Theorem fodomacn 10136

Description: A version of fodom 10602 that doesn't require the Axiom of Choice ax-ac 10538. 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 7108	. . . . 5 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐴 𝑥 = (𝐹‘𝑦))
2	1	ralrimiva 3155	. . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐴 𝑥 = (𝐹‘𝑦))
3		fveq2 6886	. . . . . 6 ⊢ (𝑦 = (𝑓‘𝑥) → (𝐹‘𝑦) = (𝐹‘(𝑓‘𝑥)))
4	3	eqeq2d 2772	. . . . 5 ⊢ (𝑦 = (𝑓‘𝑥) → (𝑥 = (𝐹‘𝑦) ↔ 𝑥 = (𝐹‘(𝑓‘𝑥))))
5	4	acni3 10127	. . . 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 605	. . 3 ⊢ ((𝐴 ∈ AC 𝐵 ∧ 𝐹:𝐴–onto→𝐵) → ∃𝑓(𝑓:𝐵⟶𝐴 ∧ ∀𝑥 ∈ 𝐵 𝑥 = (𝐹‘(𝑓‘𝑥))))
7		simpll 779	. . . . 5 ⊢ (((𝐴 ∈ AC 𝐵 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑓:𝐵⟶𝐴 ∧ ∀𝑥 ∈ 𝐵 𝑥 = (𝐹‘(𝑓‘𝑥)))) → 𝐴 ∈ AC 𝐵)
8		acnrcl 10122	. . . . 5 ⊢ (𝐴 ∈ AC 𝐵 → 𝐵 ∈ V)
9	7, 8	syl 18	. . . 4 ⊢ (((𝐴 ∈ AC 𝐵 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑓:𝐵⟶𝐴 ∧ ∀𝑥 ∈ 𝐵 𝑥 = (𝐹‘(𝑓‘𝑥)))) → 𝐵 ∈ V)
10		simprl 783	. . . . 5 ⊢ (((𝐴 ∈ AC 𝐵 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑓:𝐵⟶𝐴 ∧ ∀𝑥 ∈ 𝐵 𝑥 = (𝐹‘(𝑓‘𝑥)))) → 𝑓:𝐵⟶𝐴)
11		fveq2 6886	. . . . . . 7 ⊢ ((𝑓‘𝑦) = (𝑓‘𝑧) → (𝐹‘(𝑓‘𝑦)) = (𝐹‘(𝑓‘𝑧)))
12		simprr 785	. . . . . . . 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 23	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
14		2fveq3 6891	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (𝐹‘(𝑓‘𝑥)) = (𝐹‘(𝑓‘𝑦)))
15	13, 14	eqeq12d 2777	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝑥 = (𝐹‘(𝑓‘𝑥)) ↔ 𝑦 = (𝐹‘(𝑓‘𝑦))))
16	15	rspccva 3576	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ 𝐵 𝑥 = (𝐹‘(𝑓‘𝑥)) ∧ 𝑦 ∈ 𝐵) → 𝑦 = (𝐹‘(𝑓‘𝑦)))
17		id 23	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → 𝑥 = 𝑧)
18		2fveq3 6891	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝐹‘(𝑓‘𝑥)) = (𝐹‘(𝑓‘𝑧)))
19	17, 18	eqeq12d 2777	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (𝑥 = (𝐹‘(𝑓‘𝑥)) ↔ 𝑧 = (𝐹‘(𝑓‘𝑧))))
20	19	rspccva 3576	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ 𝐵 𝑥 = (𝐹‘(𝑓‘𝑥)) ∧ 𝑧 ∈ 𝐵) → 𝑧 = (𝐹‘(𝑓‘𝑧)))
21	16, 20	eqeqan12d 2775	. . . . . . . . 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 691	. . . . . . . 8 ⊢ ((∀𝑥 ∈ 𝐵 𝑥 = (𝐹‘(𝑓‘𝑥)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑦 = 𝑧 ↔ (𝐹‘(𝑓‘𝑦)) = (𝐹‘(𝑓‘𝑧))))
23	12, 22	sylan 592	. . . . . . 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 249	. . . . . 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 3206	. . . . 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 7259	. . . . 5 ⊢ (𝑓:𝐵–1-1→𝐴 ↔ (𝑓:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑓‘𝑦) = (𝑓‘𝑧) → 𝑦 = 𝑧)))
27	10, 25, 26	sylanbrc 595	. . . 4 ⊢ (((𝐴 ∈ AC 𝐵 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑓:𝐵⟶𝐴 ∧ ∀𝑥 ∈ 𝐵 𝑥 = (𝐹‘(𝑓‘𝑥)))) → 𝑓:𝐵–1-1→𝐴)
28		f1dom2g 8997	. . . 4 ⊢ ((𝐵 ∈ V ∧ 𝐴 ∈ AC 𝐵 ∧ 𝑓:𝐵–1-1→𝐴) → 𝐵 ≼ 𝐴)
29	9, 7, 27, 28	syl3anc 1398	. . 3 ⊢ (((𝐴 ∈ AC 𝐵 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑓:𝐵⟶𝐴 ∧ ∀𝑥 ∈ 𝐵 𝑥 = (𝐹‘(𝑓‘𝑥)))) → 𝐵 ≼ 𝐴)
30	6, 29	exlimddv 1968	. 2 ⊢ ((𝐴 ∈ AC 𝐵 ∧ 𝐹:𝐴–onto→𝐵) → 𝐵 ≼ 𝐴)
31	30	ex 418	1 ⊢ (𝐴 ∈ AC 𝐵 → (𝐹:𝐴–onto→𝐵 → 𝐵 ≼ 𝐴))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  ∀wral 3077  ∃wrex 3087  Vcvv 3451   class class class wbr 5103  ⟶wf 6535  –1-1→wf1 6536  –onto→wfo 6537  ‘cfv 6539   ≼ cdom 8972  AC wacn 10020

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7752

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3900  df-un 3902  df-in 3904  df-ss 3915  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6495  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-fv 6547  df-ov 7424  df-oprab 7425  df-mpo 7426  df-map 8850  df-dom 8976  df-acn 10024

This theorem is used by:  fodomnum  10137  iundomg  10626