Theorem acndom 9738

Description: A set with long choice sequences also has shorter choice sequences, where "shorter" here means the new index set is dominated by the old index set. (Contributed by Mario Carneiro, 31-Aug-2015.)

Assertion

Ref	Expression
acndom	⊢ (𝐴 ≼ 𝐵 → (𝑋 ∈ AC 𝐵 → 𝑋 ∈ AC 𝐴))

Proof of Theorem acndom

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

Step	Hyp	Ref	Expression
1		brdomi 8704	. 2 ⊢ (𝐴 ≼ 𝐵 → ∃𝑓 𝑓:𝐴–1-1→𝐵)
2		neq0 4276	. . . . 5 ⊢ (¬ 𝐴 = ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
3		simpl3 1191	. . . . . . . . . . 11 ⊢ (((𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑋 ∈ AC 𝐵) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → 𝑋 ∈ AC 𝐵)
4		elmapi 8595	. . . . . . . . . . . . . . 15 ⊢ (𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴) → 𝑔:𝐴⟶(𝒫 𝑋 ∖ {∅}))
5	4	ad2antlr 723	. . . . . . . . . . . . . 14 ⊢ ((((𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑋 ∈ AC 𝐵) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ 𝑦 ∈ 𝐵) → 𝑔:𝐴⟶(𝒫 𝑋 ∖ {∅}))
6		simpll1 1210	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑋 ∈ AC 𝐵) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ 𝑦 ∈ 𝐵) → 𝑓:𝐴–1-1→𝐵)
7		f1f1orn 6711	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓:𝐴–1-1→𝐵 → 𝑓:𝐴–1-1-onto→ran 𝑓)
8		f1ocnv 6712	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓:𝐴–1-1-onto→ran 𝑓 → ◡𝑓:ran 𝑓–1-1-onto→𝐴)
9		f1of 6700	. . . . . . . . . . . . . . . . 17 ⊢ (◡𝑓:ran 𝑓–1-1-onto→𝐴 → ◡𝑓:ran 𝑓⟶𝐴)
10	6, 7, 8, 9	4syl 19	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑋 ∈ AC 𝐵) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ 𝑦 ∈ 𝐵) → ◡𝑓:ran 𝑓⟶𝐴)
11	10	ffvelrnda 6943	. . . . . . . . . . . . . . 15 ⊢ (((((𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑋 ∈ AC 𝐵) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ 𝑦 ∈ 𝐵) ∧ 𝑦 ∈ ran 𝑓) → (◡𝑓‘𝑦) ∈ 𝐴)
12		simpl2 1190	. . . . . . . . . . . . . . . 16 ⊢ (((𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑋 ∈ AC 𝐵) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → 𝑥 ∈ 𝐴)
13	12	ad2antrr 722	. . . . . . . . . . . . . . 15 ⊢ (((((𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑋 ∈ AC 𝐵) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ 𝑦 ∈ 𝐵) ∧ ¬ 𝑦 ∈ ran 𝑓) → 𝑥 ∈ 𝐴)
14	11, 13	ifclda 4491	. . . . . . . . . . . . . 14 ⊢ ((((𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑋 ∈ AC 𝐵) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ 𝑦 ∈ 𝐵) → if(𝑦 ∈ ran 𝑓, (◡𝑓‘𝑦), 𝑥) ∈ 𝐴)
15	5, 14	ffvelrnd 6944	. . . . . . . . . . . . 13 ⊢ ((((𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑋 ∈ AC 𝐵) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ 𝑦 ∈ 𝐵) → (𝑔‘if(𝑦 ∈ ran 𝑓, (◡𝑓‘𝑦), 𝑥)) ∈ (𝒫 𝑋 ∖ {∅}))
16		eldifsn 4717	. . . . . . . . . . . . . 14 ⊢ ((𝑔‘if(𝑦 ∈ ran 𝑓, (◡𝑓‘𝑦), 𝑥)) ∈ (𝒫 𝑋 ∖ {∅}) ↔ ((𝑔‘if(𝑦 ∈ ran 𝑓, (◡𝑓‘𝑦), 𝑥)) ∈ 𝒫 𝑋 ∧ (𝑔‘if(𝑦 ∈ ran 𝑓, (◡𝑓‘𝑦), 𝑥)) ≠ ∅))
17		elpwi 4539	. . . . . . . . . . . . . . 15 ⊢ ((𝑔‘if(𝑦 ∈ ran 𝑓, (◡𝑓‘𝑦), 𝑥)) ∈ 𝒫 𝑋 → (𝑔‘if(𝑦 ∈ ran 𝑓, (◡𝑓‘𝑦), 𝑥)) ⊆ 𝑋)
18	17	anim1i 614	. . . . . . . . . . . . . 14 ⊢ (((𝑔‘if(𝑦 ∈ ran 𝑓, (◡𝑓‘𝑦), 𝑥)) ∈ 𝒫 𝑋 ∧ (𝑔‘if(𝑦 ∈ ran 𝑓, (◡𝑓‘𝑦), 𝑥)) ≠ ∅) → ((𝑔‘if(𝑦 ∈ ran 𝑓, (◡𝑓‘𝑦), 𝑥)) ⊆ 𝑋 ∧ (𝑔‘if(𝑦 ∈ ran 𝑓, (◡𝑓‘𝑦), 𝑥)) ≠ ∅))
19	16, 18	sylbi 216	. . . . . . . . . . . . 13 ⊢ ((𝑔‘if(𝑦 ∈ ran 𝑓, (◡𝑓‘𝑦), 𝑥)) ∈ (𝒫 𝑋 ∖ {∅}) → ((𝑔‘if(𝑦 ∈ ran 𝑓, (◡𝑓‘𝑦), 𝑥)) ⊆ 𝑋 ∧ (𝑔‘if(𝑦 ∈ ran 𝑓, (◡𝑓‘𝑦), 𝑥)) ≠ ∅))
20	15, 19	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑋 ∈ AC 𝐵) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ 𝑦 ∈ 𝐵) → ((𝑔‘if(𝑦 ∈ ran 𝑓, (◡𝑓‘𝑦), 𝑥)) ⊆ 𝑋 ∧ (𝑔‘if(𝑦 ∈ ran 𝑓, (◡𝑓‘𝑦), 𝑥)) ≠ ∅))
21	20	ralrimiva 3107	. . . . . . . . . . 11 ⊢ (((𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑋 ∈ AC 𝐵) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → ∀𝑦 ∈ 𝐵 ((𝑔‘if(𝑦 ∈ ran 𝑓, (◡𝑓‘𝑦), 𝑥)) ⊆ 𝑋 ∧ (𝑔‘if(𝑦 ∈ ran 𝑓, (◡𝑓‘𝑦), 𝑥)) ≠ ∅))
22		acni2 9733	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ AC 𝐵 ∧ ∀𝑦 ∈ 𝐵 ((𝑔‘if(𝑦 ∈ ran 𝑓, (◡𝑓‘𝑦), 𝑥)) ⊆ 𝑋 ∧ (𝑔‘if(𝑦 ∈ ran 𝑓, (◡𝑓‘𝑦), 𝑥)) ≠ ∅)) → ∃𝑘(𝑘:𝐵⟶𝑋 ∧ ∀𝑦 ∈ 𝐵 (𝑘‘𝑦) ∈ (𝑔‘if(𝑦 ∈ ran 𝑓, (◡𝑓‘𝑦), 𝑥))))
23	3, 21, 22	syl2anc 583	. . . . . . . . . 10 ⊢ (((𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑋 ∈ AC 𝐵) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → ∃𝑘(𝑘:𝐵⟶𝑋 ∧ ∀𝑦 ∈ 𝐵 (𝑘‘𝑦) ∈ (𝑔‘if(𝑦 ∈ ran 𝑓, (◡𝑓‘𝑦), 𝑥))))
24		f1dm 6658	. . . . . . . . . . . . . 14 ⊢ (𝑓:𝐴–1-1→𝐵 → dom 𝑓 = 𝐴)
25		vex 3426	. . . . . . . . . . . . . . 15 ⊢ 𝑓 ∈ V
26	25	dmex 7732	. . . . . . . . . . . . . 14 ⊢ dom 𝑓 ∈ V
27	24, 26	eqeltrrdi 2848	. . . . . . . . . . . . 13 ⊢ (𝑓:𝐴–1-1→𝐵 → 𝐴 ∈ V)
28	27	3ad2ant1 1131	. . . . . . . . . . . 12 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑋 ∈ AC 𝐵) → 𝐴 ∈ V)
29	28	ad2antrr 722	. . . . . . . . . . 11 ⊢ ((((𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑋 ∈ AC 𝐵) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ (𝑘:𝐵⟶𝑋 ∧ ∀𝑦 ∈ 𝐵 (𝑘‘𝑦) ∈ (𝑔‘if(𝑦 ∈ ran 𝑓, (◡𝑓‘𝑦), 𝑥)))) → 𝐴 ∈ V)
30		simpll1 1210	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑋 ∈ AC 𝐵) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ 𝑘:𝐵⟶𝑋) → 𝑓:𝐴–1-1→𝐵)
31		f1f 6654	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:𝐴–1-1→𝐵 → 𝑓:𝐴⟶𝐵)
32		frn 6591	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:𝐴⟶𝐵 → ran 𝑓 ⊆ 𝐵)
33		ssralv 3983	. . . . . . . . . . . . . . . 16 ⊢ (ran 𝑓 ⊆ 𝐵 → (∀𝑦 ∈ 𝐵 (𝑘‘𝑦) ∈ (𝑔‘if(𝑦 ∈ ran 𝑓, (◡𝑓‘𝑦), 𝑥)) → ∀𝑦 ∈ ran 𝑓(𝑘‘𝑦) ∈ (𝑔‘if(𝑦 ∈ ran 𝑓, (◡𝑓‘𝑦), 𝑥))))
34	30, 31, 32, 33	4syl 19	. . . . . . . . . . . . . . 15 ⊢ ((((𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑋 ∈ AC 𝐵) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ 𝑘:𝐵⟶𝑋) → (∀𝑦 ∈ 𝐵 (𝑘‘𝑦) ∈ (𝑔‘if(𝑦 ∈ ran 𝑓, (◡𝑓‘𝑦), 𝑥)) → ∀𝑦 ∈ ran 𝑓(𝑘‘𝑦) ∈ (𝑔‘if(𝑦 ∈ ran 𝑓, (◡𝑓‘𝑦), 𝑥))))
35		iftrue 4462	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ran 𝑓 → if(𝑦 ∈ ran 𝑓, (◡𝑓‘𝑦), 𝑥) = (◡𝑓‘𝑦))
36	35	fveq2d 6760	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ran 𝑓 → (𝑔‘if(𝑦 ∈ ran 𝑓, (◡𝑓‘𝑦), 𝑥)) = (𝑔‘(◡𝑓‘𝑦)))
37	36	eleq2d 2824	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ran 𝑓 → ((𝑘‘𝑦) ∈ (𝑔‘if(𝑦 ∈ ran 𝑓, (◡𝑓‘𝑦), 𝑥)) ↔ (𝑘‘𝑦) ∈ (𝑔‘(◡𝑓‘𝑦))))
38	37	ralbiia 3089	. . . . . . . . . . . . . . 15 ⊢ (∀𝑦 ∈ ran 𝑓(𝑘‘𝑦) ∈ (𝑔‘if(𝑦 ∈ ran 𝑓, (◡𝑓‘𝑦), 𝑥)) ↔ ∀𝑦 ∈ ran 𝑓(𝑘‘𝑦) ∈ (𝑔‘(◡𝑓‘𝑦)))
39	34, 38	syl6ib 250	. . . . . . . . . . . . . 14 ⊢ ((((𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑋 ∈ AC 𝐵) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ 𝑘:𝐵⟶𝑋) → (∀𝑦 ∈ 𝐵 (𝑘‘𝑦) ∈ (𝑔‘if(𝑦 ∈ ran 𝑓, (◡𝑓‘𝑦), 𝑥)) → ∀𝑦 ∈ ran 𝑓(𝑘‘𝑦) ∈ (𝑔‘(◡𝑓‘𝑦))))
40		f1fn 6655	. . . . . . . . . . . . . . 15 ⊢ (𝑓:𝐴–1-1→𝐵 → 𝑓 Fn 𝐴)
41		fveq2 6756	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = (𝑓‘𝑧) → (𝑘‘𝑦) = (𝑘‘(𝑓‘𝑧)))
42		2fveq3 6761	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = (𝑓‘𝑧) → (𝑔‘(◡𝑓‘𝑦)) = (𝑔‘(◡𝑓‘(𝑓‘𝑧))))
43	41, 42	eleq12d 2833	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = (𝑓‘𝑧) → ((𝑘‘𝑦) ∈ (𝑔‘(◡𝑓‘𝑦)) ↔ (𝑘‘(𝑓‘𝑧)) ∈ (𝑔‘(◡𝑓‘(𝑓‘𝑧)))))
44	43	ralrn 6946	. . . . . . . . . . . . . . 15 ⊢ (𝑓 Fn 𝐴 → (∀𝑦 ∈ ran 𝑓(𝑘‘𝑦) ∈ (𝑔‘(◡𝑓‘𝑦)) ↔ ∀𝑧 ∈ 𝐴 (𝑘‘(𝑓‘𝑧)) ∈ (𝑔‘(◡𝑓‘(𝑓‘𝑧)))))
45	30, 40, 44	3syl 18	. . . . . . . . . . . . . 14 ⊢ ((((𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑋 ∈ AC 𝐵) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ 𝑘:𝐵⟶𝑋) → (∀𝑦 ∈ ran 𝑓(𝑘‘𝑦) ∈ (𝑔‘(◡𝑓‘𝑦)) ↔ ∀𝑧 ∈ 𝐴 (𝑘‘(𝑓‘𝑧)) ∈ (𝑔‘(◡𝑓‘(𝑓‘𝑧)))))
46	39, 45	sylibd 238	. . . . . . . . . . . . 13 ⊢ ((((𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑋 ∈ AC 𝐵) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ 𝑘:𝐵⟶𝑋) → (∀𝑦 ∈ 𝐵 (𝑘‘𝑦) ∈ (𝑔‘if(𝑦 ∈ ran 𝑓, (◡𝑓‘𝑦), 𝑥)) → ∀𝑧 ∈ 𝐴 (𝑘‘(𝑓‘𝑧)) ∈ (𝑔‘(◡𝑓‘(𝑓‘𝑧)))))
47	30, 7	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑋 ∈ AC 𝐵) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ 𝑘:𝐵⟶𝑋) → 𝑓:𝐴–1-1-onto→ran 𝑓)
48		f1ocnvfv1 7129	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:𝐴–1-1-onto→ran 𝑓 ∧ 𝑧 ∈ 𝐴) → (◡𝑓‘(𝑓‘𝑧)) = 𝑧)
49	47, 48	sylan 579	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑋 ∈ AC 𝐵) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ 𝑘:𝐵⟶𝑋) ∧ 𝑧 ∈ 𝐴) → (◡𝑓‘(𝑓‘𝑧)) = 𝑧)
50	49	fveq2d 6760	. . . . . . . . . . . . . . 15 ⊢ (((((𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑋 ∈ AC 𝐵) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ 𝑘:𝐵⟶𝑋) ∧ 𝑧 ∈ 𝐴) → (𝑔‘(◡𝑓‘(𝑓‘𝑧))) = (𝑔‘𝑧))
51	50	eleq2d 2824	. . . . . . . . . . . . . 14 ⊢ (((((𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑋 ∈ AC 𝐵) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ 𝑘:𝐵⟶𝑋) ∧ 𝑧 ∈ 𝐴) → ((𝑘‘(𝑓‘𝑧)) ∈ (𝑔‘(◡𝑓‘(𝑓‘𝑧))) ↔ (𝑘‘(𝑓‘𝑧)) ∈ (𝑔‘𝑧)))
52	51	ralbidva 3119	. . . . . . . . . . . . 13 ⊢ ((((𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑋 ∈ AC 𝐵) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ 𝑘:𝐵⟶𝑋) → (∀𝑧 ∈ 𝐴 (𝑘‘(𝑓‘𝑧)) ∈ (𝑔‘(◡𝑓‘(𝑓‘𝑧))) ↔ ∀𝑧 ∈ 𝐴 (𝑘‘(𝑓‘𝑧)) ∈ (𝑔‘𝑧)))
53	46, 52	sylibd 238	. . . . . . . . . . . 12 ⊢ ((((𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑋 ∈ AC 𝐵) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ 𝑘:𝐵⟶𝑋) → (∀𝑦 ∈ 𝐵 (𝑘‘𝑦) ∈ (𝑔‘if(𝑦 ∈ ran 𝑓, (◡𝑓‘𝑦), 𝑥)) → ∀𝑧 ∈ 𝐴 (𝑘‘(𝑓‘𝑧)) ∈ (𝑔‘𝑧)))
54	53	impr 454	. . . . . . . . . . 11 ⊢ ((((𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑋 ∈ AC 𝐵) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ (𝑘:𝐵⟶𝑋 ∧ ∀𝑦 ∈ 𝐵 (𝑘‘𝑦) ∈ (𝑔‘if(𝑦 ∈ ran 𝑓, (◡𝑓‘𝑦), 𝑥)))) → ∀𝑧 ∈ 𝐴 (𝑘‘(𝑓‘𝑧)) ∈ (𝑔‘𝑧))
55		acnlem 9735	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ V ∧ ∀𝑧 ∈ 𝐴 (𝑘‘(𝑓‘𝑧)) ∈ (𝑔‘𝑧)) → ∃ℎ∀𝑧 ∈ 𝐴 (ℎ‘𝑧) ∈ (𝑔‘𝑧))
56	29, 54, 55	syl2anc 583	. . . . . . . . . 10 ⊢ ((((𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑋 ∈ AC 𝐵) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ (𝑘:𝐵⟶𝑋 ∧ ∀𝑦 ∈ 𝐵 (𝑘‘𝑦) ∈ (𝑔‘if(𝑦 ∈ ran 𝑓, (◡𝑓‘𝑦), 𝑥)))) → ∃ℎ∀𝑧 ∈ 𝐴 (ℎ‘𝑧) ∈ (𝑔‘𝑧))
57	23, 56	exlimddv 1939	. . . . . . . . 9 ⊢ (((𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑋 ∈ AC 𝐵) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → ∃ℎ∀𝑧 ∈ 𝐴 (ℎ‘𝑧) ∈ (𝑔‘𝑧))
58	57	ralrimiva 3107	. . . . . . . 8 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑋 ∈ AC 𝐵) → ∀𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)∃ℎ∀𝑧 ∈ 𝐴 (ℎ‘𝑧) ∈ (𝑔‘𝑧))
59		elex 3440	. . . . . . . . . 10 ⊢ (𝑋 ∈ AC 𝐵 → 𝑋 ∈ V)
60		isacn 9731	. . . . . . . . . 10 ⊢ ((𝑋 ∈ V ∧ 𝐴 ∈ V) → (𝑋 ∈ AC 𝐴 ↔ ∀𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)∃ℎ∀𝑧 ∈ 𝐴 (ℎ‘𝑧) ∈ (𝑔‘𝑧)))
61	59, 27, 60	syl2anr 596	. . . . . . . . 9 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝑋 ∈ AC 𝐵) → (𝑋 ∈ AC 𝐴 ↔ ∀𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)∃ℎ∀𝑧 ∈ 𝐴 (ℎ‘𝑧) ∈ (𝑔‘𝑧)))
62	61	3adant2 1129	. . . . . . . 8 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑋 ∈ AC 𝐵) → (𝑋 ∈ AC 𝐴 ↔ ∀𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)∃ℎ∀𝑧 ∈ 𝐴 (ℎ‘𝑧) ∈ (𝑔‘𝑧)))
63	58, 62	mpbird 256	. . . . . . 7 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑋 ∈ AC 𝐵) → 𝑋 ∈ AC 𝐴)
64	63	3exp 1117	. . . . . 6 ⊢ (𝑓:𝐴–1-1→𝐵 → (𝑥 ∈ 𝐴 → (𝑋 ∈ AC 𝐵 → 𝑋 ∈ AC 𝐴)))
65	64	exlimdv 1937	. . . . 5 ⊢ (𝑓:𝐴–1-1→𝐵 → (∃𝑥 𝑥 ∈ 𝐴 → (𝑋 ∈ AC 𝐵 → 𝑋 ∈ AC 𝐴)))
66	2, 65	syl5bi 241	. . . 4 ⊢ (𝑓:𝐴–1-1→𝐵 → (¬ 𝐴 = ∅ → (𝑋 ∈ AC 𝐵 → 𝑋 ∈ AC 𝐴)))
67		acneq 9730	. . . . . . 7 ⊢ (𝐴 = ∅ → AC 𝐴 = AC ∅)
68		0fin 8916	. . . . . . . 8 ⊢ ∅ ∈ Fin
69		finacn 9737	. . . . . . . 8 ⊢ (∅ ∈ Fin → AC ∅ = V)
70	68, 69	ax-mp 5	. . . . . . 7 ⊢ AC ∅ = V
71	67, 70	eqtrdi 2795	. . . . . 6 ⊢ (𝐴 = ∅ → AC 𝐴 = V)
72	71	eleq2d 2824	. . . . 5 ⊢ (𝐴 = ∅ → (𝑋 ∈ AC 𝐴 ↔ 𝑋 ∈ V))
73	59, 72	syl5ibr 245	. . . 4 ⊢ (𝐴 = ∅ → (𝑋 ∈ AC 𝐵 → 𝑋 ∈ AC 𝐴))
74	66, 73	pm2.61d2 181	. . 3 ⊢ (𝑓:𝐴–1-1→𝐵 → (𝑋 ∈ AC 𝐵 → 𝑋 ∈ AC 𝐴))
75	74	exlimiv 1934	. 2 ⊢ (∃𝑓 𝑓:𝐴–1-1→𝐵 → (𝑋 ∈ AC 𝐵 → 𝑋 ∈ AC 𝐴))
76	1, 75	syl 17	1 ⊢ (𝐴 ≼ 𝐵 → (𝑋 ∈ AC 𝐵 → 𝑋 ∈ AC 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539  ∃wex 1783   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063  Vcvv 3422   ∖ cdif 3880   ⊆ wss 3883  ∅c0 4253  ifcif 4456  𝒫 cpw 4530  {csn 4558   class class class wbr 5070  ^◡ccnv 5579  dom cdm 5580  ran crn 5581   Fn wfn 6413  ⟶wf 6414  –1-1→wf1 6415  –1-1-onto→wf1o 6417  ‘cfv 6418  (class class class)co 7255   ↑_m cmap 8573   ≼ cdom 8689  Fincfn 8691  AC wacn 9627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-map 8575  df-en 8692  df-dom 8693  df-fin 8695  df-acn 9631

This theorem is referenced by:  acnnum  9739  acnen  9740  iunctb  10261