Theorem acndom 8874

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 7966	. 2 ⊢ (𝐴 ≼ 𝐵 → ∃𝑓 𝑓:𝐴–1-1→𝐵)
2		neq0 3930	. . . . 5 ⊢ (¬ 𝐴 = ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
3		simpl3 1066	. . . . . . . . . . 11 ⊢ (((𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑋 ∈ AC 𝐵) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑𝑚 𝐴)) → 𝑋 ∈ AC 𝐵)
4		elmapi 7879	. . . . . . . . . . . . . . 15 ⊢ (𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑𝑚 𝐴) → 𝑔:𝐴⟶(𝒫 𝑋 ∖ {∅}))
5	4	ad2antlr 763	. . . . . . . . . . . . . 14 ⊢ ((((𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑋 ∈ AC 𝐵) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑𝑚 𝐴)) ∧ 𝑦 ∈ 𝐵) → 𝑔:𝐴⟶(𝒫 𝑋 ∖ {∅}))
6		simpll1 1100	. . . . . . . . . . . . . . . . 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 6148	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓:𝐴–1-1→𝐵 → 𝑓:𝐴–1-1-onto→ran 𝑓)
8		f1ocnv 6149	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓:𝐴–1-1-onto→ran 𝑓 → ◡𝑓:ran 𝑓–1-1-onto→𝐴)
9		f1of 6137	. . . . . . . . . . . . . . . . 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 6359	. . . . . . . . . . . . . . 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 1065	. . . . . . . . . . . . . . . 16 ⊢ (((𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑋 ∈ AC 𝐵) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑𝑚 𝐴)) → 𝑥 ∈ 𝐴)
13	12	ad2antrr 762	. . . . . . . . . . . . . . 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 4120	. . . . . . . . . . . . . 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 6360	. . . . . . . . . . . . 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 4317	. . . . . . . . . . . . . 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 4168	. . . . . . . . . . . . . . 15 ⊢ ((𝑔‘if(𝑦 ∈ ran 𝑓, (◡𝑓‘𝑦), 𝑥)) ∈ 𝒫 𝑋 → (𝑔‘if(𝑦 ∈ ran 𝑓, (◡𝑓‘𝑦), 𝑥)) ⊆ 𝑋)
18	17	anim1i 592	. . . . . . . . . . . . . 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 207	. . . . . . . . . . . . 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 2966	. . . . . . . . . . 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 8869	. . . . . . . . . . 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 693	. . . . . . . . . 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 6105	. . . . . . . . . . . . . 14 ⊢ (𝑓:𝐴–1-1→𝐵 → dom 𝑓 = 𝐴)
25		vex 3203	. . . . . . . . . . . . . . 15 ⊢ 𝑓 ∈ V
26	25	dmex 7099	. . . . . . . . . . . . . 14 ⊢ dom 𝑓 ∈ V
27	24, 26	syl6eqelr 2710	. . . . . . . . . . . . 13 ⊢ (𝑓:𝐴–1-1→𝐵 → 𝐴 ∈ V)
28	27	3ad2ant1 1082	. . . . . . . . . . . 12 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑋 ∈ AC 𝐵) → 𝐴 ∈ V)
29	28	ad2antrr 762	. . . . . . . . . . 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 1100	. . . . . . . . . . . . . . . 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 6101	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:𝐴–1-1→𝐵 → 𝑓:𝐴⟶𝐵)
32		frn 6053	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:𝐴⟶𝐵 → ran 𝑓 ⊆ 𝐵)
33		ssralv 3666	. . . . . . . . . . . . . . . 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 4092	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ran 𝑓 → if(𝑦 ∈ ran 𝑓, (◡𝑓‘𝑦), 𝑥) = (◡𝑓‘𝑦))
36	35	fveq2d 6195	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ran 𝑓 → (𝑔‘if(𝑦 ∈ ran 𝑓, (◡𝑓‘𝑦), 𝑥)) = (𝑔‘(◡𝑓‘𝑦)))
37	36	eleq2d 2687	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ran 𝑓 → ((𝑘‘𝑦) ∈ (𝑔‘if(𝑦 ∈ ran 𝑓, (◡𝑓‘𝑦), 𝑥)) ↔ (𝑘‘𝑦) ∈ (𝑔‘(◡𝑓‘𝑦))))
38	37	ralbiia 2979	. . . . . . . . . . . . . . 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 241	. . . . . . . . . . . . . 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 6102	. . . . . . . . . . . . . . 15 ⊢ (𝑓:𝐴–1-1→𝐵 → 𝑓 Fn 𝐴)
41		fveq2 6191	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = (𝑓‘𝑧) → (𝑘‘𝑦) = (𝑘‘(𝑓‘𝑧)))
42		fveq2 6191	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = (𝑓‘𝑧) → (◡𝑓‘𝑦) = (◡𝑓‘(𝑓‘𝑧)))
43	42	fveq2d 6195	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = (𝑓‘𝑧) → (𝑔‘(◡𝑓‘𝑦)) = (𝑔‘(◡𝑓‘(𝑓‘𝑧))))
44	41, 43	eleq12d 2695	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = (𝑓‘𝑧) → ((𝑘‘𝑦) ∈ (𝑔‘(◡𝑓‘𝑦)) ↔ (𝑘‘(𝑓‘𝑧)) ∈ (𝑔‘(◡𝑓‘(𝑓‘𝑧)))))
45	44	ralrn 6362	. . . . . . . . . . . . . . 15 ⊢ (𝑓 Fn 𝐴 → (∀𝑦 ∈ ran 𝑓(𝑘‘𝑦) ∈ (𝑔‘(◡𝑓‘𝑦)) ↔ ∀𝑧 ∈ 𝐴 (𝑘‘(𝑓‘𝑧)) ∈ (𝑔‘(◡𝑓‘(𝑓‘𝑧)))))
46	30, 40, 45	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)))))
47	39, 46	sylibd 229	. . . . . . . . . . . . 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)))))
48	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)
49		f1ocnvfv1 6532	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:𝐴–1-1-onto→ran 𝑓 ∧ 𝑧 ∈ 𝐴) → (◡𝑓‘(𝑓‘𝑧)) = 𝑧)
50	48, 49	sylan 488	. . . . . . . . . . . . . . . 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)
51	50	fveq2d 6195	. . . . . . . . . . . . . . 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))
52	51	eleq2d 2687	. . . . . . . . . . . . . 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)))
53	52	ralbidva 2985	. . . . . . . . . . . . 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)))
54	47, 53	sylibd 229	. . . . . . . . . . . 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)))
55	54	impr 649	. . . . . . . . . . 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))
56		acnlem 8871	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ V ∧ ∀𝑧 ∈ 𝐴 (𝑘‘(𝑓‘𝑧)) ∈ (𝑔‘𝑧)) → ∃ℎ∀𝑧 ∈ 𝐴 (ℎ‘𝑧) ∈ (𝑔‘𝑧))
57	29, 55, 56	syl2anc 693	. . . . . . . . . 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))
58	23, 57	exlimddv 1863	. . . . . . . . 9 ⊢ (((𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑋 ∈ AC 𝐵) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑𝑚 𝐴)) → ∃ℎ∀𝑧 ∈ 𝐴 (ℎ‘𝑧) ∈ (𝑔‘𝑧))
59	58	ralrimiva 2966	. . . . . . . 8 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑋 ∈ AC 𝐵) → ∀𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑𝑚 𝐴)∃ℎ∀𝑧 ∈ 𝐴 (ℎ‘𝑧) ∈ (𝑔‘𝑧))
60		elex 3212	. . . . . . . . . 10 ⊢ (𝑋 ∈ AC 𝐵 → 𝑋 ∈ V)
61		isacn 8867	. . . . . . . . . 10 ⊢ ((𝑋 ∈ V ∧ 𝐴 ∈ V) → (𝑋 ∈ AC 𝐴 ↔ ∀𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑𝑚 𝐴)∃ℎ∀𝑧 ∈ 𝐴 (ℎ‘𝑧) ∈ (𝑔‘𝑧)))
62	60, 27, 61	syl2anr 495	. . . . . . . . 9 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝑋 ∈ AC 𝐵) → (𝑋 ∈ AC 𝐴 ↔ ∀𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑𝑚 𝐴)∃ℎ∀𝑧 ∈ 𝐴 (ℎ‘𝑧) ∈ (𝑔‘𝑧)))
63	62	3adant2 1080	. . . . . . . 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)))
64	59, 63	mpbird 247	. . . . . . 7 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑋 ∈ AC 𝐵) → 𝑋 ∈ AC 𝐴)
65	64	3exp 1264	. . . . . 6 ⊢ (𝑓:𝐴–1-1→𝐵 → (𝑥 ∈ 𝐴 → (𝑋 ∈ AC 𝐵 → 𝑋 ∈ AC 𝐴)))
66	65	exlimdv 1861	. . . . 5 ⊢ (𝑓:𝐴–1-1→𝐵 → (∃𝑥 𝑥 ∈ 𝐴 → (𝑋 ∈ AC 𝐵 → 𝑋 ∈ AC 𝐴)))
67	2, 66	syl5bi 232	. . . 4 ⊢ (𝑓:𝐴–1-1→𝐵 → (¬ 𝐴 = ∅ → (𝑋 ∈ AC 𝐵 → 𝑋 ∈ AC 𝐴)))
68		acneq 8866	. . . . . . 7 ⊢ (𝐴 = ∅ → AC 𝐴 = AC ∅)
69		0fin 8188	. . . . . . . 8 ⊢ ∅ ∈ Fin
70		finacn 8873	. . . . . . . 8 ⊢ (∅ ∈ Fin → AC ∅ = V)
71	69, 70	ax-mp 5	. . . . . . 7 ⊢ AC ∅ = V
72	68, 71	syl6eq 2672	. . . . . 6 ⊢ (𝐴 = ∅ → AC 𝐴 = V)
73	72	eleq2d 2687	. . . . 5 ⊢ (𝐴 = ∅ → (𝑋 ∈ AC 𝐴 ↔ 𝑋 ∈ V))
74	60, 73	syl5ibr 236	. . . 4 ⊢ (𝐴 = ∅ → (𝑋 ∈ AC 𝐵 → 𝑋 ∈ AC 𝐴))
75	67, 74	pm2.61d2 172	. . 3 ⊢ (𝑓:𝐴–1-1→𝐵 → (𝑋 ∈ AC 𝐵 → 𝑋 ∈ AC 𝐴))
76	75	exlimiv 1858	. 2 ⊢ (∃𝑓 𝑓:𝐴–1-1→𝐵 → (𝑋 ∈ AC 𝐵 → 𝑋 ∈ AC 𝐴))
77	1, 76	syl 17	1 ⊢ (𝐴 ≼ 𝐵 → (𝑋 ∈ AC 𝐵 → 𝑋 ∈ AC 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483  ∃wex 1704   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  Vcvv 3200   ∖ cdif 3571   ⊆ wss 3574  ∅c0 3915  ifcif 4086  𝒫 cpw 4158  {csn 4177   class class class wbr 4653  ^◡ccnv 5113  dom cdm 5114  ran crn 5115   Fn wfn 5883  ⟶wf 5884  –1-1→wf1 5885  –1-1-onto→wf1o 5887  ‘cfv 5888  (class class class)co 6650   ↑_𝑚 cmap 7857   ≼ cdom 7953  Fincfn 7955  AC wacn 8764

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-1o 7560  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-fin 7959  df-acn 8768

This theorem is referenced by:  acnnum  8875  acnen  8876  iunctb  9396