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Theorem fodomacn 10051
Description: A version of fodom 10518 that doesn't require the Axiom of Choice ax-ac 10454. If 𝐴 has choice sequences of length 𝐵, then any surjection from 𝐴 to 𝐵 can be inverted to an injection the other way. (Contributed by Mario Carneiro, 31-Aug-2015.)
Assertion
Ref Expression
fodomacn (𝐴AC 𝐵 → (𝐹:𝐴onto𝐵𝐵𝐴))

Proof of Theorem fodomacn
Dummy variables 𝑥 𝑓 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 foelrn 7108 . . . . 5 ((𝐹:𝐴onto𝐵𝑥𝐵) → ∃𝑦𝐴 𝑥 = (𝐹𝑦))
21ralrimiva 3147 . . . 4 (𝐹:𝐴onto𝐵 → ∀𝑥𝐵𝑦𝐴 𝑥 = (𝐹𝑦))
3 fveq2 6892 . . . . . 6 (𝑦 = (𝑓𝑥) → (𝐹𝑦) = (𝐹‘(𝑓𝑥)))
43eqeq2d 2744 . . . . 5 (𝑦 = (𝑓𝑥) → (𝑥 = (𝐹𝑦) ↔ 𝑥 = (𝐹‘(𝑓𝑥))))
54acni3 10042 . . . 4 ((𝐴AC 𝐵 ∧ ∀𝑥𝐵𝑦𝐴 𝑥 = (𝐹𝑦)) → ∃𝑓(𝑓:𝐵𝐴 ∧ ∀𝑥𝐵 𝑥 = (𝐹‘(𝑓𝑥))))
62, 5sylan2 594 . . 3 ((𝐴AC 𝐵𝐹:𝐴onto𝐵) → ∃𝑓(𝑓:𝐵𝐴 ∧ ∀𝑥𝐵 𝑥 = (𝐹‘(𝑓𝑥))))
7 simpll 766 . . . . 5 (((𝐴AC 𝐵𝐹:𝐴onto𝐵) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑥𝐵 𝑥 = (𝐹‘(𝑓𝑥)))) → 𝐴AC 𝐵)
8 acnrcl 10037 . . . . 5 (𝐴AC 𝐵𝐵 ∈ V)
97, 8syl 17 . . . 4 (((𝐴AC 𝐵𝐹:𝐴onto𝐵) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑥𝐵 𝑥 = (𝐹‘(𝑓𝑥)))) → 𝐵 ∈ V)
10 simprl 770 . . . . 5 (((𝐴AC 𝐵𝐹:𝐴onto𝐵) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑥𝐵 𝑥 = (𝐹‘(𝑓𝑥)))) → 𝑓:𝐵𝐴)
11 fveq2 6892 . . . . . . 7 ((𝑓𝑦) = (𝑓𝑧) → (𝐹‘(𝑓𝑦)) = (𝐹‘(𝑓𝑧)))
12 simprr 772 . . . . . . . 8 (((𝐴AC 𝐵𝐹:𝐴onto𝐵) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑥𝐵 𝑥 = (𝐹‘(𝑓𝑥)))) → ∀𝑥𝐵 𝑥 = (𝐹‘(𝑓𝑥)))
13 id 22 . . . . . . . . . . . 12 (𝑥 = 𝑦𝑥 = 𝑦)
14 2fveq3 6897 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (𝐹‘(𝑓𝑥)) = (𝐹‘(𝑓𝑦)))
1513, 14eqeq12d 2749 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝑥 = (𝐹‘(𝑓𝑥)) ↔ 𝑦 = (𝐹‘(𝑓𝑦))))
1615rspccva 3612 . . . . . . . . . 10 ((∀𝑥𝐵 𝑥 = (𝐹‘(𝑓𝑥)) ∧ 𝑦𝐵) → 𝑦 = (𝐹‘(𝑓𝑦)))
17 id 22 . . . . . . . . . . . 12 (𝑥 = 𝑧𝑥 = 𝑧)
18 2fveq3 6897 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (𝐹‘(𝑓𝑥)) = (𝐹‘(𝑓𝑧)))
1917, 18eqeq12d 2749 . . . . . . . . . . 11 (𝑥 = 𝑧 → (𝑥 = (𝐹‘(𝑓𝑥)) ↔ 𝑧 = (𝐹‘(𝑓𝑧))))
2019rspccva 3612 . . . . . . . . . 10 ((∀𝑥𝐵 𝑥 = (𝐹‘(𝑓𝑥)) ∧ 𝑧𝐵) → 𝑧 = (𝐹‘(𝑓𝑧)))
2116, 20eqeqan12d 2747 . . . . . . . . 9 (((∀𝑥𝐵 𝑥 = (𝐹‘(𝑓𝑥)) ∧ 𝑦𝐵) ∧ (∀𝑥𝐵 𝑥 = (𝐹‘(𝑓𝑥)) ∧ 𝑧𝐵)) → (𝑦 = 𝑧 ↔ (𝐹‘(𝑓𝑦)) = (𝐹‘(𝑓𝑧))))
2221anandis 677 . . . . . . . 8 ((∀𝑥𝐵 𝑥 = (𝐹‘(𝑓𝑥)) ∧ (𝑦𝐵𝑧𝐵)) → (𝑦 = 𝑧 ↔ (𝐹‘(𝑓𝑦)) = (𝐹‘(𝑓𝑧))))
2312, 22sylan 581 . . . . . . 7 ((((𝐴AC 𝐵𝐹:𝐴onto𝐵) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑥𝐵 𝑥 = (𝐹‘(𝑓𝑥)))) ∧ (𝑦𝐵𝑧𝐵)) → (𝑦 = 𝑧 ↔ (𝐹‘(𝑓𝑦)) = (𝐹‘(𝑓𝑧))))
2411, 23imbitrrid 245 . . . . . 6 ((((𝐴AC 𝐵𝐹:𝐴onto𝐵) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑥𝐵 𝑥 = (𝐹‘(𝑓𝑥)))) ∧ (𝑦𝐵𝑧𝐵)) → ((𝑓𝑦) = (𝑓𝑧) → 𝑦 = 𝑧))
2524ralrimivva 3201 . . . . 5 (((𝐴AC 𝐵𝐹:𝐴onto𝐵) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑥𝐵 𝑥 = (𝐹‘(𝑓𝑥)))) → ∀𝑦𝐵𝑧𝐵 ((𝑓𝑦) = (𝑓𝑧) → 𝑦 = 𝑧))
26 dff13 7254 . . . . 5 (𝑓:𝐵1-1𝐴 ↔ (𝑓:𝐵𝐴 ∧ ∀𝑦𝐵𝑧𝐵 ((𝑓𝑦) = (𝑓𝑧) → 𝑦 = 𝑧)))
2710, 25, 26sylanbrc 584 . . . 4 (((𝐴AC 𝐵𝐹:𝐴onto𝐵) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑥𝐵 𝑥 = (𝐹‘(𝑓𝑥)))) → 𝑓:𝐵1-1𝐴)
28 f1dom2g 8965 . . . 4 ((𝐵 ∈ V ∧ 𝐴AC 𝐵𝑓:𝐵1-1𝐴) → 𝐵𝐴)
299, 7, 27, 28syl3anc 1372 . . 3 (((𝐴AC 𝐵𝐹:𝐴onto𝐵) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑥𝐵 𝑥 = (𝐹‘(𝑓𝑥)))) → 𝐵𝐴)
306, 29exlimddv 1939 . 2 ((𝐴AC 𝐵𝐹:𝐴onto𝐵) → 𝐵𝐴)
3130ex 414 1 (𝐴AC 𝐵 → (𝐹:𝐴onto𝐵𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wex 1782  wcel 2107  wral 3062  wrex 3071  Vcvv 3475   class class class wbr 5149  wf 6540  1-1wf1 6541  ontowfo 6542  cfv 6544  cdom 8937  AC wacn 9933
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-map 8822  df-dom 8941  df-acn 9937
This theorem is referenced by:  fodomnum  10052  iundomg  10536
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