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Theorem foresf1o 32523
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 7980 . . . 4 (𝐴𝑉 → (𝐹:𝐴onto𝐵𝐵 ∈ V))
21imp 406 . . 3 ((𝐴𝑉𝐹:𝐴onto𝐵) → 𝐵 ∈ V)
3 foelrn 7127 . . . . . 6 ((𝐹:𝐴onto𝐵𝑦𝐵) → ∃𝑧𝐴 𝑦 = (𝐹𝑧))
4 fofn 6822 . . . . . . . . . 10 (𝐹:𝐴onto𝐵𝐹 Fn 𝐴)
5 eqcom 2744 . . . . . . . . . . 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 3168 . . . . . . 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 3146 . . 3 ((𝐴𝑉𝐹:𝐴onto𝐵) → ∀𝑦𝐵𝑧𝐴 𝑧 ∈ (𝐹 “ {𝑦}))
17 eleq1 2829 . . . 4 (𝑧 = (𝑔𝑦) → (𝑧 ∈ (𝐹 “ {𝑦}) ↔ (𝑔𝑦) ∈ (𝐹 “ {𝑦})))
1817ac6sg 10528 . . 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 6743 . . . . 5 (𝑔:𝐵𝐴 → ran 𝑔𝐴)
2120ad2antrl 728 . . . 4 (((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) → ran 𝑔𝐴)
22 vex 3484 . . . . . 6 𝑔 ∈ V
2322rnex 7932 . . . . 5 ran 𝑔 ∈ V
2423elpw 4604 . . . 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 6820 . . . . . 6 (𝐹:𝐴onto𝐵𝐹:𝐴𝐵)
2726ad2antlr 727 . . . . 5 (((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) → 𝐹:𝐴𝐵)
2827, 21fssresd 6775 . . . 4 (((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) → (𝐹 ↾ ran 𝑔):ran 𝑔𝐵)
29 ffn 6736 . . . . . 6 (𝑔:𝐵𝐴𝑔 Fn 𝐵)
3029ad2antrl 728 . . . . 5 (((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) → 𝑔 Fn 𝐵)
31 dffn3 6748 . . . . 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 6925 . . . . . . . 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 6910 . . . . . 6 ((((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) → (𝑔‘((𝐹 ↾ ran 𝑔)‘𝑧)) = (𝑔‘(𝐹𝑧)))
36 nfv 1914 . . . . . . . . 9 𝑦(𝐴𝑉𝐹:𝐴onto𝐵)
37 nfv 1914 . . . . . . . . . 10 𝑦 𝑔:𝐵𝐴
38 nfra1 3284 . . . . . . . . . 10 𝑦𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦})
3937, 38nfan 1899 . . . . . . . . 9 𝑦(𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))
4036, 39nfan 1899 . . . . . . . 8 𝑦((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦})))
41 nfv 1914 . . . . . . . 8 𝑦 𝑧 ∈ ran 𝑔
4240, 41nfan 1899 . . . . . . 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 6910 . . . . . . . . . 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 3248 . . . . . . . . . . . 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 2779 . . . . . . . . 9 ((((((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) ∧ 𝑦𝐵) ∧ (𝑔𝑦) = 𝑧) → (𝐹𝑧) = 𝑦)
5554fveq2d 6910 . . . . . . . 8 ((((((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) ∧ 𝑦𝐵) ∧ (𝑔𝑦) = 𝑧) → (𝑔‘(𝐹𝑧)) = (𝑔𝑦))
5655, 43eqtrd 2777 . . . . . . 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 3268 . . . . . 6 ((((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) → (𝑔‘(𝐹𝑧)) = 𝑧)
6135, 60eqtrd 2777 . . . . 5 ((((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) → (𝑔‘((𝐹 ↾ ran 𝑔)‘𝑧)) = 𝑧)
6261ralrimiva 3146 . . . 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 6925 . . . . . . . 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 2777 . . . . . 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 3257 . . . 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 5992 . . . . 5 (𝑥 = ran 𝑔 → (𝐹𝑥) = (𝐹 ↾ ran 𝑔))
76 id 22 . . . . 5 (𝑥 = ran 𝑔𝑥 = ran 𝑔)
77 eqidd 2738 . . . . 5 (𝑥 = ran 𝑔𝐵 = 𝐵)
7875, 76, 77f1oeq123d 6842 . . . 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 1935 1 ((𝐴𝑉𝐹:𝐴onto𝐵) → ∃𝑥 ∈ 𝒫 𝐴(𝐹𝑥):𝑥1-1-onto𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wex 1779  wcel 2108  wral 3061  wrex 3070  Vcvv 3480  wss 3951  𝒫 cpw 4600  {csn 4626  ccnv 5684  ran crn 5686  cres 5687  cima 5688   Fn wfn 6556  wf 6557  ontowfo 6559  1-1-ontowf1o 6560  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-reg 9632  ax-inf2 9681  ax-ac2 10503
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-en 8986  df-r1 9804  df-rank 9805  df-card 9979  df-ac 10156
This theorem is referenced by:  rabfodom  32524
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