Theorem acndom 9462

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 8503	. 2 ⊢ (𝐴 ≼ 𝐵 → ∃𝑓 𝑓:𝐴–1-1→𝐵)
2		neq0 4259	. . . . 5 ⊢ (¬ 𝐴 = ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
3		simpl3 1190	. . . . . . . . . . 11 ⊢ (((𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑋 ∈ AC 𝐵) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → 𝑋 ∈ AC 𝐵)
4		elmapi 8411	. . . . . . . . . . . . . . 15 ⊢ (𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴) → 𝑔:𝐴⟶(𝒫 𝑋 ∖ {∅}))
5	4	ad2antlr 726	. . . . . . . . . . . . . 14 ⊢ ((((𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑋 ∈ AC 𝐵) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ 𝑦 ∈ 𝐵) → 𝑔:𝐴⟶(𝒫 𝑋 ∖ {∅}))
6		simpll1 1209	. . . . . . . . . . . . . . . . 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 6601	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓:𝐴–1-1→𝐵 → 𝑓:𝐴–1-1-onto→ran 𝑓)
8		f1ocnv 6602	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓:𝐴–1-1-onto→ran 𝑓 → ◡𝑓:ran 𝑓–1-1-onto→𝐴)
9		f1of 6590	. . . . . . . . . . . . . . . . 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 6828	. . . . . . . . . . . . . . 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 1189	. . . . . . . . . . . . . . . 16 ⊢ (((𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑋 ∈ AC 𝐵) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → 𝑥 ∈ 𝐴)
13	12	ad2antrr 725	. . . . . . . . . . . . . . 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 4459	. . . . . . . . . . . . . 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 6829	. . . . . . . . . . . . 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 4680	. . . . . . . . . . . . . 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 4506	. . . . . . . . . . . . . . 15 ⊢ ((𝑔‘if(𝑦 ∈ ran 𝑓, (◡𝑓‘𝑦), 𝑥)) ∈ 𝒫 𝑋 → (𝑔‘if(𝑦 ∈ ran 𝑓, (◡𝑓‘𝑦), 𝑥)) ⊆ 𝑋)
18	17	anim1i 617	. . . . . . . . . . . . . 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 220	. . . . . . . . . . . . 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 3149	. . . . . . . . . . 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 9457	. . . . . . . . . . 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 587	. . . . . . . . . 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 6553	. . . . . . . . . . . . . 14 ⊢ (𝑓:𝐴–1-1→𝐵 → dom 𝑓 = 𝐴)
25		vex 3444	. . . . . . . . . . . . . . 15 ⊢ 𝑓 ∈ V
26	25	dmex 7598	. . . . . . . . . . . . . 14 ⊢ dom 𝑓 ∈ V
27	24, 26	eqeltrrdi 2899	. . . . . . . . . . . . 13 ⊢ (𝑓:𝐴–1-1→𝐵 → 𝐴 ∈ V)
28	27	3ad2ant1 1130	. . . . . . . . . . . 12 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑋 ∈ AC 𝐵) → 𝐴 ∈ V)
29	28	ad2antrr 725	. . . . . . . . . . 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 1209	. . . . . . . . . . . . . . . 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 6549	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:𝐴–1-1→𝐵 → 𝑓:𝐴⟶𝐵)
32		frn 6493	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:𝐴⟶𝐵 → ran 𝑓 ⊆ 𝐵)
33		ssralv 3981	. . . . . . . . . . . . . . . 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 4431	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ran 𝑓 → if(𝑦 ∈ ran 𝑓, (◡𝑓‘𝑦), 𝑥) = (◡𝑓‘𝑦))
36	35	fveq2d 6649	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ran 𝑓 → (𝑔‘if(𝑦 ∈ ran 𝑓, (◡𝑓‘𝑦), 𝑥)) = (𝑔‘(◡𝑓‘𝑦)))
37	36	eleq2d 2875	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ran 𝑓 → ((𝑘‘𝑦) ∈ (𝑔‘if(𝑦 ∈ ran 𝑓, (◡𝑓‘𝑦), 𝑥)) ↔ (𝑘‘𝑦) ∈ (𝑔‘(◡𝑓‘𝑦))))
38	37	ralbiia 3132	. . . . . . . . . . . . . . 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 254	. . . . . . . . . . . . . 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 6550	. . . . . . . . . . . . . . 15 ⊢ (𝑓:𝐴–1-1→𝐵 → 𝑓 Fn 𝐴)
41		fveq2 6645	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = (𝑓‘𝑧) → (𝑘‘𝑦) = (𝑘‘(𝑓‘𝑧)))
42		2fveq3 6650	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = (𝑓‘𝑧) → (𝑔‘(◡𝑓‘𝑦)) = (𝑔‘(◡𝑓‘(𝑓‘𝑧))))
43	41, 42	eleq12d 2884	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = (𝑓‘𝑧) → ((𝑘‘𝑦) ∈ (𝑔‘(◡𝑓‘𝑦)) ↔ (𝑘‘(𝑓‘𝑧)) ∈ (𝑔‘(◡𝑓‘(𝑓‘𝑧)))))
44	43	ralrn 6831	. . . . . . . . . . . . . . 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 242	. . . . . . . . . . . . 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 7011	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:𝐴–1-1-onto→ran 𝑓 ∧ 𝑧 ∈ 𝐴) → (◡𝑓‘(𝑓‘𝑧)) = 𝑧)
49	47, 48	sylan 583	. . . . . . . . . . . . . . . 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 6649	. . . . . . . . . . . . . . 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 2875	. . . . . . . . . . . . . 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 3161	. . . . . . . . . . . . 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 242	. . . . . . . . . . . 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 458	. . . . . . . . . . 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 9459	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ V ∧ ∀𝑧 ∈ 𝐴 (𝑘‘(𝑓‘𝑧)) ∈ (𝑔‘𝑧)) → ∃ℎ∀𝑧 ∈ 𝐴 (ℎ‘𝑧) ∈ (𝑔‘𝑧))
56	29, 54, 55	syl2anc 587	. . . . . . . . . 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 1936	. . . . . . . . 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 3149	. . . . . . . 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 3459	. . . . . . . . . 10 ⊢ (𝑋 ∈ AC 𝐵 → 𝑋 ∈ V)
60		isacn 9455	. . . . . . . . . 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 599	. . . . . . . . 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 1128	. . . . . . . 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 260	. . . . . . 7 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑋 ∈ AC 𝐵) → 𝑋 ∈ AC 𝐴)
64	63	3exp 1116	. . . . . 6 ⊢ (𝑓:𝐴–1-1→𝐵 → (𝑥 ∈ 𝐴 → (𝑋 ∈ AC 𝐵 → 𝑋 ∈ AC 𝐴)))
65	64	exlimdv 1934	. . . . 5 ⊢ (𝑓:𝐴–1-1→𝐵 → (∃𝑥 𝑥 ∈ 𝐴 → (𝑋 ∈ AC 𝐵 → 𝑋 ∈ AC 𝐴)))
66	2, 65	syl5bi 245	. . . 4 ⊢ (𝑓:𝐴–1-1→𝐵 → (¬ 𝐴 = ∅ → (𝑋 ∈ AC 𝐵 → 𝑋 ∈ AC 𝐴)))
67		acneq 9454	. . . . . . 7 ⊢ (𝐴 = ∅ → AC 𝐴 = AC ∅)
68		0fin 8730	. . . . . . . 8 ⊢ ∅ ∈ Fin
69		finacn 9461	. . . . . . . 8 ⊢ (∅ ∈ Fin → AC ∅ = V)
70	68, 69	ax-mp 5	. . . . . . 7 ⊢ AC ∅ = V
71	67, 70	eqtrdi 2849	. . . . . 6 ⊢ (𝐴 = ∅ → AC 𝐴 = V)
72	71	eleq2d 2875	. . . . 5 ⊢ (𝐴 = ∅ → (𝑋 ∈ AC 𝐴 ↔ 𝑋 ∈ V))
73	59, 72	syl5ibr 249	. . . 4 ⊢ (𝐴 = ∅ → (𝑋 ∈ AC 𝐵 → 𝑋 ∈ AC 𝐴))
74	66, 73	pm2.61d2 184	. . 3 ⊢ (𝑓:𝐴–1-1→𝐵 → (𝑋 ∈ AC 𝐵 → 𝑋 ∈ AC 𝐴))
75	74	exlimiv 1931	. 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 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538  ∃wex 1781   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  Vcvv 3441   ∖ cdif 3878   ⊆ wss 3881  ∅c0 4243  ifcif 4425  𝒫 cpw 4497  {csn 4525   class class class wbr 5030  ^◡ccnv 5518  dom cdm 5519  ran crn 5520   Fn wfn 6319  ⟶wf 6320  –1-1→wf1 6321  –1-1-onto→wf1o 6323  ‘cfv 6324  (class class class)co 7135   ↑_m cmap 8389   ≼ cdom 8490  Fincfn 8492  AC wacn 9351

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-1o 8085  df-er 8272  df-map 8391  df-en 8493  df-dom 8494  df-fin 8496  df-acn 9355

This theorem is referenced by:  acnnum  9463  acnen  9464  iunctb  9985