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Theorem rnelfm 22277
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 22179 . . . . . . 7 (𝐿 ∈ (Fil‘𝑋) → 𝑋𝐿)
213ad2ant2 1114 . . . . . 6 ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → 𝑋𝐿)
3 simp1 1116 . . . . . 6 ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → 𝑌𝐴)
4 simp3 1118 . . . . . 6 ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → 𝐹:𝑌𝑋)
5 fmf 22269 . . . . . 6 ((𝑋𝐿𝑌𝐴𝐹:𝑌𝑋) → (𝑋 FilMap 𝐹):(fBas‘𝑌)⟶(Fil‘𝑋))
62, 3, 4, 5syl3anc 1351 . . . . 5 ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → (𝑋 FilMap 𝐹):(fBas‘𝑌)⟶(Fil‘𝑋))
76ffnd 6342 . . . 4 ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → (𝑋 FilMap 𝐹) Fn (fBas‘𝑌))
8 fvelrnb 6553 . . . 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 6341 . . . . . . . . . . . 12 (𝐹:𝑌𝑋𝐹 Fn 𝑌)
11 dffn4 6422 . . . . . . . . . . . 12 (𝐹 Fn 𝑌𝐹:𝑌onto→ran 𝐹)
1210, 11sylib 210 . . . . . . . . . . 11 (𝐹:𝑌𝑋𝐹:𝑌onto→ran 𝐹)
13 foima 6421 . . . . . . . . . . 11 (𝐹:𝑌onto→ran 𝐹 → (𝐹𝑌) = ran 𝐹)
1412, 13syl 17 . . . . . . . . . 10 (𝐹:𝑌𝑋 → (𝐹𝑌) = ran 𝐹)
1514ad2antlr 714 . . . . . . . . 9 (((𝑋𝐿𝐹:𝑌𝑋) ∧ 𝑏 ∈ (fBas‘𝑌)) → (𝐹𝑌) = ran 𝐹)
16 simpll 754 . . . . . . . . . 10 (((𝑋𝐿𝐹:𝑌𝑋) ∧ 𝑏 ∈ (fBas‘𝑌)) → 𝑋𝐿)
17 simpr 477 . . . . . . . . . 10 (((𝑋𝐿𝐹:𝑌𝑋) ∧ 𝑏 ∈ (fBas‘𝑌)) → 𝑏 ∈ (fBas‘𝑌))
18 simplr 756 . . . . . . . . . 10 (((𝑋𝐿𝐹:𝑌𝑋) ∧ 𝑏 ∈ (fBas‘𝑌)) → 𝐹:𝑌𝑋)
19 fgcl 22202 . . . . . . . . . . . 12 (𝑏 ∈ (fBas‘𝑌) → (𝑌filGen𝑏) ∈ (Fil‘𝑌))
20 filtop 22179 . . . . . . . . . . . 12 ((𝑌filGen𝑏) ∈ (Fil‘𝑌) → 𝑌 ∈ (𝑌filGen𝑏))
2119, 20syl 17 . . . . . . . . . . 11 (𝑏 ∈ (fBas‘𝑌) → 𝑌 ∈ (𝑌filGen𝑏))
2221adantl 474 . . . . . . . . . 10 (((𝑋𝐿𝐹:𝑌𝑋) ∧ 𝑏 ∈ (fBas‘𝑌)) → 𝑌 ∈ (𝑌filGen𝑏))
23 eqid 2772 . . . . . . . . . . 11 (𝑌filGen𝑏) = (𝑌filGen𝑏)
2423imaelfm 22275 . . . . . . . . . 10 (((𝑋𝐿𝑏 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑌 ∈ (𝑌filGen𝑏)) → (𝐹𝑌) ∈ ((𝑋 FilMap 𝐹)‘𝑏))
2516, 17, 18, 22, 24syl31anc 1353 . . . . . . . . 9 (((𝑋𝐿𝐹:𝑌𝑋) ∧ 𝑏 ∈ (fBas‘𝑌)) → (𝐹𝑌) ∈ ((𝑋 FilMap 𝐹)‘𝑏))
2615, 25eqeltrrd 2861 . . . . . . . 8 (((𝑋𝐿𝐹:𝑌𝑋) ∧ 𝑏 ∈ (fBas‘𝑌)) → ran 𝐹 ∈ ((𝑋 FilMap 𝐹)‘𝑏))
27 eleq2 2848 . . . . . . . 8 (((𝑋 FilMap 𝐹)‘𝑏) = 𝐿 → (ran 𝐹 ∈ ((𝑋 FilMap 𝐹)‘𝑏) ↔ ran 𝐹𝐿))
2826, 27syl5ibcom 237 . . . . . . 7 (((𝑋𝐿𝐹:𝑌𝑋) ∧ 𝑏 ∈ (fBas‘𝑌)) → (((𝑋 FilMap 𝐹)‘𝑏) = 𝐿 → ran 𝐹𝐿))
2928ex 405 . . . . . 6 ((𝑋𝐿𝐹:𝑌𝑋) → (𝑏 ∈ (fBas‘𝑌) → (((𝑋 FilMap 𝐹)‘𝑏) = 𝐿 → ran 𝐹𝐿)))
301, 29sylan 572 . . . . 5 ((𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → (𝑏 ∈ (fBas‘𝑌) → (((𝑋 FilMap 𝐹)‘𝑏) = 𝐿 → ran 𝐹𝐿)))
31303adant1 1110 . . . 4 ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → (𝑏 ∈ (fBas‘𝑌) → (((𝑋 FilMap 𝐹)‘𝑏) = 𝐿 → ran 𝐹𝐿)))
3231rexlimdv 3222 . . 3 ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → (∃𝑏 ∈ (fBas‘𝑌)((𝑋 FilMap 𝐹)‘𝑏) = 𝐿 → ran 𝐹𝐿))
339, 32sylbid 232 . 2 ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → (𝐿 ∈ ran (𝑋 FilMap 𝐹) → ran 𝐹𝐿))
34 simpl2 1172 . . . . . . . . 9 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → 𝐿 ∈ (Fil‘𝑋))
35 filelss 22176 . . . . . . . . . 10 ((𝐿 ∈ (Fil‘𝑋) ∧ 𝑡𝐿) → 𝑡𝑋)
3635ex 405 . . . . . . . . 9 (𝐿 ∈ (Fil‘𝑋) → (𝑡𝐿𝑡𝑋))
3734, 36syl 17 . . . . . . . 8 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → (𝑡𝐿𝑡𝑋))
38 simpr 477 . . . . . . . . . . . 12 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑡𝐿) → 𝑡𝐿)
39 eqidd 2773 . . . . . . . . . . . 12 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑡𝐿) → (𝐹𝑡) = (𝐹𝑡))
40 imaeq2 5763 . . . . . . . . . . . . 13 (𝑥 = 𝑡 → (𝐹𝑥) = (𝐹𝑡))
4140rspceeqv 3547 . . . . . . . . . . . 12 ((𝑡𝐿 ∧ (𝐹𝑡) = (𝐹𝑡)) → ∃𝑥𝐿 (𝐹𝑡) = (𝐹𝑥))
4238, 39, 41syl2anc 576 . . . . . . . . . . 11 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑡𝐿) → ∃𝑥𝐿 (𝐹𝑡) = (𝐹𝑥))
43 simpl1 1171 . . . . . . . . . . . . . 14 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → 𝑌𝐴)
44 cnvimass 5786 . . . . . . . . . . . . . . . . 17 (𝐹𝑡) ⊆ dom 𝐹
45 fdm 6349 . . . . . . . . . . . . . . . . 17 (𝐹:𝑌𝑋 → dom 𝐹 = 𝑌)
4644, 45syl5sseq 3903 . . . . . . . . . . . . . . . 16 (𝐹:𝑌𝑋 → (𝐹𝑡) ⊆ 𝑌)
47463ad2ant3 1115 . . . . . . . . . . . . . . 15 ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → (𝐹𝑡) ⊆ 𝑌)
4847adantr 473 . . . . . . . . . . . . . 14 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → (𝐹𝑡) ⊆ 𝑌)
4943, 48ssexd 5080 . . . . . . . . . . . . 13 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → (𝐹𝑡) ∈ V)
50 eqid 2772 . . . . . . . . . . . . . 14 (𝑥𝐿 ↦ (𝐹𝑥)) = (𝑥𝐿 ↦ (𝐹𝑥))
5150elrnmpt 5668 . . . . . . . . . . . . 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 473 . . . . . . . . . . 11 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑡𝐿) → ((𝐹𝑡) ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐿 (𝐹𝑡) = (𝐹𝑥)))
5442, 53mpbird 249 . . . . . . . . . 10 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑡𝐿) → (𝐹𝑡) ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)))
55 ssid 3873 . . . . . . . . . . 11 (𝐹𝑡) ⊆ (𝐹𝑡)
56 ffun 6344 . . . . . . . . . . . . . 14 (𝐹:𝑌𝑋 → Fun 𝐹)
57563ad2ant3 1115 . . . . . . . . . . . . 13 ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → Fun 𝐹)
5857ad2antrr 713 . . . . . . . . . . . 12 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑡𝐿) → Fun 𝐹)
59 funimass3 6647 . . . . . . . . . . . 12 ((Fun 𝐹 ∧ (𝐹𝑡) ⊆ dom 𝐹) → ((𝐹 “ (𝐹𝑡)) ⊆ 𝑡 ↔ (𝐹𝑡) ⊆ (𝐹𝑡)))
6058, 44, 59sylancl 577 . . . . . . . . . . 11 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑡𝐿) → ((𝐹 “ (𝐹𝑡)) ⊆ 𝑡 ↔ (𝐹𝑡) ⊆ (𝐹𝑡)))
6155, 60mpbiri 250 . . . . . . . . . 10 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑡𝐿) → (𝐹 “ (𝐹𝑡)) ⊆ 𝑡)
62 imaeq2 5763 . . . . . . . . . . . 12 (𝑠 = (𝐹𝑡) → (𝐹𝑠) = (𝐹 “ (𝐹𝑡)))
6362sseq1d 3882 . . . . . . . . . . 11 (𝑠 = (𝐹𝑡) → ((𝐹𝑠) ⊆ 𝑡 ↔ (𝐹 “ (𝐹𝑡)) ⊆ 𝑡))
6463rspcev 3529 . . . . . . . . . 10 (((𝐹𝑡) ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ∧ (𝐹 “ (𝐹𝑡)) ⊆ 𝑡) → ∃𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))(𝐹𝑠) ⊆ 𝑡)
6554, 61, 64syl2anc 576 . . . . . . . . 9 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑡𝐿) → ∃𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))(𝐹𝑠) ⊆ 𝑡)
6665ex 405 . . . . . . . 8 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → (𝑡𝐿 → ∃𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))(𝐹𝑠) ⊆ 𝑡))
6737, 66jcad 505 . . . . . . 7 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → (𝑡𝐿 → (𝑡𝑋 ∧ ∃𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))(𝐹𝑠) ⊆ 𝑡)))
6834adantr 473 . . . . . . . . . . 11 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ ((𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ∧ (𝐹𝑠) ⊆ 𝑡) ∧ 𝑡𝑋)) → 𝐿 ∈ (Fil‘𝑋))
6950elrnmpt 5668 . . . . . . . . . . . . . 14 (𝑠 ∈ V → (𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐿 𝑠 = (𝐹𝑥)))
7069elv 3414 . . . . . . . . . . . . 13 (𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐿 𝑠 = (𝐹𝑥))
71 ssid 3873 . . . . . . . . . . . . . . . . . . . 20 (𝐹𝑥) ⊆ (𝐹𝑥)
7257ad3antrrr 717 . . . . . . . . . . . . . . . . . . . . 21 (((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → Fun 𝐹)
73 cnvimass 5786 . . . . . . . . . . . . . . . . . . . . 21 (𝐹𝑥) ⊆ dom 𝐹
74 funimass3 6647 . . . . . . . . . . . . . . . . . . . . 21 ((Fun 𝐹 ∧ (𝐹𝑥) ⊆ dom 𝐹) → ((𝐹 “ (𝐹𝑥)) ⊆ 𝑥 ↔ (𝐹𝑥) ⊆ (𝐹𝑥)))
7572, 73, 74sylancl 577 . . . . . . . . . . . . . . . . . . . 20 (((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → ((𝐹 “ (𝐹𝑥)) ⊆ 𝑥 ↔ (𝐹𝑥) ⊆ (𝐹𝑥)))
7671, 75mpbiri 250 . . . . . . . . . . . . . . . . . . 19 (((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → (𝐹 “ (𝐹𝑥)) ⊆ 𝑥)
77 imassrn 5778 . . . . . . . . . . . . . . . . . . 19 (𝐹 “ (𝐹𝑥)) ⊆ ran 𝐹
78 ssin 4088 . . . . . . . . . . . . . . . . . . 19 (((𝐹 “ (𝐹𝑥)) ⊆ 𝑥 ∧ (𝐹 “ (𝐹𝑥)) ⊆ ran 𝐹) ↔ (𝐹 “ (𝐹𝑥)) ⊆ (𝑥 ∩ ran 𝐹))
7976, 77, 78sylanblc 580 . . . . . . . . . . . . . . . . . 18 (((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → (𝐹 “ (𝐹𝑥)) ⊆ (𝑥 ∩ ran 𝐹))
80 elin 4051 . . . . . . . . . . . . . . . . . . . 20 (𝑧 ∈ (𝑥 ∩ ran 𝐹) ↔ (𝑧𝑥𝑧 ∈ ran 𝐹))
81 fvelrnb 6553 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐹 Fn 𝑌 → (𝑧 ∈ ran 𝐹 ↔ ∃𝑦𝑌 (𝐹𝑦) = 𝑧))
8210, 81syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐹:𝑌𝑋 → (𝑧 ∈ ran 𝐹 ↔ ∃𝑦𝑌 (𝐹𝑦) = 𝑧))
83823ad2ant3 1115 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → (𝑧 ∈ ran 𝐹 ↔ ∃𝑦𝑌 (𝐹𝑦) = 𝑧))
8483ad3antrrr 717 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → (𝑧 ∈ ran 𝐹 ↔ ∃𝑦𝑌 (𝐹𝑦) = 𝑧))
8572ad2antrr 713 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) ∧ 𝑦𝑌) ∧ (𝐹𝑦) ∈ 𝑥) → Fun 𝐹)
8685, 73jctir 513 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) ∧ 𝑦𝑌) ∧ (𝐹𝑦) ∈ 𝑥) → (Fun 𝐹 ∧ (𝐹𝑥) ⊆ dom 𝐹))
8757ad2antrr 713 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) → Fun 𝐹)
8887ad2antrr 713 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) ∧ 𝑦𝑌) → Fun 𝐹)
89453ad2ant3 1115 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → dom 𝐹 = 𝑌)
9089ad3antrrr 717 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → dom 𝐹 = 𝑌)
9190eleq2d 2845 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → (𝑦 ∈ dom 𝐹𝑦𝑌))
9291biimpar 470 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) ∧ 𝑦𝑌) → 𝑦 ∈ dom 𝐹)
93 fvimacnv 6646 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((Fun 𝐹𝑦 ∈ dom 𝐹) → ((𝐹𝑦) ∈ 𝑥𝑦 ∈ (𝐹𝑥)))
9488, 92, 93syl2anc 576 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) ∧ 𝑦𝑌) → ((𝐹𝑦) ∈ 𝑥𝑦 ∈ (𝐹𝑥)))
9594biimpa 469 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) ∧ 𝑦𝑌) ∧ (𝐹𝑦) ∈ 𝑥) → 𝑦 ∈ (𝐹𝑥))
96 funfvima2 6817 . . . . . . . . . . . . . . . . . . . . . . . . . 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 405 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) ∧ 𝑦𝑌) → ((𝐹𝑦) ∈ 𝑥 → (𝐹𝑦) ∈ (𝐹 “ (𝐹𝑥))))
99 eleq1 2847 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐹𝑦) = 𝑧 → ((𝐹𝑦) ∈ 𝑥𝑧𝑥))
100 eleq1 2847 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐹𝑦) = 𝑧 → ((𝐹𝑦) ∈ (𝐹 “ (𝐹𝑥)) ↔ 𝑧 ∈ (𝐹 “ (𝐹𝑥))))
10199, 100imbi12d 337 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐹𝑦) = 𝑧 → (((𝐹𝑦) ∈ 𝑥 → (𝐹𝑦) ∈ (𝐹 “ (𝐹𝑥))) ↔ (𝑧𝑥𝑧 ∈ (𝐹 “ (𝐹𝑥)))))
10298, 101syl5ibcom 237 . . . . . . . . . . . . . . . . . . . . . . 23 ((((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) ∧ 𝑦𝑌) → ((𝐹𝑦) = 𝑧 → (𝑧𝑥𝑧 ∈ (𝐹 “ (𝐹𝑥)))))
103102rexlimdva 3223 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → (∃𝑦𝑌 (𝐹𝑦) = 𝑧 → (𝑧𝑥𝑧 ∈ (𝐹 “ (𝐹𝑥)))))
10484, 103sylbid 232 . . . . . . . . . . . . . . . . . . . . 21 (((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → (𝑧 ∈ ran 𝐹 → (𝑧𝑥𝑧 ∈ (𝐹 “ (𝐹𝑥)))))
105104impcomd 403 . . . . . . . . . . . . . . . . . . . 20 (((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → ((𝑧𝑥𝑧 ∈ ran 𝐹) → 𝑧 ∈ (𝐹 “ (𝐹𝑥))))
10680, 105syl5bi 234 . . . . . . . . . . . . . . . . . . 19 (((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → (𝑧 ∈ (𝑥 ∩ ran 𝐹) → 𝑧 ∈ (𝐹 “ (𝐹𝑥))))
107106ssrdv 3858 . . . . . . . . . . . . . . . . . 18 (((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → (𝑥 ∩ ran 𝐹) ⊆ (𝐹 “ (𝐹𝑥)))
10879, 107eqssd 3869 . . . . . . . . . . . . . . . . 17 (((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → (𝐹 “ (𝐹𝑥)) = (𝑥 ∩ ran 𝐹))
109 filin 22178 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐿 ∈ (Fil‘𝑋) ∧ 𝑥𝐿 ∧ ran 𝐹𝐿) → (𝑥 ∩ ran 𝐹) ∈ 𝐿)
1101093exp 1099 . . . . . . . . . . . . . . . . . . . . 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 1114 . . . . . . . . . . . . . . . . . . 19 ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → (ran 𝐹𝐿 → (𝑥𝐿 → (𝑥 ∩ ran 𝐹) ∈ 𝐿)))
113112imp31 410 . . . . . . . . . . . . . . . . . 18 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) → (𝑥 ∩ ran 𝐹) ∈ 𝐿)
114113adantr 473 . . . . . . . . . . . . . . . . 17 (((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → (𝑥 ∩ ran 𝐹) ∈ 𝐿)
115108, 114eqeltrd 2860 . . . . . . . . . . . . . . . 16 (((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → (𝐹 “ (𝐹𝑥)) ∈ 𝐿)
116115exp32 413 . . . . . . . . . . . . . . 15 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) → ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡 → (𝑡𝑋 → (𝐹 “ (𝐹𝑥)) ∈ 𝐿)))
117 imaeq2 5763 . . . . . . . . . . . . . . . . 17 (𝑠 = (𝐹𝑥) → (𝐹𝑠) = (𝐹 “ (𝐹𝑥)))
118117sseq1d 3882 . . . . . . . . . . . . . . . 16 (𝑠 = (𝐹𝑥) → ((𝐹𝑠) ⊆ 𝑡 ↔ (𝐹 “ (𝐹𝑥)) ⊆ 𝑡))
119117eleq1d 2844 . . . . . . . . . . . . . . . . 17 (𝑠 = (𝐹𝑥) → ((𝐹𝑠) ∈ 𝐿 ↔ (𝐹 “ (𝐹𝑥)) ∈ 𝐿))
120119imbi2d 333 . . . . . . . . . . . . . . . 16 (𝑠 = (𝐹𝑥) → ((𝑡𝑋 → (𝐹𝑠) ∈ 𝐿) ↔ (𝑡𝑋 → (𝐹 “ (𝐹𝑥)) ∈ 𝐿)))
121118, 120imbi12d 337 . . . . . . . . . . . . . . 15 (𝑠 = (𝐹𝑥) → (((𝐹𝑠) ⊆ 𝑡 → (𝑡𝑋 → (𝐹𝑠) ∈ 𝐿)) ↔ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡 → (𝑡𝑋 → (𝐹 “ (𝐹𝑥)) ∈ 𝐿))))
122116, 121syl5ibrcom 239 . . . . . . . . . . . . . 14 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ 𝑥𝐿) → (𝑠 = (𝐹𝑥) → ((𝐹𝑠) ⊆ 𝑡 → (𝑡𝑋 → (𝐹𝑠) ∈ 𝐿))))
123122rexlimdva 3223 . . . . . . . . . . . . 13 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → (∃𝑥𝐿 𝑠 = (𝐹𝑥) → ((𝐹𝑠) ⊆ 𝑡 → (𝑡𝑋 → (𝐹𝑠) ∈ 𝐿))))
12470, 123syl5bi 234 . . . . . . . . . . . 12 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → (𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) → ((𝐹𝑠) ⊆ 𝑡 → (𝑡𝑋 → (𝐹𝑠) ∈ 𝐿))))
125124imp44 421 . . . . . . . . . . 11 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ ((𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ∧ (𝐹𝑠) ⊆ 𝑡) ∧ 𝑡𝑋)) → (𝐹𝑠) ∈ 𝐿)
126 simprr 760 . . . . . . . . . . 11 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ ((𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ∧ (𝐹𝑠) ⊆ 𝑡) ∧ 𝑡𝑋)) → 𝑡𝑋)
127 simprlr 767 . . . . . . . . . . 11 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ ((𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ∧ (𝐹𝑠) ⊆ 𝑡) ∧ 𝑡𝑋)) → (𝐹𝑠) ⊆ 𝑡)
128 filss 22177 . . . . . . . . . . 11 ((𝐿 ∈ (Fil‘𝑋) ∧ ((𝐹𝑠) ∈ 𝐿𝑡𝑋 ∧ (𝐹𝑠) ⊆ 𝑡)) → 𝑡𝐿)
12968, 125, 126, 127, 128syl13anc 1352 . . . . . . . . . 10 ((((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) ∧ ((𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ∧ (𝐹𝑠) ⊆ 𝑡) ∧ 𝑡𝑋)) → 𝑡𝐿)
130129exp44 430 . . . . . . . . 9 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → (𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) → ((𝐹𝑠) ⊆ 𝑡 → (𝑡𝑋𝑡𝐿))))
131130rexlimdv 3222 . . . . . . . 8 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → (∃𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))(𝐹𝑠) ⊆ 𝑡 → (𝑡𝑋𝑡𝐿)))
132131impcomd 403 . . . . . . 7 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → ((𝑡𝑋 ∧ ∃𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))(𝐹𝑠) ⊆ 𝑡) → 𝑡𝐿))
13367, 132impbid 204 . . . . . 6 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → (𝑡𝐿 ↔ (𝑡𝑋 ∧ ∃𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))(𝐹𝑠) ⊆ 𝑡)))
1342adantr 473 . . . . . . 7 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → 𝑋𝐿)
135 rnelfmlem 22276 . . . . . . 7 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → ran (𝑥𝐿 ↦ (𝐹𝑥)) ∈ (fBas‘𝑌))
136 simpl3 1173 . . . . . . 7 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → 𝐹:𝑌𝑋)
137 elfm 22271 . . . . . . 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 1351 . . . . . 6 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → (𝑡 ∈ ((𝑋 FilMap 𝐹)‘ran (𝑥𝐿 ↦ (𝐹𝑥))) ↔ (𝑡𝑋 ∧ ∃𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))(𝐹𝑠) ⊆ 𝑡)))
139133, 138bitr4d 274 . . . . 5 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → (𝑡𝐿𝑡 ∈ ((𝑋 FilMap 𝐹)‘ran (𝑥𝐿 ↦ (𝐹𝑥)))))
140139eqrdv 2770 . . . 4 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → 𝐿 = ((𝑋 FilMap 𝐹)‘ran (𝑥𝐿 ↦ (𝐹𝑥))))
1417adantr 473 . . . . 5 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → (𝑋 FilMap 𝐹) Fn (fBas‘𝑌))
142 fnfvelrn 6671 . . . . 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 576 . . . 4 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → ((𝑋 FilMap 𝐹)‘ran (𝑥𝐿 ↦ (𝐹𝑥))) ∈ ran (𝑋 FilMap 𝐹))
144140, 143eqeltrd 2860 . . 3 (((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → 𝐿 ∈ ran (𝑋 FilMap 𝐹))
145144ex 405 . 2 ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → (ran 𝐹𝐿𝐿 ∈ ran (𝑋 FilMap 𝐹)))
14633, 145impbid 204 1 ((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → (𝐿 ∈ ran (𝑋 FilMap 𝐹) ↔ ran 𝐹𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387  w3a 1068   = wceq 1507  wcel 2050  wrex 3083  Vcvv 3409  cin 3822  wss 3823  cmpt 5004  ccnv 5402  dom cdm 5403  ran crn 5404  cima 5406  Fun wfun 6179   Fn wfn 6180  wf 6181  ontowfo 6183  cfv 6185  (class class class)co 6974  fBascfbas 20247  filGencfg 20248  Filcfil 22169   FilMap cfm 22257
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-rep 5045  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182  ax-un 7277
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-nel 3068  df-ral 3087  df-rex 3088  df-reu 3089  df-rab 3091  df-v 3411  df-sbc 3676  df-csb 3781  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-iun 4790  df-br 4926  df-opab 4988  df-mpt 5005  df-id 5308  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-iota 6149  df-fun 6187  df-fn 6188  df-f 6189  df-f1 6190  df-fo 6191  df-f1o 6192  df-fv 6193  df-ov 6977  df-oprab 6978  df-mpo 6979  df-fbas 20256  df-fg 20257  df-fil 22170  df-fm 22262
This theorem is referenced by: (None)
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