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Theorem acndom 10043

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 8951	. 2 ⊢ (𝐴 ≼ 𝐵 → ∃𝑓 𝑓:𝐴–1-1→𝐵)
2		neq0 4345	. . . . 5 ⊢ (¬ 𝐴 = ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
3		simpl3 1194	. . . . . . . . . . 11 ⊢ (((𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑋 ∈ AC 𝐵) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑_m 𝐴)) → 𝑋 ∈ AC 𝐵)
4		elmapi 8840	. . . . . . . . . . . . . . 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 1213	. . . . . . . . . . . . . . . . 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 6842	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓:𝐴–1-1→𝐵 → 𝑓:𝐴–1-1-onto→ran 𝑓)
8		f1ocnv 6843	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓:𝐴–1-1-onto→ran 𝑓 → ^◡𝑓:ran 𝑓–1-1-onto→𝐴)
9		f1of 6831	. . . . . . . . . . . . . . . . 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	ffvelcdmda 7084	. . . . . . . . . . . . . . 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 1193	. . . . . . . . . . . . . . . 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 4563	. . . . . . . . . . . . . 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	ffvelcdmd 7085	. . . . . . . . . . . . 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 4790	. . . . . . . . . . . . . 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 4609	. . . . . . . . . . . . . . 15 ⊢ ((𝑔‘if(𝑦 ∈ ran 𝑓, (^◡𝑓‘𝑦), 𝑥)) ∈ 𝒫 𝑋 → (𝑔‘if(𝑦 ∈ ran 𝑓, (^◡𝑓‘𝑦), 𝑥)) ⊆ 𝑋)
18	17	anim1i 616	. . . . . . . . . . . . . 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 3147	. . . . . . . . . . 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 10038	. . . . . . . . . . 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 585	. . . . . . . . . 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 6789	. . . . . . . . . . . . . 14 ⊢ (𝑓:𝐴–1-1→𝐵 → dom 𝑓 = 𝐴)
25		vex 3479	. . . . . . . . . . . . . . 15 ⊢ 𝑓 ∈ V
26	25	dmex 7899	. . . . . . . . . . . . . 14 ⊢ dom 𝑓 ∈ V
27	24, 26	eqeltrrdi 2843	. . . . . . . . . . . . 13 ⊢ (𝑓:𝐴–1-1→𝐵 → 𝐴 ∈ V)
28	27	3ad2ant1 1134	. . . . . . . . . . . 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 1213	. . . . . . . . . . . . . . . 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 6785	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:𝐴–1-1→𝐵 → 𝑓:𝐴⟶𝐵)
32		frn 6722	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:𝐴⟶𝐵 → ran 𝑓 ⊆ 𝐵)
33		ssralv 4050	. . . . . . . . . . . . . . . 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 4534	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ran 𝑓 → if(𝑦 ∈ ran 𝑓, (^◡𝑓‘𝑦), 𝑥) = (^◡𝑓‘𝑦))
36	35	fveq2d 6893	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ran 𝑓 → (𝑔‘if(𝑦 ∈ ran 𝑓, (^◡𝑓‘𝑦), 𝑥)) = (𝑔‘(^◡𝑓‘𝑦)))
37	36	eleq2d 2820	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ran 𝑓 → ((𝑘‘𝑦) ∈ (𝑔‘if(𝑦 ∈ ran 𝑓, (^◡𝑓‘𝑦), 𝑥)) ↔ (𝑘‘𝑦) ∈ (𝑔‘(^◡𝑓‘𝑦))))
38	37	ralbiia 3092	. . . . . . . . . . . . . . 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	imbitrdi 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 6786	. . . . . . . . . . . . . . 15 ⊢ (𝑓:𝐴–1-1→𝐵 → 𝑓 Fn 𝐴)
41		fveq2 6889	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = (𝑓‘𝑧) → (𝑘‘𝑦) = (𝑘‘(𝑓‘𝑧)))
42		2fveq3 6894	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = (𝑓‘𝑧) → (𝑔‘(^◡𝑓‘𝑦)) = (𝑔‘(^◡𝑓‘(𝑓‘𝑧))))
43	41, 42	eleq12d 2828	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = (𝑓‘𝑧) → ((𝑘‘𝑦) ∈ (𝑔‘(^◡𝑓‘𝑦)) ↔ (𝑘‘(𝑓‘𝑧)) ∈ (𝑔‘(^◡𝑓‘(𝑓‘𝑧)))))
44	43	ralrn 7087	. . . . . . . . . . . . . . 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 7271	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:𝐴–1-1-onto→ran 𝑓 ∧ 𝑧 ∈ 𝐴) → (^◡𝑓‘(𝑓‘𝑧)) = 𝑧)
49	47, 48	sylan 581	. . . . . . . . . . . . . . . 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 6893	. . . . . . . . . . . . . . 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 2820	. . . . . . . . . . . . . 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 3176	. . . . . . . . . . . . 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 456	. . . . . . . . . . 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 10040	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ V ∧ ∀𝑧 ∈ 𝐴 (𝑘‘(𝑓‘𝑧)) ∈ (𝑔‘𝑧)) → ∃ℎ∀𝑧 ∈ 𝐴 (ℎ‘𝑧) ∈ (𝑔‘𝑧))
56	29, 54, 55	syl2anc 585	. . . . . . . . . 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 3147	. . . . . . . 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 3493	. . . . . . . . . 10 ⊢ (𝑋 ∈ AC 𝐵 → 𝑋 ∈ V)
60		isacn 10036	. . . . . . . . . 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 598	. . . . . . . . 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 1132	. . . . . . . 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 257	. . . . . . 7 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑋 ∈ AC 𝐵) → 𝑋 ∈ AC 𝐴)
64	63	3exp 1120	. . . . . 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	biimtrid 241	. . . 4 ⊢ (𝑓:𝐴–1-1→𝐵 → (¬ 𝐴 = ∅ → (𝑋 ∈ AC 𝐵 → 𝑋 ∈ AC 𝐴)))
67		acneq 10035	. . . . . . 7 ⊢ (𝐴 = ∅ → AC 𝐴 = AC ∅)
68		0fin 9168	. . . . . . . 8 ⊢ ∅ ∈ Fin
69		finacn 10042	. . . . . . . 8 ⊢ (∅ ∈ Fin → AC ∅ = V)
70	68, 69	ax-mp 5	. . . . . . 7 ⊢ AC ∅ = V
71	67, 70	eqtrdi 2789	. . . . . 6 ⊢ (𝐴 = ∅ → AC 𝐴 = V)
72	71	eleq2d 2820	. . . . 5 ⊢ (𝐴 = ∅ → (𝑋 ∈ AC 𝐴 ↔ 𝑋 ∈ V))
73	59, 72	imbitrrid 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))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∃wex 1782 ∈ wcel 2107 ≠ wne 2941 ∀wral 3062 Vcvv 3475 ∖ cdif 3945 ⊆ wss 3948 ∅c0 4322 ifcif 4528 𝒫 cpw 4602 {csn 4628 class class class wbr 5148 ^◡ccnv 5675 dom cdm 5676 ran crn 5677 Fn wfn 6536 ⟶wf 6537 –1-1→wf1 6538 –1-1-onto→wf1o 6540 ‘cfv 6541 (class class class)co 7406 ↑_m cmap 8817 ≼ cdom 8934 Fincfn 8936 AC wacn 9930

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-ov 7409 df-oprab 7410 df-mpo 7411 df-om 7853 df-1st 7972 df-2nd 7973 df-map 8819 df-en 8937 df-dom 8938 df-fin 8940 df-acn 9934

This theorem is referenced by: acnnum 10044 acnen 10045 iunctb 10566

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