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Theorem fodomacn 9530
 Description: A version of fodom 9997 that doesn't require the Axiom of Choice ax-ac 9933. 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 6870 . . . . 5 ((𝐹:𝐴onto𝐵𝑥𝐵) → ∃𝑦𝐴 𝑥 = (𝐹𝑦))
21ralrimiva 3114 . . . 4 (𝐹:𝐴onto𝐵 → ∀𝑥𝐵𝑦𝐴 𝑥 = (𝐹𝑦))
3 fveq2 6664 . . . . . 6 (𝑦 = (𝑓𝑥) → (𝐹𝑦) = (𝐹‘(𝑓𝑥)))
43eqeq2d 2770 . . . . 5 (𝑦 = (𝑓𝑥) → (𝑥 = (𝐹𝑦) ↔ 𝑥 = (𝐹‘(𝑓𝑥))))
54acni3 9521 . . . 4 ((𝐴AC 𝐵 ∧ ∀𝑥𝐵𝑦𝐴 𝑥 = (𝐹𝑦)) → ∃𝑓(𝑓:𝐵𝐴 ∧ ∀𝑥𝐵 𝑥 = (𝐹‘(𝑓𝑥))))
62, 5sylan2 595 . . 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 9516 . . . . 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 6664 . . . . . . 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 6669 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (𝐹‘(𝑓𝑥)) = (𝐹‘(𝑓𝑦)))
1513, 14eqeq12d 2775 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝑥 = (𝐹‘(𝑓𝑥)) ↔ 𝑦 = (𝐹‘(𝑓𝑦))))
1615rspccva 3543 . . . . . . . . . 10 ((∀𝑥𝐵 𝑥 = (𝐹‘(𝑓𝑥)) ∧ 𝑦𝐵) → 𝑦 = (𝐹‘(𝑓𝑦)))
17 id 22 . . . . . . . . . . . 12 (𝑥 = 𝑧𝑥 = 𝑧)
18 2fveq3 6669 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (𝐹‘(𝑓𝑥)) = (𝐹‘(𝑓𝑧)))
1917, 18eqeq12d 2775 . . . . . . . . . . 11 (𝑥 = 𝑧 → (𝑥 = (𝐹‘(𝑓𝑥)) ↔ 𝑧 = (𝐹‘(𝑓𝑧))))
2019rspccva 3543 . . . . . . . . . 10 ((∀𝑥𝐵 𝑥 = (𝐹‘(𝑓𝑥)) ∧ 𝑧𝐵) → 𝑧 = (𝐹‘(𝑓𝑧)))
2116, 20eqeqan12d 2776 . . . . . . . . 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 583 . . . . . . 7 ((((𝐴AC 𝐵𝐹:𝐴onto𝐵) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑥𝐵 𝑥 = (𝐹‘(𝑓𝑥)))) ∧ (𝑦𝐵𝑧𝐵)) → (𝑦 = 𝑧 ↔ (𝐹‘(𝑓𝑦)) = (𝐹‘(𝑓𝑧))))
2411, 23syl5ibr 249 . . . . . 6 ((((𝐴AC 𝐵𝐹:𝐴onto𝐵) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑥𝐵 𝑥 = (𝐹‘(𝑓𝑥)))) ∧ (𝑦𝐵𝑧𝐵)) → ((𝑓𝑦) = (𝑓𝑧) → 𝑦 = 𝑧))
2524ralrimivva 3121 . . . . 5 (((𝐴AC 𝐵𝐹:𝐴onto𝐵) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑥𝐵 𝑥 = (𝐹‘(𝑓𝑥)))) → ∀𝑦𝐵𝑧𝐵 ((𝑓𝑦) = (𝑓𝑧) → 𝑦 = 𝑧))
26 dff13 7012 . . . . 5 (𝑓:𝐵1-1𝐴 ↔ (𝑓:𝐵𝐴 ∧ ∀𝑦𝐵𝑧𝐵 ((𝑓𝑦) = (𝑓𝑧) → 𝑦 = 𝑧)))
2710, 25, 26sylanbrc 586 . . . 4 (((𝐴AC 𝐵𝐹:𝐴onto𝐵) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑥𝐵 𝑥 = (𝐹‘(𝑓𝑥)))) → 𝑓:𝐵1-1𝐴)
28 f1dom2g 8559 . . . 4 ((𝐵 ∈ V ∧ 𝐴AC 𝐵𝑓:𝐵1-1𝐴) → 𝐵𝐴)
299, 7, 27, 28syl3anc 1369 . . 3 (((𝐴AC 𝐵𝐹:𝐴onto𝐵) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑥𝐵 𝑥 = (𝐹‘(𝑓𝑥)))) → 𝐵𝐴)
306, 29exlimddv 1937 . 2 ((𝐴AC 𝐵𝐹:𝐴onto𝐵) → 𝐵𝐴)
3130ex 416 1 (𝐴AC 𝐵 → (𝐹:𝐴onto𝐵𝐵𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1539  ∃wex 1782   ∈ wcel 2112  ∀wral 3071  ∃wrex 3072  Vcvv 3410   class class class wbr 5037  ⟶wf 6337  –1-1→wf1 6338  –onto→wfo 6339  ‘cfv 6341   ≼ cdom 8539  AC wacn 9414 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 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5174  ax-nul 5181  ax-pow 5239  ax-pr 5303  ax-un 7466 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-sbc 3700  df-csb 3809  df-dif 3864  df-un 3866  df-in 3868  df-ss 3878  df-nul 4229  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4803  df-br 5038  df-opab 5100  df-mpt 5118  df-id 5435  df-xp 5535  df-rel 5536  df-cnv 5537  df-co 5538  df-dm 5539  df-rn 5540  df-res 5541  df-ima 5542  df-iota 6300  df-fun 6343  df-fn 6344  df-f 6345  df-f1 6346  df-fo 6347  df-fv 6349  df-ov 7160  df-oprab 7161  df-mpo 7162  df-map 8425  df-dom 8543  df-acn 9418 This theorem is referenced by:  fodomnum  9531  iundomg  10015
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