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Theorem fodomacn 9854
Description: A version of fodom 10321 that doesn't require the Axiom of Choice ax-ac 10257. 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 7010 . . . . 5 ((𝐹:𝐴onto𝐵𝑥𝐵) → ∃𝑦𝐴 𝑥 = (𝐹𝑦))
21ralrimiva 3140 . . . 4 (𝐹:𝐴onto𝐵 → ∀𝑥𝐵𝑦𝐴 𝑥 = (𝐹𝑦))
3 fveq2 6800 . . . . . 6 (𝑦 = (𝑓𝑥) → (𝐹𝑦) = (𝐹‘(𝑓𝑥)))
43eqeq2d 2747 . . . . 5 (𝑦 = (𝑓𝑥) → (𝑥 = (𝐹𝑦) ↔ 𝑥 = (𝐹‘(𝑓𝑥))))
54acni3 9845 . . . 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 765 . . . . 5 (((𝐴AC 𝐵𝐹:𝐴onto𝐵) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑥𝐵 𝑥 = (𝐹‘(𝑓𝑥)))) → 𝐴AC 𝐵)
8 acnrcl 9840 . . . . 5 (𝐴AC 𝐵𝐵 ∈ V)
97, 8syl 17 . . . 4 (((𝐴AC 𝐵𝐹:𝐴onto𝐵) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑥𝐵 𝑥 = (𝐹‘(𝑓𝑥)))) → 𝐵 ∈ V)
10 simprl 769 . . . . 5 (((𝐴AC 𝐵𝐹:𝐴onto𝐵) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑥𝐵 𝑥 = (𝐹‘(𝑓𝑥)))) → 𝑓:𝐵𝐴)
11 fveq2 6800 . . . . . . 7 ((𝑓𝑦) = (𝑓𝑧) → (𝐹‘(𝑓𝑦)) = (𝐹‘(𝑓𝑧)))
12 simprr 771 . . . . . . . 8 (((𝐴AC 𝐵𝐹:𝐴onto𝐵) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑥𝐵 𝑥 = (𝐹‘(𝑓𝑥)))) → ∀𝑥𝐵 𝑥 = (𝐹‘(𝑓𝑥)))
13 id 22 . . . . . . . . . . . 12 (𝑥 = 𝑦𝑥 = 𝑦)
14 2fveq3 6805 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (𝐹‘(𝑓𝑥)) = (𝐹‘(𝑓𝑦)))
1513, 14eqeq12d 2752 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝑥 = (𝐹‘(𝑓𝑥)) ↔ 𝑦 = (𝐹‘(𝑓𝑦))))
1615rspccva 3565 . . . . . . . . . 10 ((∀𝑥𝐵 𝑥 = (𝐹‘(𝑓𝑥)) ∧ 𝑦𝐵) → 𝑦 = (𝐹‘(𝑓𝑦)))
17 id 22 . . . . . . . . . . . 12 (𝑥 = 𝑧𝑥 = 𝑧)
18 2fveq3 6805 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (𝐹‘(𝑓𝑥)) = (𝐹‘(𝑓𝑧)))
1917, 18eqeq12d 2752 . . . . . . . . . . 11 (𝑥 = 𝑧 → (𝑥 = (𝐹‘(𝑓𝑥)) ↔ 𝑧 = (𝐹‘(𝑓𝑧))))
2019rspccva 3565 . . . . . . . . . 10 ((∀𝑥𝐵 𝑥 = (𝐹‘(𝑓𝑥)) ∧ 𝑧𝐵) → 𝑧 = (𝐹‘(𝑓𝑧)))
2116, 20eqeqan12d 2750 . . . . . . . . 9 (((∀𝑥𝐵 𝑥 = (𝐹‘(𝑓𝑥)) ∧ 𝑦𝐵) ∧ (∀𝑥𝐵 𝑥 = (𝐹‘(𝑓𝑥)) ∧ 𝑧𝐵)) → (𝑦 = 𝑧 ↔ (𝐹‘(𝑓𝑦)) = (𝐹‘(𝑓𝑧))))
2221anandis 676 . . . . . . . 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, 23syl5ibr 247 . . . . . 6 ((((𝐴AC 𝐵𝐹:𝐴onto𝐵) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑥𝐵 𝑥 = (𝐹‘(𝑓𝑥)))) ∧ (𝑦𝐵𝑧𝐵)) → ((𝑓𝑦) = (𝑓𝑧) → 𝑦 = 𝑧))
2524ralrimivva 3194 . . . . 5 (((𝐴AC 𝐵𝐹:𝐴onto𝐵) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑥𝐵 𝑥 = (𝐹‘(𝑓𝑥)))) → ∀𝑦𝐵𝑧𝐵 ((𝑓𝑦) = (𝑓𝑧) → 𝑦 = 𝑧))
26 dff13 7156 . . . . 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 8786 . . . 4 ((𝐵 ∈ V ∧ 𝐴AC 𝐵𝑓:𝐵1-1𝐴) → 𝐵𝐴)
299, 7, 27, 28syl3anc 1371 . . 3 (((𝐴AC 𝐵𝐹:𝐴onto𝐵) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑥𝐵 𝑥 = (𝐹‘(𝑓𝑥)))) → 𝐵𝐴)
306, 29exlimddv 1936 . 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 1539  wex 1779  wcel 2104  wral 3062  wrex 3071  Vcvv 3437   class class class wbr 5081  wf 6450  1-1wf1 6451  ontowfo 6452  cfv 6454  cdom 8758  AC wacn 9736
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-sep 5232  ax-nul 5239  ax-pow 5297  ax-pr 5361  ax-un 7616
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3287  df-v 3439  df-sbc 3722  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-br 5082  df-opab 5144  df-mpt 5165  df-id 5496  df-xp 5602  df-rel 5603  df-cnv 5604  df-co 5605  df-dm 5606  df-rn 5607  df-res 5608  df-ima 5609  df-iota 6406  df-fun 6456  df-fn 6457  df-f 6458  df-f1 6459  df-fo 6460  df-fv 6462  df-ov 7306  df-oprab 7307  df-mpo 7308  df-map 8644  df-dom 8762  df-acn 9740
This theorem is referenced by:  fodomnum  9855  iundomg  10339
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