Theorem fodomacn 9964

Description: A version of fodom 10431 that doesn't require the Axiom of Choice ax-ac 10367. 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	⊢ (𝐴 ∈ AC 𝐵 → (𝐹:𝐴–onto→𝐵 → 𝐵 ≼ 𝐴))

Proof of Theorem fodomacn

Dummy variables 𝑥 𝑓 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		foelrn 7050	. . . . 5 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐴 𝑥 = (𝐹‘𝑦))
2	1	ralrimiva 3126	. . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐴 𝑥 = (𝐹‘𝑦))
3		fveq2 6832	. . . . . 6 ⊢ (𝑦 = (𝑓‘𝑥) → (𝐹‘𝑦) = (𝐹‘(𝑓‘𝑥)))
4	3	eqeq2d 2745	. . . . 5 ⊢ (𝑦 = (𝑓‘𝑥) → (𝑥 = (𝐹‘𝑦) ↔ 𝑥 = (𝐹‘(𝑓‘𝑥))))
5	4	acni3 9955	. . . 4 ⊢ ((𝐴 ∈ AC 𝐵 ∧ ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐴 𝑥 = (𝐹‘𝑦)) → ∃𝑓(𝑓:𝐵⟶𝐴 ∧ ∀𝑥 ∈ 𝐵 𝑥 = (𝐹‘(𝑓‘𝑥))))
6	2, 5	sylan2 593	. . 3 ⊢ ((𝐴 ∈ AC 𝐵 ∧ 𝐹:𝐴–onto→𝐵) → ∃𝑓(𝑓:𝐵⟶𝐴 ∧ ∀𝑥 ∈ 𝐵 𝑥 = (𝐹‘(𝑓‘𝑥))))
7		simpll 766	. . . . 5 ⊢ (((𝐴 ∈ AC 𝐵 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑓:𝐵⟶𝐴 ∧ ∀𝑥 ∈ 𝐵 𝑥 = (𝐹‘(𝑓‘𝑥)))) → 𝐴 ∈ AC 𝐵)
8		acnrcl 9950	. . . . 5 ⊢ (𝐴 ∈ AC 𝐵 → 𝐵 ∈ V)
9	7, 8	syl 17	. . . 4 ⊢ (((𝐴 ∈ AC 𝐵 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑓:𝐵⟶𝐴 ∧ ∀𝑥 ∈ 𝐵 𝑥 = (𝐹‘(𝑓‘𝑥)))) → 𝐵 ∈ V)
10		simprl 770	. . . . 5 ⊢ (((𝐴 ∈ AC 𝐵 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑓:𝐵⟶𝐴 ∧ ∀𝑥 ∈ 𝐵 𝑥 = (𝐹‘(𝑓‘𝑥)))) → 𝑓:𝐵⟶𝐴)
11		fveq2 6832	. . . . . . 7 ⊢ ((𝑓‘𝑦) = (𝑓‘𝑧) → (𝐹‘(𝑓‘𝑦)) = (𝐹‘(𝑓‘𝑧)))
12		simprr 772	. . . . . . . 8 ⊢ (((𝐴 ∈ AC 𝐵 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑓:𝐵⟶𝐴 ∧ ∀𝑥 ∈ 𝐵 𝑥 = (𝐹‘(𝑓‘𝑥)))) → ∀𝑥 ∈ 𝐵 𝑥 = (𝐹‘(𝑓‘𝑥)))
13		id 22	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
14		2fveq3 6837	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (𝐹‘(𝑓‘𝑥)) = (𝐹‘(𝑓‘𝑦)))
15	13, 14	eqeq12d 2750	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝑥 = (𝐹‘(𝑓‘𝑥)) ↔ 𝑦 = (𝐹‘(𝑓‘𝑦))))
16	15	rspccva 3573	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ 𝐵 𝑥 = (𝐹‘(𝑓‘𝑥)) ∧ 𝑦 ∈ 𝐵) → 𝑦 = (𝐹‘(𝑓‘𝑦)))
17		id 22	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → 𝑥 = 𝑧)
18		2fveq3 6837	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝐹‘(𝑓‘𝑥)) = (𝐹‘(𝑓‘𝑧)))
19	17, 18	eqeq12d 2750	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (𝑥 = (𝐹‘(𝑓‘𝑥)) ↔ 𝑧 = (𝐹‘(𝑓‘𝑧))))
20	19	rspccva 3573	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ 𝐵 𝑥 = (𝐹‘(𝑓‘𝑥)) ∧ 𝑧 ∈ 𝐵) → 𝑧 = (𝐹‘(𝑓‘𝑧)))
21	16, 20	eqeqan12d 2748	. . . . . . . . 9 ⊢ (((∀𝑥 ∈ 𝐵 𝑥 = (𝐹‘(𝑓‘𝑥)) ∧ 𝑦 ∈ 𝐵) ∧ (∀𝑥 ∈ 𝐵 𝑥 = (𝐹‘(𝑓‘𝑥)) ∧ 𝑧 ∈ 𝐵)) → (𝑦 = 𝑧 ↔ (𝐹‘(𝑓‘𝑦)) = (𝐹‘(𝑓‘𝑧))))
22	21	anandis 678	. . . . . . . 8 ⊢ ((∀𝑥 ∈ 𝐵 𝑥 = (𝐹‘(𝑓‘𝑥)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑦 = 𝑧 ↔ (𝐹‘(𝑓‘𝑦)) = (𝐹‘(𝑓‘𝑧))))
23	12, 22	sylan 580	. . . . . . 7 ⊢ ((((𝐴 ∈ AC 𝐵 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑓:𝐵⟶𝐴 ∧ ∀𝑥 ∈ 𝐵 𝑥 = (𝐹‘(𝑓‘𝑥)))) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑦 = 𝑧 ↔ (𝐹‘(𝑓‘𝑦)) = (𝐹‘(𝑓‘𝑧))))
24	11, 23	imbitrrid 246	. . . . . 6 ⊢ ((((𝐴 ∈ AC 𝐵 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑓:𝐵⟶𝐴 ∧ ∀𝑥 ∈ 𝐵 𝑥 = (𝐹‘(𝑓‘𝑥)))) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑓‘𝑦) = (𝑓‘𝑧) → 𝑦 = 𝑧))
25	24	ralrimivva 3177	. . . . 5 ⊢ (((𝐴 ∈ AC 𝐵 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑓:𝐵⟶𝐴 ∧ ∀𝑥 ∈ 𝐵 𝑥 = (𝐹‘(𝑓‘𝑥)))) → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑓‘𝑦) = (𝑓‘𝑧) → 𝑦 = 𝑧))
26		dff13 7198	. . . . 5 ⊢ (𝑓:𝐵–1-1→𝐴 ↔ (𝑓:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑓‘𝑦) = (𝑓‘𝑧) → 𝑦 = 𝑧)))
27	10, 25, 26	sylanbrc 583	. . . 4 ⊢ (((𝐴 ∈ AC 𝐵 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑓:𝐵⟶𝐴 ∧ ∀𝑥 ∈ 𝐵 𝑥 = (𝐹‘(𝑓‘𝑥)))) → 𝑓:𝐵–1-1→𝐴)
28		f1dom2g 8904	. . . 4 ⊢ ((𝐵 ∈ V ∧ 𝐴 ∈ AC 𝐵 ∧ 𝑓:𝐵–1-1→𝐴) → 𝐵 ≼ 𝐴)
29	9, 7, 27, 28	syl3anc 1373	. . 3 ⊢ (((𝐴 ∈ AC 𝐵 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑓:𝐵⟶𝐴 ∧ ∀𝑥 ∈ 𝐵 𝑥 = (𝐹‘(𝑓‘𝑥)))) → 𝐵 ≼ 𝐴)
30	6, 29	exlimddv 1936	. 2 ⊢ ((𝐴 ∈ AC 𝐵 ∧ 𝐹:𝐴–onto→𝐵) → 𝐵 ≼ 𝐴)
31	30	ex 412	1 ⊢ (𝐴 ∈ AC 𝐵 → (𝐹:𝐴–onto→𝐵 → 𝐵 ≼ 𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2113  ∀wral 3049  ∃wrex 3058  Vcvv 3438   class class class wbr 5096  ⟶wf 6486  –1-1→wf1 6487  –onto→wfo 6488  ‘cfv 6490   ≼ cdom 8879  AC wacn 9848

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-fv 6498  df-ov 7359  df-oprab 7360  df-mpo 7361  df-map 8763  df-dom 8883  df-acn 9852

This theorem is referenced by:  fodomnum  9965  iundomg  10449