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Theorem foresf1o 32430
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 7969 . . . 4 (𝐴𝑉 → (𝐹:𝐴onto𝐵𝐵 ∈ V))
21imp 405 . . 3 ((𝐴𝑉𝐹:𝐴onto𝐵) → 𝐵 ∈ V)
3 foelrn 7121 . . . . . 6 ((𝐹:𝐴onto𝐵𝑦𝐵) → ∃𝑧𝐴 𝑦 = (𝐹𝑧))
4 fofn 6817 . . . . . . . . . 10 (𝐹:𝐴onto𝐵𝐹 Fn 𝐴)
5 eqcom 2733 . . . . . . . . . . 11 ((𝐹𝑧) = 𝑦𝑦 = (𝐹𝑧))
6 fniniseg 7073 . . . . . . . . . . . . 13 (𝐹 Fn 𝐴 → (𝑧 ∈ (𝐹 “ {𝑦}) ↔ (𝑧𝐴 ∧ (𝐹𝑧) = 𝑦)))
76biimpar 476 . . . . . . . . . . . 12 ((𝐹 Fn 𝐴 ∧ (𝑧𝐴 ∧ (𝐹𝑧) = 𝑦)) → 𝑧 ∈ (𝐹 “ {𝑦}))
87anassrs 466 . . . . . . . . . . 11 (((𝐹 Fn 𝐴𝑧𝐴) ∧ (𝐹𝑧) = 𝑦) → 𝑧 ∈ (𝐹 “ {𝑦}))
95, 8sylan2br 593 . . . . . . . . . 10 (((𝐹 Fn 𝐴𝑧𝐴) ∧ 𝑦 = (𝐹𝑧)) → 𝑧 ∈ (𝐹 “ {𝑦}))
104, 9sylanl1 678 . . . . . . . . 9 (((𝐹:𝐴onto𝐵𝑧𝐴) ∧ 𝑦 = (𝐹𝑧)) → 𝑧 ∈ (𝐹 “ {𝑦}))
1110ex 411 . . . . . . . 8 ((𝐹:𝐴onto𝐵𝑧𝐴) → (𝑦 = (𝐹𝑧) → 𝑧 ∈ (𝐹 “ {𝑦})))
1211reximdva 3158 . . . . . . 7 (𝐹:𝐴onto𝐵 → (∃𝑧𝐴 𝑦 = (𝐹𝑧) → ∃𝑧𝐴 𝑧 ∈ (𝐹 “ {𝑦})))
1312adantr 479 . . . . . 6 ((𝐹:𝐴onto𝐵𝑦𝐵) → (∃𝑧𝐴 𝑦 = (𝐹𝑧) → ∃𝑧𝐴 𝑧 ∈ (𝐹 “ {𝑦})))
143, 13mpd 15 . . . . 5 ((𝐹:𝐴onto𝐵𝑦𝐵) → ∃𝑧𝐴 𝑧 ∈ (𝐹 “ {𝑦}))
1514adantll 712 . . . 4 (((𝐴𝑉𝐹:𝐴onto𝐵) ∧ 𝑦𝐵) → ∃𝑧𝐴 𝑧 ∈ (𝐹 “ {𝑦}))
1615ralrimiva 3136 . . 3 ((𝐴𝑉𝐹:𝐴onto𝐵) → ∀𝑦𝐵𝑧𝐴 𝑧 ∈ (𝐹 “ {𝑦}))
17 eleq1 2814 . . . 4 (𝑧 = (𝑔𝑦) → (𝑧 ∈ (𝐹 “ {𝑦}) ↔ (𝑔𝑦) ∈ (𝐹 “ {𝑦})))
1817ac6sg 10531 . . 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 6735 . . . . 5 (𝑔:𝐵𝐴 → ran 𝑔𝐴)
2120ad2antrl 726 . . . 4 (((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) → ran 𝑔𝐴)
22 vex 3466 . . . . . 6 𝑔 ∈ V
2322rnex 7923 . . . . 5 ran 𝑔 ∈ V
2423elpw 4611 . . . 4 (ran 𝑔 ∈ 𝒫 𝐴 ↔ ran 𝑔𝐴)
2521, 24sylibr 233 . . 3 (((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) → ran 𝑔 ∈ 𝒫 𝐴)
26 fof 6815 . . . . . 6 (𝐹:𝐴onto𝐵𝐹:𝐴𝐵)
2726ad2antlr 725 . . . . 5 (((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) → 𝐹:𝐴𝐵)
2827, 21fssresd 6769 . . . 4 (((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) → (𝐹 ↾ ran 𝑔):ran 𝑔𝐵)
29 ffn 6728 . . . . . 6 (𝑔:𝐵𝐴𝑔 Fn 𝐵)
3029ad2antrl 726 . . . . 5 (((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) → 𝑔 Fn 𝐵)
31 dffn3 6740 . . . . 5 (𝑔 Fn 𝐵𝑔:𝐵⟶ran 𝑔)
3230, 31sylib 217 . . . 4 (((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) → 𝑔:𝐵⟶ran 𝑔)
33 fvres 6920 . . . . . . . 8 (𝑧 ∈ ran 𝑔 → ((𝐹 ↾ ran 𝑔)‘𝑧) = (𝐹𝑧))
3433adantl 480 . . . . . . 7 ((((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) → ((𝐹 ↾ ran 𝑔)‘𝑧) = (𝐹𝑧))
3534fveq2d 6905 . . . . . 6 ((((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) → (𝑔‘((𝐹 ↾ ran 𝑔)‘𝑧)) = (𝑔‘(𝐹𝑧)))
36 nfv 1910 . . . . . . . . 9 𝑦(𝐴𝑉𝐹:𝐴onto𝐵)
37 nfv 1910 . . . . . . . . . 10 𝑦 𝑔:𝐵𝐴
38 nfra1 3272 . . . . . . . . . 10 𝑦𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦})
3937, 38nfan 1895 . . . . . . . . 9 𝑦(𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))
4036, 39nfan 1895 . . . . . . . 8 𝑦((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦})))
41 nfv 1910 . . . . . . . 8 𝑦 𝑧 ∈ ran 𝑔
4240, 41nfan 1895 . . . . . . 7 𝑦(((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔)
43 simpr 483 . . . . . . . . . . 11 ((((((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) ∧ 𝑦𝐵) ∧ (𝑔𝑦) = 𝑧) → (𝑔𝑦) = 𝑧)
4443fveq2d 6905 . . . . . . . . . 10 ((((((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) ∧ 𝑦𝐵) ∧ (𝑔𝑦) = 𝑧) → (𝐹‘(𝑔𝑦)) = (𝐹𝑧))
454ad5antlr 733 . . . . . . . . . . 11 ((((((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) ∧ 𝑦𝐵) ∧ (𝑔𝑦) = 𝑧) → 𝐹 Fn 𝐴)
46 simplrr 776 . . . . . . . . . . . . 13 ((((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) → ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))
4746ad2antrr 724 . . . . . . . . . . . 12 ((((((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) ∧ 𝑦𝐵) ∧ (𝑔𝑦) = 𝑧) → ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))
48 simplr 767 . . . . . . . . . . . 12 ((((((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) ∧ 𝑦𝐵) ∧ (𝑔𝑦) = 𝑧) → 𝑦𝐵)
49 rspa 3236 . . . . . . . . . . . 12 ((∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}) ∧ 𝑦𝐵) → (𝑔𝑦) ∈ (𝐹 “ {𝑦}))
5047, 48, 49syl2anc 582 . . . . . . . . . . 11 ((((((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) ∧ 𝑦𝐵) ∧ (𝑔𝑦) = 𝑧) → (𝑔𝑦) ∈ (𝐹 “ {𝑦}))
51 fniniseg 7073 . . . . . . . . . . . 12 (𝐹 Fn 𝐴 → ((𝑔𝑦) ∈ (𝐹 “ {𝑦}) ↔ ((𝑔𝑦) ∈ 𝐴 ∧ (𝐹‘(𝑔𝑦)) = 𝑦)))
5251simplbda 498 . . . . . . . . . . 11 ((𝐹 Fn 𝐴 ∧ (𝑔𝑦) ∈ (𝐹 “ {𝑦})) → (𝐹‘(𝑔𝑦)) = 𝑦)
5345, 50, 52syl2anc 582 . . . . . . . . . 10 ((((((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) ∧ 𝑦𝐵) ∧ (𝑔𝑦) = 𝑧) → (𝐹‘(𝑔𝑦)) = 𝑦)
5444, 53eqtr3d 2768 . . . . . . . . 9 ((((((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) ∧ 𝑦𝐵) ∧ (𝑔𝑦) = 𝑧) → (𝐹𝑧) = 𝑦)
5554fveq2d 6905 . . . . . . . 8 ((((((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) ∧ 𝑦𝐵) ∧ (𝑔𝑦) = 𝑧) → (𝑔‘(𝐹𝑧)) = (𝑔𝑦))
5655, 43eqtrd 2766 . . . . . . 7 ((((((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) ∧ 𝑦𝐵) ∧ (𝑔𝑦) = 𝑧) → (𝑔‘(𝐹𝑧)) = 𝑧)
57 fvelrnb 6963 . . . . . . . . 9 (𝑔 Fn 𝐵 → (𝑧 ∈ ran 𝑔 ↔ ∃𝑦𝐵 (𝑔𝑦) = 𝑧))
5857biimpa 475 . . . . . . . 8 ((𝑔 Fn 𝐵𝑧 ∈ ran 𝑔) → ∃𝑦𝐵 (𝑔𝑦) = 𝑧)
5930, 58sylan 578 . . . . . . 7 ((((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) → ∃𝑦𝐵 (𝑔𝑦) = 𝑧)
6042, 56, 59r19.29af 3256 . . . . . 6 ((((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) → (𝑔‘(𝐹𝑧)) = 𝑧)
6135, 60eqtrd 2766 . . . . 5 ((((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) → (𝑔‘((𝐹 ↾ ran 𝑔)‘𝑧)) = 𝑧)
6261ralrimiva 3136 . . . 4 (((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) → ∀𝑧 ∈ ran 𝑔(𝑔‘((𝐹 ↾ ran 𝑔)‘𝑧)) = 𝑧)
6332ffvelcdmda 7098 . . . . . . . 8 ((((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑦𝐵) → (𝑔𝑦) ∈ ran 𝑔)
64 fvres 6920 . . . . . . . 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 729 . . . . . . . 8 ((((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑦𝐵) → 𝐹 Fn 𝐴)
67 simplrr 776 . . . . . . . . 9 ((((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑦𝐵) → ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))
68 simpr 483 . . . . . . . . 9 ((((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑦𝐵) → 𝑦𝐵)
6967, 68, 49syl2anc 582 . . . . . . . 8 ((((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑦𝐵) → (𝑔𝑦) ∈ (𝐹 “ {𝑦}))
7066, 69, 52syl2anc 582 . . . . . . 7 ((((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑦𝐵) → (𝐹‘(𝑔𝑦)) = 𝑦)
7165, 70eqtrd 2766 . . . . . 6 ((((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑦𝐵) → ((𝐹 ↾ ran 𝑔)‘(𝑔𝑦)) = 𝑦)
7271ex 411 . . . . 5 (((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) → (𝑦𝐵 → ((𝐹 ↾ ran 𝑔)‘(𝑔𝑦)) = 𝑦))
7340, 72ralrimi 3245 . . . 4 (((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) → ∀𝑦𝐵 ((𝐹 ↾ ran 𝑔)‘(𝑔𝑦)) = 𝑦)
7428, 32, 62, 732fvidf1od 7312 . . 3 (((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) → (𝐹 ↾ ran 𝑔):ran 𝑔1-1-onto𝐵)
75 reseq2 5984 . . . . 5 (𝑥 = ran 𝑔 → (𝐹𝑥) = (𝐹 ↾ ran 𝑔))
76 id 22 . . . . 5 (𝑥 = ran 𝑔𝑥 = ran 𝑔)
77 eqidd 2727 . . . . 5 (𝑥 = ran 𝑔𝐵 = 𝐵)
7875, 76, 77f1oeq123d 6837 . . . 4 (𝑥 = ran 𝑔 → ((𝐹𝑥):𝑥1-1-onto𝐵 ↔ (𝐹 ↾ ran 𝑔):ran 𝑔1-1-onto𝐵))
7978rspcev 3608 . . 3 ((ran 𝑔 ∈ 𝒫 𝐴 ∧ (𝐹 ↾ ran 𝑔):ran 𝑔1-1-onto𝐵) → ∃𝑥 ∈ 𝒫 𝐴(𝐹𝑥):𝑥1-1-onto𝐵)
8025, 74, 79syl2anc 582 . 2 (((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) → ∃𝑥 ∈ 𝒫 𝐴(𝐹𝑥):𝑥1-1-onto𝐵)
8119, 80exlimddv 1931 1 ((𝐴𝑉𝐹:𝐴onto𝐵) → ∃𝑥 ∈ 𝒫 𝐴(𝐹𝑥):𝑥1-1-onto𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1534  wex 1774  wcel 2099  wral 3051  wrex 3060  Vcvv 3462  wss 3947  𝒫 cpw 4607  {csn 4633  ccnv 5681  ran crn 5683  cres 5684  cima 5685   Fn wfn 6549  wf 6550  ontowfo 6552  1-1-ontowf1o 6553  cfv 6554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-reg 9635  ax-inf2 9684  ax-ac2 10506
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-int 4955  df-iun 5003  df-iin 5004  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-se 5638  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-isom 6563  df-riota 7380  df-ov 7427  df-om 7877  df-2nd 8004  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-en 8975  df-r1 9807  df-rank 9808  df-card 9982  df-ac 10159
This theorem is referenced by:  rabfodom  32431
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