Theorem fodomacn 8879

Description: A version of fodom 9344 that doesn't require the Axiom of Choice ax-ac 9281. 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 6378	. . . . 5 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐴 𝑥 = (𝐹‘𝑦))
2	1	ralrimiva 2966	. . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐴 𝑥 = (𝐹‘𝑦))
3		fveq2 6191	. . . . . 6 ⊢ (𝑦 = (𝑓‘𝑥) → (𝐹‘𝑦) = (𝐹‘(𝑓‘𝑥)))
4	3	eqeq2d 2632	. . . . 5 ⊢ (𝑦 = (𝑓‘𝑥) → (𝑥 = (𝐹‘𝑦) ↔ 𝑥 = (𝐹‘(𝑓‘𝑥))))
5	4	acni3 8870	. . . 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 491	. . 3 ⊢ ((𝐴 ∈ AC 𝐵 ∧ 𝐹:𝐴–onto→𝐵) → ∃𝑓(𝑓:𝐵⟶𝐴 ∧ ∀𝑥 ∈ 𝐵 𝑥 = (𝐹‘(𝑓‘𝑥))))
7		simpll 790	. . . . 5 ⊢ (((𝐴 ∈ AC 𝐵 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑓:𝐵⟶𝐴 ∧ ∀𝑥 ∈ 𝐵 𝑥 = (𝐹‘(𝑓‘𝑥)))) → 𝐴 ∈ AC 𝐵)
8		acnrcl 8865	. . . . 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 794	. . . . 5 ⊢ (((𝐴 ∈ AC 𝐵 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑓:𝐵⟶𝐴 ∧ ∀𝑥 ∈ 𝐵 𝑥 = (𝐹‘(𝑓‘𝑥)))) → 𝑓:𝐵⟶𝐴)
11		fveq2 6191	. . . . . . 7 ⊢ ((𝑓‘𝑦) = (𝑓‘𝑧) → (𝐹‘(𝑓‘𝑦)) = (𝐹‘(𝑓‘𝑧)))
12		simprr 796	. . . . . . . 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		fveq2 6191	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (𝑓‘𝑥) = (𝑓‘𝑦))
15	14	fveq2d 6195	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (𝐹‘(𝑓‘𝑥)) = (𝐹‘(𝑓‘𝑦)))
16	13, 15	eqeq12d 2637	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝑥 = (𝐹‘(𝑓‘𝑥)) ↔ 𝑦 = (𝐹‘(𝑓‘𝑦))))
17	16	rspccva 3308	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ 𝐵 𝑥 = (𝐹‘(𝑓‘𝑥)) ∧ 𝑦 ∈ 𝐵) → 𝑦 = (𝐹‘(𝑓‘𝑦)))
18		id 22	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → 𝑥 = 𝑧)
19		fveq2 6191	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → (𝑓‘𝑥) = (𝑓‘𝑧))
20	19	fveq2d 6195	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝐹‘(𝑓‘𝑥)) = (𝐹‘(𝑓‘𝑧)))
21	18, 20	eqeq12d 2637	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (𝑥 = (𝐹‘(𝑓‘𝑥)) ↔ 𝑧 = (𝐹‘(𝑓‘𝑧))))
22	21	rspccva 3308	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ 𝐵 𝑥 = (𝐹‘(𝑓‘𝑥)) ∧ 𝑧 ∈ 𝐵) → 𝑧 = (𝐹‘(𝑓‘𝑧)))
23	17, 22	eqeqan12d 2638	. . . . . . . . 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))))
24	23	anandis 873	. . . . . . . 8 ⊢ ((∀𝑥 ∈ 𝐵 𝑥 = (𝐹‘(𝑓‘𝑥)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑦 = 𝑧 ↔ (𝐹‘(𝑓‘𝑦)) = (𝐹‘(𝑓‘𝑧))))
25	12, 24	sylan 488	. . . . . . 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))))
26	11, 25	syl5ibr 236	. . . . . 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))
27	26	ralrimivva 2971	. . . . 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))
28		dff13 6512	. . . . 5 ⊢ (𝑓:𝐵–1-1→𝐴 ↔ (𝑓:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑓‘𝑦) = (𝑓‘𝑧) → 𝑦 = 𝑧)))
29	10, 27, 28	sylanbrc 698	. . . 4 ⊢ (((𝐴 ∈ AC 𝐵 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑓:𝐵⟶𝐴 ∧ ∀𝑥 ∈ 𝐵 𝑥 = (𝐹‘(𝑓‘𝑥)))) → 𝑓:𝐵–1-1→𝐴)
30		f1dom2g 7973	. . . 4 ⊢ ((𝐵 ∈ V ∧ 𝐴 ∈ AC 𝐵 ∧ 𝑓:𝐵–1-1→𝐴) → 𝐵 ≼ 𝐴)
31	9, 7, 29, 30	syl3anc 1326	. . 3 ⊢ (((𝐴 ∈ AC 𝐵 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑓:𝐵⟶𝐴 ∧ ∀𝑥 ∈ 𝐵 𝑥 = (𝐹‘(𝑓‘𝑥)))) → 𝐵 ≼ 𝐴)
32	6, 31	exlimddv 1863	. 2 ⊢ ((𝐴 ∈ AC 𝐵 ∧ 𝐹:𝐴–onto→𝐵) → 𝐵 ≼ 𝐴)
33	32	ex 450	1 ⊢ (𝐴 ∈ AC 𝐵 → (𝐹:𝐴–onto→𝐵 → 𝐵 ≼ 𝐴))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913  Vcvv 3200   class class class wbr 4653  ⟶wf 5884  –1-1→wf1 5885  –onto→wfo 5886  ‘cfv 5888   ≼ cdom 7953  AC wacn 8764

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-dom 7957  df-acn 8768

This theorem is referenced by:  fodomnum  8880  iundomg  9363