Theorem acndom2 10004

Description: A set smaller than one with choice sequences of length 𝐴 also has choice sequences of length 𝐴. (Contributed by Mario Carneiro, 31-Aug-2015.)

Assertion

Ref	Expression
acndom2	⊢ (𝑋 ≼ 𝑌 → (𝑌 ∈ AC 𝐴 → 𝑋 ∈ AC 𝐴))

Proof of Theorem acndom2

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

Step	Hyp	Ref	Expression
1		brdomi 8934	. 2 ⊢ (𝑋 ≼ 𝑌 → ∃𝑓 𝑓:𝑋–1-1→𝑌)
2		simplr 778	. . . . . . . 8 ⊢ (((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → 𝑌 ∈ AC 𝐴)
3		imassrn 6056	. . . . . . . . . . 11 ⊢ (𝑓 “ (𝑔‘𝑥)) ⊆ ran 𝑓
4		simplll 784	. . . . . . . . . . . 12 ⊢ ((((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ 𝑥 ∈ 𝐴) → 𝑓:𝑋–1-1→𝑌)
5		f1f 6755	. . . . . . . . . . . 12 ⊢ (𝑓:𝑋–1-1→𝑌 → 𝑓:𝑋⟶𝑌)
6		frn 6694	. . . . . . . . . . . 12 ⊢ (𝑓:𝑋⟶𝑌 → ran 𝑓 ⊆ 𝑌)
7	4, 5, 6	3syl 18	. . . . . . . . . . 11 ⊢ ((((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ 𝑥 ∈ 𝐴) → ran 𝑓 ⊆ 𝑌)
8	3, 7	sstrid 3945	. . . . . . . . . 10 ⊢ ((((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ 𝑥 ∈ 𝐴) → (𝑓 “ (𝑔‘𝑥)) ⊆ 𝑌)
9		elmapi 8824	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴) → 𝑔:𝐴⟶(𝒫 𝑋 ∖ {∅}))
10	9	adantl 485	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → 𝑔:𝐴⟶(𝒫 𝑋 ∖ {∅}))
11	10	ffvelcdmda 7060	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ 𝑥 ∈ 𝐴) → (𝑔‘𝑥) ∈ (𝒫 𝑋 ∖ {∅}))
12	11	eldifad 3914	. . . . . . . . . . . . . . 15 ⊢ ((((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ 𝑥 ∈ 𝐴) → (𝑔‘𝑥) ∈ 𝒫 𝑋)
13	12	elpwid 4561	. . . . . . . . . . . . . 14 ⊢ ((((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ 𝑥 ∈ 𝐴) → (𝑔‘𝑥) ⊆ 𝑋)
14		f1dm 6761	. . . . . . . . . . . . . . 15 ⊢ (𝑓:𝑋–1-1→𝑌 → dom 𝑓 = 𝑋)
15	4, 14	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ 𝑥 ∈ 𝐴) → dom 𝑓 = 𝑋)
16	13, 15	sseqtrrd 3971	. . . . . . . . . . . . 13 ⊢ ((((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ 𝑥 ∈ 𝐴) → (𝑔‘𝑥) ⊆ dom 𝑓)
17		sseqin2 4173	. . . . . . . . . . . . 13 ⊢ ((𝑔‘𝑥) ⊆ dom 𝑓 ↔ (dom 𝑓 ∩ (𝑔‘𝑥)) = (𝑔‘𝑥))
18	16, 17	sylib 220	. . . . . . . . . . . 12 ⊢ ((((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ 𝑥 ∈ 𝐴) → (dom 𝑓 ∩ (𝑔‘𝑥)) = (𝑔‘𝑥))
19		eldifsni 4747	. . . . . . . . . . . . 13 ⊢ ((𝑔‘𝑥) ∈ (𝒫 𝑋 ∖ {∅}) → (𝑔‘𝑥) ≠ ∅)
20	11, 19	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ 𝑥 ∈ 𝐴) → (𝑔‘𝑥) ≠ ∅)
21	18, 20	eqnetrd 3023	. . . . . . . . . . 11 ⊢ ((((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ 𝑥 ∈ 𝐴) → (dom 𝑓 ∩ (𝑔‘𝑥)) ≠ ∅)
22		imadisj 6065	. . . . . . . . . . . 12 ⊢ ((𝑓 “ (𝑔‘𝑥)) = ∅ ↔ (dom 𝑓 ∩ (𝑔‘𝑥)) = ∅)
23	22	necon3bii 3008	. . . . . . . . . . 11 ⊢ ((𝑓 “ (𝑔‘𝑥)) ≠ ∅ ↔ (dom 𝑓 ∩ (𝑔‘𝑥)) ≠ ∅)
24	21, 23	sylibr 236	. . . . . . . . . 10 ⊢ ((((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ 𝑥 ∈ 𝐴) → (𝑓 “ (𝑔‘𝑥)) ≠ ∅)
25	8, 24	jca 519	. . . . . . . . 9 ⊢ ((((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ 𝑥 ∈ 𝐴) → ((𝑓 “ (𝑔‘𝑥)) ⊆ 𝑌 ∧ (𝑓 “ (𝑔‘𝑥)) ≠ ∅))
26	25	ralrimiva 3153	. . . . . . . 8 ⊢ (((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → ∀𝑥 ∈ 𝐴 ((𝑓 “ (𝑔‘𝑥)) ⊆ 𝑌 ∧ (𝑓 “ (𝑔‘𝑥)) ≠ ∅))
27		acni2 9996	. . . . . . . 8 ⊢ ((𝑌 ∈ AC 𝐴 ∧ ∀𝑥 ∈ 𝐴 ((𝑓 “ (𝑔‘𝑥)) ⊆ 𝑌 ∧ (𝑓 “ (𝑔‘𝑥)) ≠ ∅)) → ∃𝑘(𝑘:𝐴⟶𝑌 ∧ ∀𝑥 ∈ 𝐴 (𝑘‘𝑥) ∈ (𝑓 “ (𝑔‘𝑥))))
28	2, 26, 27	syl2anc 593	. . . . . . 7 ⊢ (((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → ∃𝑘(𝑘:𝐴⟶𝑌 ∧ ∀𝑥 ∈ 𝐴 (𝑘‘𝑥) ∈ (𝑓 “ (𝑔‘𝑥))))
29		acnrcl 9992	. . . . . . . . 9 ⊢ (𝑌 ∈ AC 𝐴 → 𝐴 ∈ V)
30	29	ad3antlr 741	. . . . . . . 8 ⊢ ((((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ (𝑘:𝐴⟶𝑌 ∧ ∀𝑥 ∈ 𝐴 (𝑘‘𝑥) ∈ (𝑓 “ (𝑔‘𝑥)))) → 𝐴 ∈ V)
31		simp-4l 792	. . . . . . . . . . . . . . 15 ⊢ (((((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ 𝑘:𝐴⟶𝑌) ∧ (𝑥 ∈ 𝐴 ∧ (𝑘‘𝑥) ∈ (𝑓 “ (𝑔‘𝑥)))) → 𝑓:𝑋–1-1→𝑌)
32		f1f1orn 6813	. . . . . . . . . . . . . . 15 ⊢ (𝑓:𝑋–1-1→𝑌 → 𝑓:𝑋–1-1-onto→ran 𝑓)
33	31, 32	syl 17	. . . . . . . . . . . . . 14 ⊢ (((((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ 𝑘:𝐴⟶𝑌) ∧ (𝑥 ∈ 𝐴 ∧ (𝑘‘𝑥) ∈ (𝑓 “ (𝑔‘𝑥)))) → 𝑓:𝑋–1-1-onto→ran 𝑓)
34		simprr 782	. . . . . . . . . . . . . . 15 ⊢ (((((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ 𝑘:𝐴⟶𝑌) ∧ (𝑥 ∈ 𝐴 ∧ (𝑘‘𝑥) ∈ (𝑓 “ (𝑔‘𝑥)))) → (𝑘‘𝑥) ∈ (𝑓 “ (𝑔‘𝑥)))
35	3, 34	sselid 3932	. . . . . . . . . . . . . 14 ⊢ (((((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ 𝑘:𝐴⟶𝑌) ∧ (𝑥 ∈ 𝐴 ∧ (𝑘‘𝑥) ∈ (𝑓 “ (𝑔‘𝑥)))) → (𝑘‘𝑥) ∈ ran 𝑓)
36		f1ocnvfv2 7256	. . . . . . . . . . . . . 14 ⊢ ((𝑓:𝑋–1-1-onto→ran 𝑓 ∧ (𝑘‘𝑥) ∈ ran 𝑓) → (𝑓‘(◡𝑓‘(𝑘‘𝑥))) = (𝑘‘𝑥))
37	33, 35, 36	syl2anc 593	. . . . . . . . . . . . 13 ⊢ (((((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ 𝑘:𝐴⟶𝑌) ∧ (𝑥 ∈ 𝐴 ∧ (𝑘‘𝑥) ∈ (𝑓 “ (𝑔‘𝑥)))) → (𝑓‘(◡𝑓‘(𝑘‘𝑥))) = (𝑘‘𝑥))
38	37, 34	eqeltrd 2861	. . . . . . . . . . . 12 ⊢ (((((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ 𝑘:𝐴⟶𝑌) ∧ (𝑥 ∈ 𝐴 ∧ (𝑘‘𝑥) ∈ (𝑓 “ (𝑔‘𝑥)))) → (𝑓‘(◡𝑓‘(𝑘‘𝑥))) ∈ (𝑓 “ (𝑔‘𝑥)))
39		f1ocnv 6814	. . . . . . . . . . . . . . 15 ⊢ (𝑓:𝑋–1-1-onto→ran 𝑓 → ◡𝑓:ran 𝑓–1-1-onto→𝑋)
40		f1of 6801	. . . . . . . . . . . . . . 15 ⊢ (◡𝑓:ran 𝑓–1-1-onto→𝑋 → ◡𝑓:ran 𝑓⟶𝑋)
41	33, 39, 40	3syl 18	. . . . . . . . . . . . . 14 ⊢ (((((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ 𝑘:𝐴⟶𝑌) ∧ (𝑥 ∈ 𝐴 ∧ (𝑘‘𝑥) ∈ (𝑓 “ (𝑔‘𝑥)))) → ◡𝑓:ran 𝑓⟶𝑋)
42	41, 35	ffvelcdmd 7061	. . . . . . . . . . . . 13 ⊢ (((((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ 𝑘:𝐴⟶𝑌) ∧ (𝑥 ∈ 𝐴 ∧ (𝑘‘𝑥) ∈ (𝑓 “ (𝑔‘𝑥)))) → (◡𝑓‘(𝑘‘𝑥)) ∈ 𝑋)
43	13	ad2ant2r 757	. . . . . . . . . . . . 13 ⊢ (((((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ 𝑘:𝐴⟶𝑌) ∧ (𝑥 ∈ 𝐴 ∧ (𝑘‘𝑥) ∈ (𝑓 “ (𝑔‘𝑥)))) → (𝑔‘𝑥) ⊆ 𝑋)
44		f1elima 7242	. . . . . . . . . . . . 13 ⊢ ((𝑓:𝑋–1-1→𝑌 ∧ (◡𝑓‘(𝑘‘𝑥)) ∈ 𝑋 ∧ (𝑔‘𝑥) ⊆ 𝑋) → ((𝑓‘(◡𝑓‘(𝑘‘𝑥))) ∈ (𝑓 “ (𝑔‘𝑥)) ↔ (◡𝑓‘(𝑘‘𝑥)) ∈ (𝑔‘𝑥)))
45	31, 42, 43, 44	syl3anc 1389	. . . . . . . . . . . 12 ⊢ (((((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ 𝑘:𝐴⟶𝑌) ∧ (𝑥 ∈ 𝐴 ∧ (𝑘‘𝑥) ∈ (𝑓 “ (𝑔‘𝑥)))) → ((𝑓‘(◡𝑓‘(𝑘‘𝑥))) ∈ (𝑓 “ (𝑔‘𝑥)) ↔ (◡𝑓‘(𝑘‘𝑥)) ∈ (𝑔‘𝑥)))
46	38, 45	mpbid 234	. . . . . . . . . . 11 ⊢ (((((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ 𝑘:𝐴⟶𝑌) ∧ (𝑥 ∈ 𝐴 ∧ (𝑘‘𝑥) ∈ (𝑓 “ (𝑔‘𝑥)))) → (◡𝑓‘(𝑘‘𝑥)) ∈ (𝑔‘𝑥))
47	46	expr 460	. . . . . . . . . 10 ⊢ (((((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ 𝑘:𝐴⟶𝑌) ∧ 𝑥 ∈ 𝐴) → ((𝑘‘𝑥) ∈ (𝑓 “ (𝑔‘𝑥)) → (◡𝑓‘(𝑘‘𝑥)) ∈ (𝑔‘𝑥)))
48	47	ralimdva 3173	. . . . . . . . 9 ⊢ ((((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ 𝑘:𝐴⟶𝑌) → (∀𝑥 ∈ 𝐴 (𝑘‘𝑥) ∈ (𝑓 “ (𝑔‘𝑥)) → ∀𝑥 ∈ 𝐴 (◡𝑓‘(𝑘‘𝑥)) ∈ (𝑔‘𝑥)))
49	48	impr 458	. . . . . . . 8 ⊢ ((((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ (𝑘:𝐴⟶𝑌 ∧ ∀𝑥 ∈ 𝐴 (𝑘‘𝑥) ∈ (𝑓 “ (𝑔‘𝑥)))) → ∀𝑥 ∈ 𝐴 (◡𝑓‘(𝑘‘𝑥)) ∈ (𝑔‘𝑥))
50		acnlem 9998	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ ∀𝑥 ∈ 𝐴 (◡𝑓‘(𝑘‘𝑥)) ∈ (𝑔‘𝑥)) → ∃ℎ∀𝑥 ∈ 𝐴 (ℎ‘𝑥) ∈ (𝑔‘𝑥))
51	30, 49, 50	syl2anc 593	. . . . . . 7 ⊢ ((((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ (𝑘:𝐴⟶𝑌 ∧ ∀𝑥 ∈ 𝐴 (𝑘‘𝑥) ∈ (𝑓 “ (𝑔‘𝑥)))) → ∃ℎ∀𝑥 ∈ 𝐴 (ℎ‘𝑥) ∈ (𝑔‘𝑥))
52	28, 51	exlimddv 1954	. . . . . 6 ⊢ (((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → ∃ℎ∀𝑥 ∈ 𝐴 (ℎ‘𝑥) ∈ (𝑔‘𝑥))
53	52	ralrimiva 3153	. . . . 5 ⊢ ((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) → ∀𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)∃ℎ∀𝑥 ∈ 𝐴 (ℎ‘𝑥) ∈ (𝑔‘𝑥))
54		vex 3457	. . . . . . . 8 ⊢ 𝑓 ∈ V
55	54	dmex 7885	. . . . . . 7 ⊢ dom 𝑓 ∈ V
56	14, 55	eqeltrrdi 2870	. . . . . 6 ⊢ (𝑓:𝑋–1-1→𝑌 → 𝑋 ∈ V)
57		isacn 9994	. . . . . 6 ⊢ ((𝑋 ∈ V ∧ 𝐴 ∈ V) → (𝑋 ∈ AC 𝐴 ↔ ∀𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)∃ℎ∀𝑥 ∈ 𝐴 (ℎ‘𝑥) ∈ (𝑔‘𝑥)))
58	56, 29, 57	syl2an 605	. . . . 5 ⊢ ((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) → (𝑋 ∈ AC 𝐴 ↔ ∀𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)∃ℎ∀𝑥 ∈ 𝐴 (ℎ‘𝑥) ∈ (𝑔‘𝑥)))
59	53, 58	mpbird 259	. . . 4 ⊢ ((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) → 𝑋 ∈ AC 𝐴)
60	59	ex 416	. . 3 ⊢ (𝑓:𝑋–1-1→𝑌 → (𝑌 ∈ AC 𝐴 → 𝑋 ∈ AC 𝐴))
61	60	exlimiv 1949	. 2 ⊢ (∃𝑓 𝑓:𝑋–1-1→𝑌 → (𝑌 ∈ AC 𝐴 → 𝑋 ∈ AC 𝐴))
62	1, 61	syl 17	1 ⊢ (𝑋 ≼ 𝑌 → (𝑌 ∈ AC 𝐴 → 𝑋 ∈ AC 𝐴))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1559  ∃wex 1798   ∈ wcel 2141   ≠ wne 2956  ∀wral 3075  Vcvv 3453   ∖ cdif 3899   ∩ cin 3901   ⊆ wss 3902  ∅c0 4283  𝒫 cpw 4552  {csn 4579   class class class wbr 5097  ^◡ccnv 5642  dom cdm 5643  ran crn 5644   “ cima 5646  ⟶wf 6512  –1-1→wf1 6513  –1-1-onto→wf1o 6515  ‘cfv 6516  (class class class)co 7391   ↑_m cmap 8802   ≼ cdom 8919  AC wacn 9890

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7713

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-ov 7394  df-oprab 7395  df-mpo 7396  df-1st 7965  df-2nd 7966  df-map 8804  df-dom 8923  df-acn 9894

This theorem is referenced by:  acnen2  10005  dfac13  10093  iundomg  10492  iunctb  10526