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Theorem fodomacn 8824
Description: A version of fodom 9289 that doesn't require the Axiom of Choice ax-ac 9226. 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 6335 . . . . 5 ((𝐹:𝐴onto𝐵𝑥𝐵) → ∃𝑦𝐴 𝑥 = (𝐹𝑦))
21ralrimiva 2965 . . . 4 (𝐹:𝐴onto𝐵 → ∀𝑥𝐵𝑦𝐴 𝑥 = (𝐹𝑦))
3 fveq2 6150 . . . . . 6 (𝑦 = (𝑓𝑥) → (𝐹𝑦) = (𝐹‘(𝑓𝑥)))
43eqeq2d 2636 . . . . 5 (𝑦 = (𝑓𝑥) → (𝑥 = (𝐹𝑦) ↔ 𝑥 = (𝐹‘(𝑓𝑥))))
54acni3 8815 . . . 4 ((𝐴AC 𝐵 ∧ ∀𝑥𝐵𝑦𝐴 𝑥 = (𝐹𝑦)) → ∃𝑓(𝑓:𝐵𝐴 ∧ ∀𝑥𝐵 𝑥 = (𝐹‘(𝑓𝑥))))
62, 5sylan2 491 . . 3 ((𝐴AC 𝐵𝐹:𝐴onto𝐵) → ∃𝑓(𝑓:𝐵𝐴 ∧ ∀𝑥𝐵 𝑥 = (𝐹‘(𝑓𝑥))))
7 simpll 789 . . . . 5 (((𝐴AC 𝐵𝐹:𝐴onto𝐵) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑥𝐵 𝑥 = (𝐹‘(𝑓𝑥)))) → 𝐴AC 𝐵)
8 acnrcl 8810 . . . . 5 (𝐴AC 𝐵𝐵 ∈ V)
97, 8syl 17 . . . 4 (((𝐴AC 𝐵𝐹:𝐴onto𝐵) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑥𝐵 𝑥 = (𝐹‘(𝑓𝑥)))) → 𝐵 ∈ V)
10 simprl 793 . . . . 5 (((𝐴AC 𝐵𝐹:𝐴onto𝐵) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑥𝐵 𝑥 = (𝐹‘(𝑓𝑥)))) → 𝑓:𝐵𝐴)
11 fveq2 6150 . . . . . . 7 ((𝑓𝑦) = (𝑓𝑧) → (𝐹‘(𝑓𝑦)) = (𝐹‘(𝑓𝑧)))
12 simprr 795 . . . . . . . 8 (((𝐴AC 𝐵𝐹:𝐴onto𝐵) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑥𝐵 𝑥 = (𝐹‘(𝑓𝑥)))) → ∀𝑥𝐵 𝑥 = (𝐹‘(𝑓𝑥)))
13 id 22 . . . . . . . . . . . 12 (𝑥 = 𝑦𝑥 = 𝑦)
14 fveq2 6150 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (𝑓𝑥) = (𝑓𝑦))
1514fveq2d 6154 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (𝐹‘(𝑓𝑥)) = (𝐹‘(𝑓𝑦)))
1613, 15eqeq12d 2641 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝑥 = (𝐹‘(𝑓𝑥)) ↔ 𝑦 = (𝐹‘(𝑓𝑦))))
1716rspccva 3299 . . . . . . . . . 10 ((∀𝑥𝐵 𝑥 = (𝐹‘(𝑓𝑥)) ∧ 𝑦𝐵) → 𝑦 = (𝐹‘(𝑓𝑦)))
18 id 22 . . . . . . . . . . . 12 (𝑥 = 𝑧𝑥 = 𝑧)
19 fveq2 6150 . . . . . . . . . . . . 13 (𝑥 = 𝑧 → (𝑓𝑥) = (𝑓𝑧))
2019fveq2d 6154 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (𝐹‘(𝑓𝑥)) = (𝐹‘(𝑓𝑧)))
2118, 20eqeq12d 2641 . . . . . . . . . . 11 (𝑥 = 𝑧 → (𝑥 = (𝐹‘(𝑓𝑥)) ↔ 𝑧 = (𝐹‘(𝑓𝑧))))
2221rspccva 3299 . . . . . . . . . 10 ((∀𝑥𝐵 𝑥 = (𝐹‘(𝑓𝑥)) ∧ 𝑧𝐵) → 𝑧 = (𝐹‘(𝑓𝑧)))
2317, 22eqeqan12d 2642 . . . . . . . . 9 (((∀𝑥𝐵 𝑥 = (𝐹‘(𝑓𝑥)) ∧ 𝑦𝐵) ∧ (∀𝑥𝐵 𝑥 = (𝐹‘(𝑓𝑥)) ∧ 𝑧𝐵)) → (𝑦 = 𝑧 ↔ (𝐹‘(𝑓𝑦)) = (𝐹‘(𝑓𝑧))))
2423anandis 872 . . . . . . . 8 ((∀𝑥𝐵 𝑥 = (𝐹‘(𝑓𝑥)) ∧ (𝑦𝐵𝑧𝐵)) → (𝑦 = 𝑧 ↔ (𝐹‘(𝑓𝑦)) = (𝐹‘(𝑓𝑧))))
2512, 24sylan 488 . . . . . . 7 ((((𝐴AC 𝐵𝐹:𝐴onto𝐵) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑥𝐵 𝑥 = (𝐹‘(𝑓𝑥)))) ∧ (𝑦𝐵𝑧𝐵)) → (𝑦 = 𝑧 ↔ (𝐹‘(𝑓𝑦)) = (𝐹‘(𝑓𝑧))))
2611, 25syl5ibr 236 . . . . . 6 ((((𝐴AC 𝐵𝐹:𝐴onto𝐵) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑥𝐵 𝑥 = (𝐹‘(𝑓𝑥)))) ∧ (𝑦𝐵𝑧𝐵)) → ((𝑓𝑦) = (𝑓𝑧) → 𝑦 = 𝑧))
2726ralrimivva 2970 . . . . 5 (((𝐴AC 𝐵𝐹:𝐴onto𝐵) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑥𝐵 𝑥 = (𝐹‘(𝑓𝑥)))) → ∀𝑦𝐵𝑧𝐵 ((𝑓𝑦) = (𝑓𝑧) → 𝑦 = 𝑧))
28 dff13 6467 . . . . 5 (𝑓:𝐵1-1𝐴 ↔ (𝑓:𝐵𝐴 ∧ ∀𝑦𝐵𝑧𝐵 ((𝑓𝑦) = (𝑓𝑧) → 𝑦 = 𝑧)))
2910, 27, 28sylanbrc 697 . . . 4 (((𝐴AC 𝐵𝐹:𝐴onto𝐵) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑥𝐵 𝑥 = (𝐹‘(𝑓𝑥)))) → 𝑓:𝐵1-1𝐴)
30 f1dom2g 7918 . . . 4 ((𝐵 ∈ V ∧ 𝐴AC 𝐵𝑓:𝐵1-1𝐴) → 𝐵𝐴)
319, 7, 29, 30syl3anc 1323 . . 3 (((𝐴AC 𝐵𝐹:𝐴onto𝐵) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑥𝐵 𝑥 = (𝐹‘(𝑓𝑥)))) → 𝐵𝐴)
326, 31exlimddv 1865 . 2 ((𝐴AC 𝐵𝐹:𝐴onto𝐵) → 𝐵𝐴)
3332ex 450 1 (𝐴AC 𝐵 → (𝐹:𝐴onto𝐵𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wex 1701  wcel 1992  wral 2912  wrex 2913  Vcvv 3191   class class class wbr 4618  wf 5846  1-1wf1 5847  ontowfo 5848  cfv 5850  cdom 7898  AC wacn 8709
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-map 7805  df-dom 7902  df-acn 8713
This theorem is referenced by:  fodomnum  8825  iundomg  9308
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