Theorem fodomacn 10079

Description: A version of fodom 10546 that doesn't require the Axiom of Choice ax-ac 10482. 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 7117	. . . . 5 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐴 𝑥 = (𝐹‘𝑦))
2	1	ralrimiva 3143	. . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐴 𝑥 = (𝐹‘𝑦))
3		fveq2 6897	. . . . . 6 ⊢ (𝑦 = (𝑓‘𝑥) → (𝐹‘𝑦) = (𝐹‘(𝑓‘𝑥)))
4	3	eqeq2d 2739	. . . . 5 ⊢ (𝑦 = (𝑓‘𝑥) → (𝑥 = (𝐹‘𝑦) ↔ 𝑥 = (𝐹‘(𝑓‘𝑥))))
5	4	acni3 10070	. . . 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 592	. . 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 10065	. . . . 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 6897	. . . . . . 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 6902	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (𝐹‘(𝑓‘𝑥)) = (𝐹‘(𝑓‘𝑦)))
15	13, 14	eqeq12d 2744	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝑥 = (𝐹‘(𝑓‘𝑥)) ↔ 𝑦 = (𝐹‘(𝑓‘𝑦))))
16	15	rspccva 3608	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ 𝐵 𝑥 = (𝐹‘(𝑓‘𝑥)) ∧ 𝑦 ∈ 𝐵) → 𝑦 = (𝐹‘(𝑓‘𝑦)))
17		id 22	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → 𝑥 = 𝑧)
18		2fveq3 6902	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝐹‘(𝑓‘𝑥)) = (𝐹‘(𝑓‘𝑧)))
19	17, 18	eqeq12d 2744	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (𝑥 = (𝐹‘(𝑓‘𝑥)) ↔ 𝑧 = (𝐹‘(𝑓‘𝑧))))
20	19	rspccva 3608	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ 𝐵 𝑥 = (𝐹‘(𝑓‘𝑥)) ∧ 𝑧 ∈ 𝐵) → 𝑧 = (𝐹‘(𝑓‘𝑧)))
21	16, 20	eqeqan12d 2742	. . . . . . . . 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 677	. . . . . . . 8 ⊢ ((∀𝑥 ∈ 𝐵 𝑥 = (𝐹‘(𝑓‘𝑥)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑦 = 𝑧 ↔ (𝐹‘(𝑓‘𝑦)) = (𝐹‘(𝑓‘𝑧))))
23	12, 22	sylan 579	. . . . . . 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 245	. . . . . 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 3197	. . . . 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 7265	. . . . 5 ⊢ (𝑓:𝐵–1-1→𝐴 ↔ (𝑓:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑓‘𝑦) = (𝑓‘𝑧) → 𝑦 = 𝑧)))
27	10, 25, 26	sylanbrc 582	. . . 4 ⊢ (((𝐴 ∈ AC 𝐵 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑓:𝐵⟶𝐴 ∧ ∀𝑥 ∈ 𝐵 𝑥 = (𝐹‘(𝑓‘𝑥)))) → 𝑓:𝐵–1-1→𝐴)
28		f1dom2g 8989	. . . 4 ⊢ ((𝐵 ∈ V ∧ 𝐴 ∈ AC 𝐵 ∧ 𝑓:𝐵–1-1→𝐴) → 𝐵 ≼ 𝐴)
29	9, 7, 27, 28	syl3anc 1369	. . 3 ⊢ (((𝐴 ∈ AC 𝐵 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑓:𝐵⟶𝐴 ∧ ∀𝑥 ∈ 𝐵 𝑥 = (𝐹‘(𝑓‘𝑥)))) → 𝐵 ≼ 𝐴)
30	6, 29	exlimddv 1931	. 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 205   ∧ wa 395   = wceq 1534  ∃wex 1774   ∈ wcel 2099  ∀wral 3058  ∃wrex 3067  Vcvv 3471   class class class wbr 5148  ⟶wf 6544  –1-1→wf1 6545  –onto→wfo 6546  ‘cfv 6548   ≼ cdom 8961  AC wacn 9961

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-map 8846  df-dom 8965  df-acn 9965

This theorem is referenced by:  fodomnum  10080  iundomg  10564

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