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Theorem rnelfm 23962
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 23864 . . . . . . 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 23954 . . . . . 6 ((𝑋𝐿𝑌𝐴𝐹:𝑌𝑋) → (𝑋 FilMap 𝐹):(fBas‘𝑌)⟶(Fil‘𝑋))
62, 3, 4, 5syl3anc 1372 . . . . 5 ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → (𝑋 FilMap 𝐹):(fBas‘𝑌)⟶(Fil‘𝑋))
76ffnd 6736 . . . 4 ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → (𝑋 FilMap 𝐹) Fn (fBas‘𝑌))
8 fvelrnb 6968 . . . 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 6735 . . . . . . . . . . . 12 (𝐹:𝑌𝑋𝐹 Fn 𝑌)
11 dffn4 6825 . . . . . . . . . . . 12 (𝐹 Fn 𝑌𝐹:𝑌onto→ran 𝐹)
1210, 11sylib 218 . . . . . . . . . . 11 (𝐹:𝑌𝑋𝐹:𝑌onto→ran 𝐹)
13 foima 6824 . . . . . . . . . . 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 766 . . . . . . . . . 10 (((𝑋𝐿𝐹:𝑌𝑋) ∧ 𝑏 ∈ (fBas‘𝑌)) → 𝑋𝐿)
17 simpr 484 . . . . . . . . . 10 (((𝑋𝐿𝐹:𝑌𝑋) ∧ 𝑏 ∈ (fBas‘𝑌)) → 𝑏 ∈ (fBas‘𝑌))
18 simplr 768 . . . . . . . . . 10 (((𝑋𝐿𝐹:𝑌𝑋) ∧ 𝑏 ∈ (fBas‘𝑌)) → 𝐹:𝑌𝑋)
19 fgcl 23887 . . . . . . . . . . . 12 (𝑏 ∈ (fBas‘𝑌) → (𝑌filGen𝑏) ∈ (Fil‘𝑌))
20 filtop 23864 . . . . . . . . . . . 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 2736 . . . . . . . . . . 11 (𝑌filGen𝑏) = (𝑌filGen𝑏)
2423imaelfm 23960 . . . . . . . . . 10 (((𝑋𝐿𝑏 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑌 ∈ (𝑌filGen𝑏)) → (𝐹𝑌) ∈ ((𝑋 FilMap 𝐹)‘𝑏))
2516, 17, 18, 22, 24syl31anc 1374 . . . . . . . . 9 (((𝑋𝐿𝐹:𝑌𝑋) ∧ 𝑏 ∈ (fBas‘𝑌)) → (𝐹𝑌) ∈ ((𝑋 FilMap 𝐹)‘𝑏))
2615, 25eqeltrrd 2841 . . . . . . . 8 (((𝑋𝐿𝐹:𝑌𝑋) ∧ 𝑏 ∈ (fBas‘𝑌)) → ran 𝐹 ∈ ((𝑋 FilMap 𝐹)‘𝑏))
27 eleq2 2829 . . . . . . . 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 1130 . . . 4 ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → (𝑏 ∈ (fBas‘𝑌) → (((𝑋 FilMap 𝐹)‘𝑏) = 𝐿 → ran 𝐹𝐿)))
3231rexlimdv 3152 . . 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 1192 . . . . . . . . 9 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → 𝐿 ∈ (Fil‘𝑋))
35 filelss 23861 . . . . . . . . . 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 2737 . . . . . . . . . . . 12 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑡𝐿) → (𝐹𝑡) = (𝐹𝑡))
40 imaeq2 6073 . . . . . . . . . . . . 13 (𝑥 = 𝑡 → (𝐹𝑥) = (𝐹𝑡))
4140rspceeqv 3644 . . . . . . . . . . . 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 6099 . . . . . . . . . . . . . . . . 17 (𝐹𝑡) ⊆ dom 𝐹
45 fdm 6744 . . . . . . . . . . . . . . . . 17 (𝐹:𝑌𝑋 → dom 𝐹 = 𝑌)
4644, 45sseqtrid 4025 . . . . . . . . . . . . . . . 16 (𝐹:𝑌𝑋 → (𝐹𝑡) ⊆ 𝑌)
47463ad2ant3 1135 . . . . . . . . . . . . . . 15 ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → (𝐹𝑡) ⊆ 𝑌)
4847adantr 480 . . . . . . . . . . . . . 14 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → (𝐹𝑡) ⊆ 𝑌)
4943, 48ssexd 5323 . . . . . . . . . . . . 13 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → (𝐹𝑡) ∈ V)
50 eqid 2736 . . . . . . . . . . . . . 14 (𝑥𝐿 ↦ (𝐹𝑥)) = (𝑥𝐿 ↦ (𝐹𝑥))
5150elrnmpt 5968 . . . . . . . . . . . . 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 4005 . . . . . . . . . . 11 (𝐹𝑡) ⊆ (𝐹𝑡)
56 ffun 6738 . . . . . . . . . . . . . 14 (𝐹:𝑌𝑋 → Fun 𝐹)
57563ad2ant3 1135 . . . . . . . . . . . . 13 ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → Fun 𝐹)
5857ad2antrr 726 . . . . . . . . . . . 12 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑡𝐿) → Fun 𝐹)
59 funimass3 7073 . . . . . . . . . . . 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 6073 . . . . . . . . . . . 12 (𝑠 = (𝐹𝑡) → (𝐹𝑠) = (𝐹 “ (𝐹𝑡)))
6362sseq1d 4014 . . . . . . . . . . 11 (𝑠 = (𝐹𝑡) → ((𝐹𝑠) ⊆ 𝑡 ↔ (𝐹 “ (𝐹𝑡)) ⊆ 𝑡))
6463rspcev 3621 . . . . . . . . . 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 5968 . . . . . . . . . . . . . 14 (𝑠 ∈ V → (𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐿 𝑠 = (𝐹𝑥)))
7069elv 3484 . . . . . . . . . . . . 13 (𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐿 𝑠 = (𝐹𝑥))
71 ssid 4005 . . . . . . . . . . . . . . . . . . . 20 (𝐹𝑥) ⊆ (𝐹𝑥)
7257ad3antrrr 730 . . . . . . . . . . . . . . . . . . . . 21 (((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → Fun 𝐹)
73 cnvimass 6099 . . . . . . . . . . . . . . . . . . . . 21 (𝐹𝑥) ⊆ dom 𝐹
74 funimass3 7073 . . . . . . . . . . . . . . . . . . . . 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 6088 . . . . . . . . . . . . . . . . . . 19 (𝐹 “ (𝐹𝑥)) ⊆ ran 𝐹
78 ssin 4238 . . . . . . . . . . . . . . . . . . 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 3966 . . . . . . . . . . . . . . . . . . . 20 (𝑧 ∈ (𝑥 ∩ ran 𝐹) ↔ (𝑧𝑥𝑧 ∈ ran 𝐹))
81 fvelrnb 6968 . . . . . . . . . . . . . . . . . . . . . . . . 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 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 1135 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2826 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7072 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7252 . . . . . . . . . . . . . . . . . . . . . . . . . 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 2828 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐹𝑦) = 𝑧 → ((𝐹𝑦) ∈ 𝑥𝑧𝑥))
100 eleq1 2828 . . . . . . . . . . . . . . . . . . . . . . . . 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 3154 . . . . . . . . . . . . . . . . . . . . . 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 3988 . . . . . . . . . . . . . . . . . 18 (((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → (𝑥 ∩ ran 𝐹) ⊆ (𝐹 “ (𝐹𝑥)))
10879, 107eqssd 4000 . . . . . . . . . . . . . . . . 17 (((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → (𝐹 “ (𝐹𝑥)) = (𝑥 ∩ ran 𝐹))
109 filin 23863 . . . . . . . . . . . . . . . . . . . . . 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 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 2840 . . . . . . . . . . . . . . . 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 6073 . . . . . . . . . . . . . . . . 17 (𝑠 = (𝐹𝑥) → (𝐹𝑠) = (𝐹 “ (𝐹𝑥)))
118117sseq1d 4014 . . . . . . . . . . . . . . . 16 (𝑠 = (𝐹𝑥) → ((𝐹𝑠) ⊆ 𝑡 ↔ (𝐹 “ (𝐹𝑥)) ⊆ 𝑡))
119117eleq1d 2825 . . . . . . . . . . . . . . . . 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 3154 . . . . . . . . . . . . 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 772 . . . . . . . . . . 11 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ ((𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ∧ (𝐹𝑠) ⊆ 𝑡) ∧ 𝑡𝑋)) → 𝑡𝑋)
127 simprlr 779 . . . . . . . . . . 11 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ ((𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ∧ (𝐹𝑠) ⊆ 𝑡) ∧ 𝑡𝑋)) → (𝐹𝑠) ⊆ 𝑡)
128 filss 23862 . . . . . . . . . . 11 ((𝐿 ∈ (Fil‘𝑋) ∧ ((𝐹𝑠) ∈ 𝐿𝑡𝑋 ∧ (𝐹𝑠) ⊆ 𝑡)) → 𝑡𝐿)
12968, 125, 126, 127, 128syl13anc 1373 . . . . . . . . . 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 3152 . . . . . . . 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 23961 . . . . . . 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 23956 . . . . . . 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 1372 . . . . . 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 2734 . . . 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 7099 . . . . 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 2840 . . 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 1539  wcel 2107  wrex 3069  Vcvv 3479  cin 3949  wss 3950  cmpt 5224  ccnv 5683  dom cdm 5684  ran crn 5685  cima 5687  Fun wfun 6554   Fn wfn 6555  wf 6556  ontowfo 6558  cfv 6560  (class class class)co 7432  fBascfbas 21353  filGencfg 21354  Filcfil 23854   FilMap cfm 23942
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-fbas 21362  df-fg 21363  df-fil 23855  df-fm 23947
This theorem is referenced by: (None)
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