 rnelfmlem 23976 . . . . . . 7 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → ran (𝑥𝐿 ↦ (𝐹𝑥)) ∈ (fBas‘𝑌))
136 simpl3 1192 . . . . . . 7 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → 𝐹:𝑌𝑋)
137 elfm 23971 . . . . . . 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 1370 . . . . . 6 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → (𝑡 ∈ ((𝑋 FilMap 𝐹)‘ran (𝑥𝐿 ↦ (𝐹𝑥))) ↔ (𝑡𝑋 ∧ ∃𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))(𝐹𝑠) ⊆ 𝑡)))
139133, 138bitr4d 282 . . . . 5 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → (𝑡𝐿𝑡 ∈ ((𝑋 FilMap 𝐹)‘ran (𝑥𝐿 ↦ (𝐹𝑥)))))
140139eqrdv 2733 . . . 4 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → 𝐿 = ((𝑋 FilMap 𝐹)‘ran (𝑥𝐿 ↦ (𝐹𝑥))))
1417adantr 480 . . . . 5 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → (𝑋 FilMap 𝐹) Fn (fBas‘𝑌))
142 fnfvelrn 7100 . . . . 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 2839 . . 3 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → 𝐿 ∈ ran (𝑋 FilMap 𝐹))
145144ex 412 . 2 ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → (ran 𝐹𝐿𝐿 ∈ ran (𝑋 FilMap 𝐹)))
14633, 145impbid 212 1 ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → (𝐿 ∈ ran (𝑋 FilMap 𝐹) ↔ ran 𝐹𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wrex 3068  Vcvv 3478  cin 3962  wss 3963  cmpt 5231  ccnv 5688  dom cdm 5689  ran crn 5690  cima 5692  Fun wfun 6557   Fn wfn 6558  wf 6559  ontowfo 6561  cfv 6563  (class class class)co 7431  fBascfbas 21370  filGencfg 21371  Filcfil 23869   FilMap cfm 23957
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-fbas 21379  df-fg 21380  df-fil 23870  df-fm 23962
This theorem is referenced by: (None)
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