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Theorem foresf1o 32532
Description: From a surjective function, *choose* a subset of the domain, such that the restricted function is bijective. (Contributed by Thierry Arnoux, 27-Jan-2020.)
Assertion
Ref Expression
foresf1o ((𝐴𝑉𝐹:𝐴onto𝐵) → ∃𝑥 ∈ 𝒫 𝐴(𝐹𝑥):𝑥1-1-onto𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐹
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem foresf1o
Dummy variables 𝑔 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 focdmex 7979 . . . 4 (𝐴𝑉 → (𝐹:𝐴onto𝐵𝐵 ∈ V))
21imp 406 . . 3 ((𝐴𝑉𝐹:𝐴onto𝐵) → 𝐵 ∈ V)
3 foelrn 7127 . . . . . 6 ((𝐹:𝐴onto𝐵𝑦𝐵) → ∃𝑧𝐴 𝑦 = (𝐹𝑧))
4 fofn 6823 . . . . . . . . . 10 (𝐹:𝐴onto𝐵𝐹 Fn 𝐴)
5 eqcom 2742 . . . . . . . . . . 11 ((𝐹𝑧) = 𝑦𝑦 = (𝐹𝑧))
6 fniniseg 7080 . . . . . . . . . . . . 13 (𝐹 Fn 𝐴 → (𝑧 ∈ (𝐹 “ {𝑦}) ↔ (𝑧𝐴 ∧ (𝐹𝑧) = 𝑦)))
76biimpar 477 . . . . . . . . . . . 12 ((𝐹 Fn 𝐴 ∧ (𝑧𝐴 ∧ (𝐹𝑧) = 𝑦)) → 𝑧 ∈ (𝐹 “ {𝑦}))
87anassrs 467 . . . . . . . . . . 11 (((𝐹 Fn 𝐴𝑧𝐴) ∧ (𝐹𝑧) = 𝑦) → 𝑧 ∈ (𝐹 “ {𝑦}))
95, 8sylan2br 595 . . . . . . . . . 10 (((𝐹 Fn 𝐴𝑧𝐴) ∧ 𝑦 = (𝐹𝑧)) → 𝑧 ∈ (𝐹 “ {𝑦}))
104, 9sylanl1 680 . . . . . . . . 9 (((𝐹:𝐴onto𝐵𝑧𝐴) ∧ 𝑦 = (𝐹𝑧)) → 𝑧 ∈ (𝐹 “ {𝑦}))
1110ex 412 . . . . . . . 8 ((𝐹:𝐴onto𝐵𝑧𝐴) → (𝑦 = (𝐹𝑧) → 𝑧 ∈ (𝐹 “ {𝑦})))
1211reximdva 3166 . . . . . . 7 (𝐹:𝐴onto𝐵 → (∃𝑧𝐴 𝑦 = (𝐹𝑧) → ∃𝑧𝐴 𝑧 ∈ (𝐹 “ {𝑦})))
1312adantr 480 . . . . . 6 ((𝐹:𝐴onto𝐵𝑦𝐵) → (∃𝑧𝐴 𝑦 = (𝐹𝑧) → ∃𝑧𝐴 𝑧 ∈ (𝐹 “ {𝑦})))
143, 13mpd 15 . . . . 5 ((𝐹:𝐴onto𝐵𝑦𝐵) → ∃𝑧𝐴 𝑧 ∈ (𝐹 “ {𝑦}))
1514adantll 714 . . . 4 (((𝐴𝑉𝐹:𝐴onto𝐵) ∧ 𝑦𝐵) → ∃𝑧𝐴 𝑧 ∈ (𝐹 “ {𝑦}))
1615ralrimiva 3144 . . 3 ((𝐴𝑉𝐹:𝐴onto𝐵) → ∀𝑦𝐵𝑧𝐴 𝑧 ∈ (𝐹 “ {𝑦}))
17 eleq1 2827 . . . 4 (𝑧 = (𝑔𝑦) → (𝑧 ∈ (𝐹 “ {𝑦}) ↔ (𝑔𝑦) ∈ (𝐹 “ {𝑦})))
1817ac6sg 10526 . . 3 (𝐵 ∈ V → (∀𝑦𝐵𝑧𝐴 𝑧 ∈ (𝐹 “ {𝑦}) → ∃𝑔(𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))))
192, 16, 18sylc 65 . 2 ((𝐴𝑉𝐹:𝐴onto𝐵) → ∃𝑔(𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦})))
20 frn 6744 . . . . 5 (𝑔:𝐵𝐴 → ran 𝑔𝐴)
2120ad2antrl 728 . . . 4 (((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) → ran 𝑔𝐴)
22 vex 3482 . . . . . 6 𝑔 ∈ V
2322rnex 7933 . . . . 5 ran 𝑔 ∈ V
2423elpw 4609 . . . 4 (ran 𝑔 ∈ 𝒫 𝐴 ↔ ran 𝑔𝐴)
2521, 24sylibr 234 . . 3 (((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) → ran 𝑔 ∈ 𝒫 𝐴)
26 fof 6821 . . . . . 6 (𝐹:𝐴onto𝐵𝐹:𝐴𝐵)
2726ad2antlr 727 . . . . 5 (((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) → 𝐹:𝐴𝐵)
2827, 21fssresd 6776 . . . 4 (((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) → (𝐹 ↾ ran 𝑔):ran 𝑔𝐵)
29 ffn 6737 . . . . . 6 (𝑔:𝐵𝐴𝑔 Fn 𝐵)
3029ad2antrl 728 . . . . 5 (((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) → 𝑔 Fn 𝐵)
31 dffn3 6749 . . . . 5 (𝑔 Fn 𝐵𝑔:𝐵⟶ran 𝑔)
3230, 31sylib 218 . . . 4 (((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) → 𝑔:𝐵⟶ran 𝑔)
33 fvres 6926 . . . . . . . 8 (𝑧 ∈ ran 𝑔 → ((𝐹 ↾ ran 𝑔)‘𝑧) = (𝐹𝑧))
3433adantl 481 . . . . . . 7 ((((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) → ((𝐹 ↾ ran 𝑔)‘𝑧) = (𝐹𝑧))
3534fveq2d 6911 . . . . . 6 ((((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) → (𝑔‘((𝐹 ↾ ran 𝑔)‘𝑧)) = (𝑔‘(𝐹𝑧)))
36 nfv 1912 . . . . . . . . 9 𝑦(𝐴𝑉𝐹:𝐴onto𝐵)
37 nfv 1912 . . . . . . . . . 10 𝑦 𝑔:𝐵𝐴
38 nfra1 3282 . . . . . . . . . 10 𝑦𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦})
3937, 38nfan 1897 . . . . . . . . 9 𝑦(𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))
4036, 39nfan 1897 . . . . . . . 8 𝑦((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦})))
41 nfv 1912 . . . . . . . 8 𝑦 𝑧 ∈ ran 𝑔
4240, 41nfan 1897 . . . . . . 7 𝑦(((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔)
43 simpr 484 . . . . . . . . . . 11 ((((((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) ∧ 𝑦𝐵) ∧ (𝑔𝑦) = 𝑧) → (𝑔𝑦) = 𝑧)
4443fveq2d 6911 . . . . . . . . . 10 ((((((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) ∧ 𝑦𝐵) ∧ (𝑔𝑦) = 𝑧) → (𝐹‘(𝑔𝑦)) = (𝐹𝑧))
454ad5antlr 735 . . . . . . . . . . 11 ((((((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) ∧ 𝑦𝐵) ∧ (𝑔𝑦) = 𝑧) → 𝐹 Fn 𝐴)
46 simplrr 778 . . . . . . . . . . . . 13 ((((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) → ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))
4746ad2antrr 726 . . . . . . . . . . . 12 ((((((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) ∧ 𝑦𝐵) ∧ (𝑔𝑦) = 𝑧) → ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))
48 simplr 769 . . . . . . . . . . . 12 ((((((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) ∧ 𝑦𝐵) ∧ (𝑔𝑦) = 𝑧) → 𝑦𝐵)
49 rspa 3246 . . . . . . . . . . . 12 ((∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}) ∧ 𝑦𝐵) → (𝑔𝑦) ∈ (𝐹 “ {𝑦}))
5047, 48, 49syl2anc 584 . . . . . . . . . . 11 ((((((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) ∧ 𝑦𝐵) ∧ (𝑔𝑦) = 𝑧) → (𝑔𝑦) ∈ (𝐹 “ {𝑦}))
51 fniniseg 7080 . . . . . . . . . . . 12 (𝐹 Fn 𝐴 → ((𝑔𝑦) ∈ (𝐹 “ {𝑦}) ↔ ((𝑔𝑦) ∈ 𝐴 ∧ (𝐹‘(𝑔𝑦)) = 𝑦)))
5251simplbda 499 . . . . . . . . . . 11 ((𝐹 Fn 𝐴 ∧ (𝑔𝑦) ∈ (𝐹 “ {𝑦})) → (𝐹‘(𝑔𝑦)) = 𝑦)
5345, 50, 52syl2anc 584 . . . . . . . . . 10 ((((((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) ∧ 𝑦𝐵) ∧ (𝑔𝑦) = 𝑧) → (𝐹‘(𝑔𝑦)) = 𝑦)
5444, 53eqtr3d 2777 . . . . . . . . 9 ((((((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) ∧ 𝑦𝐵) ∧ (𝑔𝑦) = 𝑧) → (𝐹𝑧) = 𝑦)
5554fveq2d 6911 . . . . . . . 8 ((((((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) ∧ 𝑦𝐵) ∧ (𝑔𝑦) = 𝑧) → (𝑔‘(𝐹𝑧)) = (𝑔𝑦))
5655, 43eqtrd 2775 . . . . . . 7 ((((((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) ∧ 𝑦𝐵) ∧ (𝑔𝑦) = 𝑧) → (𝑔‘(𝐹𝑧)) = 𝑧)
57 fvelrnb 6969 . . . . . . . . 9 (𝑔 Fn 𝐵 → (𝑧 ∈ ran 𝑔 ↔ ∃𝑦𝐵 (𝑔𝑦) = 𝑧))
5857biimpa 476 . . . . . . . 8 ((𝑔 Fn 𝐵𝑧 ∈ ran 𝑔) → ∃𝑦𝐵 (𝑔𝑦) = 𝑧)
5930, 58sylan 580 . . . . . . 7 ((((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) → ∃𝑦𝐵 (𝑔𝑦) = 𝑧)
6042, 56, 59r19.29af 3266 . . . . . 6 ((((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) → (𝑔‘(𝐹𝑧)) = 𝑧)
6135, 60eqtrd 2775 . . . . 5 ((((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) → (𝑔‘((𝐹 ↾ ran 𝑔)‘𝑧)) = 𝑧)
6261ralrimiva 3144 . . . 4 (((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) → ∀𝑧 ∈ ran 𝑔(𝑔‘((𝐹 ↾ ran 𝑔)‘𝑧)) = 𝑧)
6332ffvelcdmda 7104 . . . . . . . 8 ((((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑦𝐵) → (𝑔𝑦) ∈ ran 𝑔)
64 fvres 6926 . . . . . . . 8 ((𝑔𝑦) ∈ ran 𝑔 → ((𝐹 ↾ ran 𝑔)‘(𝑔𝑦)) = (𝐹‘(𝑔𝑦)))
6563, 64syl 17 . . . . . . 7 ((((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑦𝐵) → ((𝐹 ↾ ran 𝑔)‘(𝑔𝑦)) = (𝐹‘(𝑔𝑦)))
664ad3antlr 731 . . . . . . . 8 ((((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑦𝐵) → 𝐹 Fn 𝐴)
67 simplrr 778 . . . . . . . . 9 ((((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑦𝐵) → ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))
68 simpr 484 . . . . . . . . 9 ((((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑦𝐵) → 𝑦𝐵)
6967, 68, 49syl2anc 584 . . . . . . . 8 ((((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑦𝐵) → (𝑔𝑦) ∈ (𝐹 “ {𝑦}))
7066, 69, 52syl2anc 584 . . . . . . 7 ((((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑦𝐵) → (𝐹‘(𝑔𝑦)) = 𝑦)
7165, 70eqtrd 2775 . . . . . 6 ((((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑦𝐵) → ((𝐹 ↾ ran 𝑔)‘(𝑔𝑦)) = 𝑦)
7271ex 412 . . . . 5 (((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) → (𝑦𝐵 → ((𝐹 ↾ ran 𝑔)‘(𝑔𝑦)) = 𝑦))
7340, 72ralrimi 3255 . . . 4 (((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) → ∀𝑦𝐵 ((𝐹 ↾ ran 𝑔)‘(𝑔𝑦)) = 𝑦)
7428, 32, 62, 732fvidf1od 7318 . . 3 (((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) → (𝐹 ↾ ran 𝑔):ran 𝑔1-1-onto𝐵)
75 reseq2 5995 . . . . 5 (𝑥 = ran 𝑔 → (𝐹𝑥) = (𝐹 ↾ ran 𝑔))
76 id 22 . . . . 5 (𝑥 = ran 𝑔𝑥 = ran 𝑔)
77 eqidd 2736 . . . . 5 (𝑥 = ran 𝑔𝐵 = 𝐵)
7875, 76, 77f1oeq123d 6843 . . . 4 (𝑥 = ran 𝑔 → ((𝐹𝑥):𝑥1-1-onto𝐵 ↔ (𝐹 ↾ ran 𝑔):ran 𝑔1-1-onto𝐵))
7978rspcev 3622 . . 3 ((ran 𝑔 ∈ 𝒫 𝐴 ∧ (𝐹 ↾ ran 𝑔):ran 𝑔1-1-onto𝐵) → ∃𝑥 ∈ 𝒫 𝐴(𝐹𝑥):𝑥1-1-onto𝐵)
8025, 74, 79syl2anc 584 . 2 (((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) → ∃𝑥 ∈ 𝒫 𝐴(𝐹𝑥):𝑥1-1-onto𝐵)
8119, 80exlimddv 1933 1 ((𝐴𝑉𝐹:𝐴onto𝐵) → ∃𝑥 ∈ 𝒫 𝐴(𝐹𝑥):𝑥1-1-onto𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wex 1776  wcel 2106  wral 3059  wrex 3068  Vcvv 3478  wss 3963  𝒫 cpw 4605  {csn 4631  ccnv 5688  ran crn 5690  cres 5691  cima 5692   Fn wfn 6558  wf 6559  ontowfo 6561  1-1-ontowf1o 6562  cfv 6563
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  ax-reg 9630  ax-inf2 9679  ax-ac2 10501
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  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-ral 3060  df-rex 3069  df-rmo 3378  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-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  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-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  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-isom 6572  df-riota 7388  df-ov 7434  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-en 8985  df-r1 9802  df-rank 9803  df-card 9977  df-ac 10154
This theorem is referenced by:  rabfodom  32533
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