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Theorem foresf1o 28727
 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 fornex 7028 . . . 4 (𝐴𝑉 → (𝐹:𝐴onto𝐵𝐵 ∈ V))
21imp 444 . . 3 ((𝐴𝑉𝐹:𝐴onto𝐵) → 𝐵 ∈ V)
3 foelrn 6286 . . . . . 6 ((𝐹:𝐴onto𝐵𝑦𝐵) → ∃𝑧𝐴 𝑦 = (𝐹𝑧))
4 fofn 6030 . . . . . . . . . 10 (𝐹:𝐴onto𝐵𝐹 Fn 𝐴)
5 eqcom 2617 . . . . . . . . . . 11 ((𝐹𝑧) = 𝑦𝑦 = (𝐹𝑧))
6 fniniseg 6246 . . . . . . . . . . . . 13 (𝐹 Fn 𝐴 → (𝑧 ∈ (𝐹 “ {𝑦}) ↔ (𝑧𝐴 ∧ (𝐹𝑧) = 𝑦)))
76biimpar 501 . . . . . . . . . . . 12 ((𝐹 Fn 𝐴 ∧ (𝑧𝐴 ∧ (𝐹𝑧) = 𝑦)) → 𝑧 ∈ (𝐹 “ {𝑦}))
87anassrs 678 . . . . . . . . . . 11 (((𝐹 Fn 𝐴𝑧𝐴) ∧ (𝐹𝑧) = 𝑦) → 𝑧 ∈ (𝐹 “ {𝑦}))
95, 8sylan2br 492 . . . . . . . . . 10 (((𝐹 Fn 𝐴𝑧𝐴) ∧ 𝑦 = (𝐹𝑧)) → 𝑧 ∈ (𝐹 “ {𝑦}))
104, 9sylanl1 680 . . . . . . . . 9 (((𝐹:𝐴onto𝐵𝑧𝐴) ∧ 𝑦 = (𝐹𝑧)) → 𝑧 ∈ (𝐹 “ {𝑦}))
1110ex 449 . . . . . . . 8 ((𝐹:𝐴onto𝐵𝑧𝐴) → (𝑦 = (𝐹𝑧) → 𝑧 ∈ (𝐹 “ {𝑦})))
1211reximdva 3000 . . . . . . 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 746 . . . 4 (((𝐴𝑉𝐹:𝐴onto𝐵) ∧ 𝑦𝐵) → ∃𝑧𝐴 𝑧 ∈ (𝐹 “ {𝑦}))
1615ralrimiva 2949 . . 3 ((𝐴𝑉𝐹:𝐴onto𝐵) → ∀𝑦𝐵𝑧𝐴 𝑧 ∈ (𝐹 “ {𝑦}))
17 eleq1 2676 . . . 4 (𝑧 = (𝑔𝑦) → (𝑧 ∈ (𝐹 “ {𝑦}) ↔ (𝑔𝑦) ∈ (𝐹 “ {𝑦})))
1817ac6sg 9193 . . 3 (𝐵 ∈ V → (∀𝑦𝐵𝑧𝐴 𝑧 ∈ (𝐹 “ {𝑦}) → ∃𝑔(𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))))
192, 16, 18sylc 63 . 2 ((𝐴𝑉𝐹:𝐴onto𝐵) → ∃𝑔(𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦})))
20 frn 5966 . . . . 5 (𝑔:𝐵𝐴 → ran 𝑔𝐴)
2120ad2antrl 760 . . . 4 (((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) → ran 𝑔𝐴)
22 vex 3176 . . . . . 6 𝑔 ∈ V
2322rnex 6992 . . . . 5 ran 𝑔 ∈ V
2423elpw 4114 . . . 4 (ran 𝑔 ∈ 𝒫 𝐴 ↔ ran 𝑔𝐴)
2521, 24sylibr 223 . . 3 (((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) → ran 𝑔 ∈ 𝒫 𝐴)
26 fof 6028 . . . . . 6 (𝐹:𝐴onto𝐵𝐹:𝐴𝐵)
2726ad2antlr 759 . . . . 5 (((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) → 𝐹:𝐴𝐵)
2827, 21fssresd 5984 . . . 4 (((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) → (𝐹 ↾ ran 𝑔):ran 𝑔𝐵)
29 ffn 5958 . . . . . 6 (𝑔:𝐵𝐴𝑔 Fn 𝐵)
3029ad2antrl 760 . . . . 5 (((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) → 𝑔 Fn 𝐵)
31 dffn3 5967 . . . . 5 (𝑔 Fn 𝐵𝑔:𝐵⟶ran 𝑔)
3230, 31sylib 207 . . . 4 (((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) → 𝑔:𝐵⟶ran 𝑔)
33 fvres 6117 . . . . . . . 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 6107 . . . . . 6 ((((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) → (𝑔‘((𝐹 ↾ ran 𝑔)‘𝑧)) = (𝑔‘(𝐹𝑧)))
36 nfv 1830 . . . . . . . . 9 𝑦(𝐴𝑉𝐹:𝐴onto𝐵)
37 nfv 1830 . . . . . . . . . 10 𝑦 𝑔:𝐵𝐴
38 nfra1 2925 . . . . . . . . . 10 𝑦𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦})
3937, 38nfan 1816 . . . . . . . . 9 𝑦(𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))
4036, 39nfan 1816 . . . . . . . 8 𝑦((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦})))
41 nfv 1830 . . . . . . . 8 𝑦 𝑧 ∈ ran 𝑔
4240, 41nfan 1816 . . . . . . 7 𝑦(((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔)
43 simpr 476 . . . . . . . . . . 11 ((((((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) ∧ 𝑦𝐵) ∧ (𝑔𝑦) = 𝑧) → (𝑔𝑦) = 𝑧)
4443fveq2d 6107 . . . . . . . . . 10 ((((((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) ∧ 𝑦𝐵) ∧ (𝑔𝑦) = 𝑧) → (𝐹‘(𝑔𝑦)) = (𝐹𝑧))
454ad5antlr 767 . . . . . . . . . . 11 ((((((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) ∧ 𝑦𝐵) ∧ (𝑔𝑦) = 𝑧) → 𝐹 Fn 𝐴)
46 simplrr 797 . . . . . . . . . . . . 13 ((((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) → ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))
4746ad2antrr 758 . . . . . . . . . . . 12 ((((((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) ∧ 𝑦𝐵) ∧ (𝑔𝑦) = 𝑧) → ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))
48 simplr 788 . . . . . . . . . . . 12 ((((((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) ∧ 𝑦𝐵) ∧ (𝑔𝑦) = 𝑧) → 𝑦𝐵)
49 rspa 2914 . . . . . . . . . . . 12 ((∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}) ∧ 𝑦𝐵) → (𝑔𝑦) ∈ (𝐹 “ {𝑦}))
5047, 48, 49syl2anc 691 . . . . . . . . . . 11 ((((((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) ∧ 𝑦𝐵) ∧ (𝑔𝑦) = 𝑧) → (𝑔𝑦) ∈ (𝐹 “ {𝑦}))
51 fniniseg 6246 . . . . . . . . . . . 12 (𝐹 Fn 𝐴 → ((𝑔𝑦) ∈ (𝐹 “ {𝑦}) ↔ ((𝑔𝑦) ∈ 𝐴 ∧ (𝐹‘(𝑔𝑦)) = 𝑦)))
5251simplbda 652 . . . . . . . . . . 11 ((𝐹 Fn 𝐴 ∧ (𝑔𝑦) ∈ (𝐹 “ {𝑦})) → (𝐹‘(𝑔𝑦)) = 𝑦)
5345, 50, 52syl2anc 691 . . . . . . . . . 10 ((((((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) ∧ 𝑦𝐵) ∧ (𝑔𝑦) = 𝑧) → (𝐹‘(𝑔𝑦)) = 𝑦)
5444, 53eqtr3d 2646 . . . . . . . . 9 ((((((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) ∧ 𝑦𝐵) ∧ (𝑔𝑦) = 𝑧) → (𝐹𝑧) = 𝑦)
5554fveq2d 6107 . . . . . . . 8 ((((((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) ∧ 𝑦𝐵) ∧ (𝑔𝑦) = 𝑧) → (𝑔‘(𝐹𝑧)) = (𝑔𝑦))
5655, 43eqtrd 2644 . . . . . . 7 ((((((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) ∧ 𝑦𝐵) ∧ (𝑔𝑦) = 𝑧) → (𝑔‘(𝐹𝑧)) = 𝑧)
57 fvelrnb 6153 . . . . . . . . 9 (𝑔 Fn 𝐵 → (𝑧 ∈ ran 𝑔 ↔ ∃𝑦𝐵 (𝑔𝑦) = 𝑧))
5857biimpa 500 . . . . . . . 8 ((𝑔 Fn 𝐵𝑧 ∈ ran 𝑔) → ∃𝑦𝐵 (𝑔𝑦) = 𝑧)
5930, 58sylan 487 . . . . . . 7 ((((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) → ∃𝑦𝐵 (𝑔𝑦) = 𝑧)
6042, 56, 59r19.29af 3058 . . . . . 6 ((((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) → (𝑔‘(𝐹𝑧)) = 𝑧)
6135, 60eqtrd 2644 . . . . 5 ((((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) → (𝑔‘((𝐹 ↾ ran 𝑔)‘𝑧)) = 𝑧)
6261ralrimiva 2949 . . . 4 (((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) → ∀𝑧 ∈ ran 𝑔(𝑔‘((𝐹 ↾ ran 𝑔)‘𝑧)) = 𝑧)
6332ffvelrnda 6267 . . . . . . . 8 ((((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑦𝐵) → (𝑔𝑦) ∈ ran 𝑔)
64 fvres 6117 . . . . . . . 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 763 . . . . . . . 8 ((((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑦𝐵) → 𝐹 Fn 𝐴)
67 simplrr 797 . . . . . . . . 9 ((((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑦𝐵) → ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))
68 simpr 476 . . . . . . . . 9 ((((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑦𝐵) → 𝑦𝐵)
6967, 68, 49syl2anc 691 . . . . . . . 8 ((((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑦𝐵) → (𝑔𝑦) ∈ (𝐹 “ {𝑦}))
7066, 69, 52syl2anc 691 . . . . . . 7 ((((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑦𝐵) → (𝐹‘(𝑔𝑦)) = 𝑦)
7165, 70eqtrd 2644 . . . . . 6 ((((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) ∧ 𝑦𝐵) → ((𝐹 ↾ ran 𝑔)‘(𝑔𝑦)) = 𝑦)
7271ex 449 . . . . 5 (((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) → (𝑦𝐵 → ((𝐹 ↾ ran 𝑔)‘(𝑔𝑦)) = 𝑦))
7340, 72ralrimi 2940 . . . 4 (((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) → ∀𝑦𝐵 ((𝐹 ↾ ran 𝑔)‘(𝑔𝑦)) = 𝑦)
7428, 32, 62, 732fvidf1od 6453 . . 3 (((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) → (𝐹 ↾ ran 𝑔):ran 𝑔1-1-onto𝐵)
75 reseq2 5312 . . . . 5 (𝑥 = ran 𝑔 → (𝐹𝑥) = (𝐹 ↾ ran 𝑔))
76 id 22 . . . . 5 (𝑥 = ran 𝑔𝑥 = ran 𝑔)
77 eqidd 2611 . . . . 5 (𝑥 = ran 𝑔𝐵 = 𝐵)
7875, 76, 77f1oeq123d 6046 . . . 4 (𝑥 = ran 𝑔 → ((𝐹𝑥):𝑥1-1-onto𝐵 ↔ (𝐹 ↾ ran 𝑔):ran 𝑔1-1-onto𝐵))
7978rspcev 3282 . . 3 ((ran 𝑔 ∈ 𝒫 𝐴 ∧ (𝐹 ↾ ran 𝑔):ran 𝑔1-1-onto𝐵) → ∃𝑥 ∈ 𝒫 𝐴(𝐹𝑥):𝑥1-1-onto𝐵)
8025, 74, 79syl2anc 691 . 2 (((𝐴𝑉𝐹:𝐴onto𝐵) ∧ (𝑔:𝐵𝐴 ∧ ∀𝑦𝐵 (𝑔𝑦) ∈ (𝐹 “ {𝑦}))) → ∃𝑥 ∈ 𝒫 𝐴(𝐹𝑥):𝑥1-1-onto𝐵)
8119, 80exlimddv 1850 1 ((𝐴𝑉𝐹:𝐴onto𝐵) → ∃𝑥 ∈ 𝒫 𝐴(𝐹𝑥):𝑥1-1-onto𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ⊆ wss 3540  𝒫 cpw 4108  {csn 4125  ◡ccnv 5037  ran crn 5039   ↾ cres 5040   “ cima 5041   Fn wfn 5799  ⟶wf 5800  –onto→wfo 5802  –1-1-onto→wf1o 5803  ‘cfv 5804 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-reg 8380  ax-inf2 8421  ax-ac2 9168 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-en 7842  df-r1 8510  df-rank 8511  df-card 8648  df-ac 8822 This theorem is referenced by:  rabfodom  28728
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