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Theorem fodomacn 9089
Description: A version of fodom 9556 that doesn't require the Axiom of Choice ax-ac 9493. 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 6542 . . . . 5 ((𝐹:𝐴onto𝐵𝑥𝐵) → ∃𝑦𝐴 𝑥 = (𝐹𝑦))
21ralrimiva 3104 . . . 4 (𝐹:𝐴onto𝐵 → ∀𝑥𝐵𝑦𝐴 𝑥 = (𝐹𝑦))
3 fveq2 6353 . . . . . 6 (𝑦 = (𝑓𝑥) → (𝐹𝑦) = (𝐹‘(𝑓𝑥)))
43eqeq2d 2770 . . . . 5 (𝑦 = (𝑓𝑥) → (𝑥 = (𝐹𝑦) ↔ 𝑥 = (𝐹‘(𝑓𝑥))))
54acni3 9080 . . . 4 ((𝐴AC 𝐵 ∧ ∀𝑥𝐵𝑦𝐴 𝑥 = (𝐹𝑦)) → ∃𝑓(𝑓:𝐵𝐴 ∧ ∀𝑥𝐵 𝑥 = (𝐹‘(𝑓𝑥))))
62, 5sylan2 492 . . 3 ((𝐴AC 𝐵𝐹:𝐴onto𝐵) → ∃𝑓(𝑓:𝐵𝐴 ∧ ∀𝑥𝐵 𝑥 = (𝐹‘(𝑓𝑥))))
7 simpll 807 . . . . 5 (((𝐴AC 𝐵𝐹:𝐴onto𝐵) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑥𝐵 𝑥 = (𝐹‘(𝑓𝑥)))) → 𝐴AC 𝐵)
8 acnrcl 9075 . . . . 5 (𝐴AC 𝐵𝐵 ∈ V)
97, 8syl 17 . . . 4 (((𝐴AC 𝐵𝐹:𝐴onto𝐵) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑥𝐵 𝑥 = (𝐹‘(𝑓𝑥)))) → 𝐵 ∈ V)
10 simprl 811 . . . . 5 (((𝐴AC 𝐵𝐹:𝐴onto𝐵) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑥𝐵 𝑥 = (𝐹‘(𝑓𝑥)))) → 𝑓:𝐵𝐴)
11 fveq2 6353 . . . . . . 7 ((𝑓𝑦) = (𝑓𝑧) → (𝐹‘(𝑓𝑦)) = (𝐹‘(𝑓𝑧)))
12 simprr 813 . . . . . . . 8 (((𝐴AC 𝐵𝐹:𝐴onto𝐵) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑥𝐵 𝑥 = (𝐹‘(𝑓𝑥)))) → ∀𝑥𝐵 𝑥 = (𝐹‘(𝑓𝑥)))
13 id 22 . . . . . . . . . . . 12 (𝑥 = 𝑦𝑥 = 𝑦)
14 fveq2 6353 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (𝑓𝑥) = (𝑓𝑦))
1514fveq2d 6357 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (𝐹‘(𝑓𝑥)) = (𝐹‘(𝑓𝑦)))
1613, 15eqeq12d 2775 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝑥 = (𝐹‘(𝑓𝑥)) ↔ 𝑦 = (𝐹‘(𝑓𝑦))))
1716rspccva 3448 . . . . . . . . . 10 ((∀𝑥𝐵 𝑥 = (𝐹‘(𝑓𝑥)) ∧ 𝑦𝐵) → 𝑦 = (𝐹‘(𝑓𝑦)))
18 id 22 . . . . . . . . . . . 12 (𝑥 = 𝑧𝑥 = 𝑧)
19 fveq2 6353 . . . . . . . . . . . . 13 (𝑥 = 𝑧 → (𝑓𝑥) = (𝑓𝑧))
2019fveq2d 6357 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (𝐹‘(𝑓𝑥)) = (𝐹‘(𝑓𝑧)))
2118, 20eqeq12d 2775 . . . . . . . . . . 11 (𝑥 = 𝑧 → (𝑥 = (𝐹‘(𝑓𝑥)) ↔ 𝑧 = (𝐹‘(𝑓𝑧))))
2221rspccva 3448 . . . . . . . . . 10 ((∀𝑥𝐵 𝑥 = (𝐹‘(𝑓𝑥)) ∧ 𝑧𝐵) → 𝑧 = (𝐹‘(𝑓𝑧)))
2317, 22eqeqan12d 2776 . . . . . . . . 9 (((∀𝑥𝐵 𝑥 = (𝐹‘(𝑓𝑥)) ∧ 𝑦𝐵) ∧ (∀𝑥𝐵 𝑥 = (𝐹‘(𝑓𝑥)) ∧ 𝑧𝐵)) → (𝑦 = 𝑧 ↔ (𝐹‘(𝑓𝑦)) = (𝐹‘(𝑓𝑧))))
2423anandis 908 . . . . . . . 8 ((∀𝑥𝐵 𝑥 = (𝐹‘(𝑓𝑥)) ∧ (𝑦𝐵𝑧𝐵)) → (𝑦 = 𝑧 ↔ (𝐹‘(𝑓𝑦)) = (𝐹‘(𝑓𝑧))))
2512, 24sylan 489 . . . . . . 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 3109 . . . . 5 (((𝐴AC 𝐵𝐹:𝐴onto𝐵) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑥𝐵 𝑥 = (𝐹‘(𝑓𝑥)))) → ∀𝑦𝐵𝑧𝐵 ((𝑓𝑦) = (𝑓𝑧) → 𝑦 = 𝑧))
28 dff13 6676 . . . . 5 (𝑓:𝐵1-1𝐴 ↔ (𝑓:𝐵𝐴 ∧ ∀𝑦𝐵𝑧𝐵 ((𝑓𝑦) = (𝑓𝑧) → 𝑦 = 𝑧)))
2910, 27, 28sylanbrc 701 . . . 4 (((𝐴AC 𝐵𝐹:𝐴onto𝐵) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑥𝐵 𝑥 = (𝐹‘(𝑓𝑥)))) → 𝑓:𝐵1-1𝐴)
30 f1dom2g 8141 . . . 4 ((𝐵 ∈ V ∧ 𝐴AC 𝐵𝑓:𝐵1-1𝐴) → 𝐵𝐴)
319, 7, 29, 30syl3anc 1477 . . 3 (((𝐴AC 𝐵𝐹:𝐴onto𝐵) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑥𝐵 𝑥 = (𝐹‘(𝑓𝑥)))) → 𝐵𝐴)
326, 31exlimddv 2012 . 2 ((𝐴AC 𝐵𝐹:𝐴onto𝐵) → 𝐵𝐴)
3332ex 449 1 (𝐴AC 𝐵 → (𝐹:𝐴onto𝐵𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wex 1853  wcel 2139  wral 3050  wrex 3051  Vcvv 3340   class class class wbr 4804  wf 6045  1-1wf1 6046  ontowfo 6047  cfv 6049  cdom 8121  AC wacn 8974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-map 8027  df-dom 8125  df-acn 8978
This theorem is referenced by:  fodomnum  9090  iundomg  9575
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