Theorem acndom2 9741

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 8704	. 2 ⊢ (𝑋 ≼ 𝑌 → ∃𝑓 𝑓:𝑋–1-1→𝑌)
2		simplr 765	. . . . . . . 8 ⊢ (((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → 𝑌 ∈ AC 𝐴)
3		imassrn 5969	. . . . . . . . . . 11 ⊢ (𝑓 “ (𝑔‘𝑥)) ⊆ ran 𝑓
4		simplll 771	. . . . . . . . . . . 12 ⊢ ((((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ 𝑥 ∈ 𝐴) → 𝑓:𝑋–1-1→𝑌)
5		f1f 6654	. . . . . . . . . . . 12 ⊢ (𝑓:𝑋–1-1→𝑌 → 𝑓:𝑋⟶𝑌)
6		frn 6591	. . . . . . . . . . . 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 3928	. . . . . . . . . 10 ⊢ ((((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ 𝑥 ∈ 𝐴) → (𝑓 “ (𝑔‘𝑥)) ⊆ 𝑌)
9		elmapi 8595	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴) → 𝑔:𝐴⟶(𝒫 𝑋 ∖ {∅}))
10	9	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → 𝑔:𝐴⟶(𝒫 𝑋 ∖ {∅}))
11	10	ffvelrnda 6943	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ 𝑥 ∈ 𝐴) → (𝑔‘𝑥) ∈ (𝒫 𝑋 ∖ {∅}))
12	11	eldifad 3895	. . . . . . . . . . . . . . 15 ⊢ ((((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ 𝑥 ∈ 𝐴) → (𝑔‘𝑥) ∈ 𝒫 𝑋)
13	12	elpwid 4541	. . . . . . . . . . . . . 14 ⊢ ((((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ 𝑥 ∈ 𝐴) → (𝑔‘𝑥) ⊆ 𝑋)
14		f1dm 6658	. . . . . . . . . . . . . . 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 3958	. . . . . . . . . . . . 13 ⊢ ((((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ 𝑥 ∈ 𝐴) → (𝑔‘𝑥) ⊆ dom 𝑓)
17		sseqin2 4146	. . . . . . . . . . . . 13 ⊢ ((𝑔‘𝑥) ⊆ dom 𝑓 ↔ (dom 𝑓 ∩ (𝑔‘𝑥)) = (𝑔‘𝑥))
18	16, 17	sylib 217	. . . . . . . . . . . 12 ⊢ ((((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ 𝑥 ∈ 𝐴) → (dom 𝑓 ∩ (𝑔‘𝑥)) = (𝑔‘𝑥))
19		eldifsni 4720	. . . . . . . . . . . . 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 3010	. . . . . . . . . . 11 ⊢ ((((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ 𝑥 ∈ 𝐴) → (dom 𝑓 ∩ (𝑔‘𝑥)) ≠ ∅)
22		imadisj 5977	. . . . . . . . . . . 12 ⊢ ((𝑓 “ (𝑔‘𝑥)) = ∅ ↔ (dom 𝑓 ∩ (𝑔‘𝑥)) = ∅)
23	22	necon3bii 2995	. . . . . . . . . . 11 ⊢ ((𝑓 “ (𝑔‘𝑥)) ≠ ∅ ↔ (dom 𝑓 ∩ (𝑔‘𝑥)) ≠ ∅)
24	21, 23	sylibr 233	. . . . . . . . . 10 ⊢ ((((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ 𝑥 ∈ 𝐴) → (𝑓 “ (𝑔‘𝑥)) ≠ ∅)
25	8, 24	jca 511	. . . . . . . . 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 3107	. . . . . . . 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 9733	. . . . . . . 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 583	. . . . . . 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 9729	. . . . . . . . 9 ⊢ (𝑌 ∈ AC 𝐴 → 𝐴 ∈ V)
30	29	ad3antlr 727	. . . . . . . 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 779	. . . . . . . . . . . . . . 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 6711	. . . . . . . . . . . . . . 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 769	. . . . . . . . . . . . . . 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 3915	. . . . . . . . . . . . . 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 7130	. . . . . . . . . . . . . 14 ⊢ ((𝑓:𝑋–1-1-onto→ran 𝑓 ∧ (𝑘‘𝑥) ∈ ran 𝑓) → (𝑓‘(◡𝑓‘(𝑘‘𝑥))) = (𝑘‘𝑥))
37	33, 35, 36	syl2anc 583	. . . . . . . . . . . . 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 2839	. . . . . . . . . . . 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 6712	. . . . . . . . . . . . . . 15 ⊢ (𝑓:𝑋–1-1-onto→ran 𝑓 → ◡𝑓:ran 𝑓–1-1-onto→𝑋)
40		f1of 6700	. . . . . . . . . . . . . . 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	ffvelrnd 6944	. . . . . . . . . . . . 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 743	. . . . . . . . . . . . 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 7117	. . . . . . . . . . . . 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 1369	. . . . . . . . . . . 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 231	. . . . . . . . . . 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 456	. . . . . . . . . 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 3102	. . . . . . . . 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 454	. . . . . . . 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 9735	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ ∀𝑥 ∈ 𝐴 (◡𝑓‘(𝑘‘𝑥)) ∈ (𝑔‘𝑥)) → ∃ℎ∀𝑥 ∈ 𝐴 (ℎ‘𝑥) ∈ (𝑔‘𝑥))
51	30, 49, 50	syl2anc 583	. . . . . . 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 1939	. . . . . 6 ⊢ (((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → ∃ℎ∀𝑥 ∈ 𝐴 (ℎ‘𝑥) ∈ (𝑔‘𝑥))
53	52	ralrimiva 3107	. . . . 5 ⊢ ((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) → ∀𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)∃ℎ∀𝑥 ∈ 𝐴 (ℎ‘𝑥) ∈ (𝑔‘𝑥))
54		vex 3426	. . . . . . . 8 ⊢ 𝑓 ∈ V
55	54	dmex 7732	. . . . . . 7 ⊢ dom 𝑓 ∈ V
56	14, 55	eqeltrrdi 2848	. . . . . 6 ⊢ (𝑓:𝑋–1-1→𝑌 → 𝑋 ∈ V)
57		isacn 9731	. . . . . 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 595	. . . . 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 256	. . . 4 ⊢ ((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) → 𝑋 ∈ AC 𝐴)
60	59	ex 412	. . 3 ⊢ (𝑓:𝑋–1-1→𝑌 → (𝑌 ∈ AC 𝐴 → 𝑋 ∈ AC 𝐴))
61	60	exlimiv 1934	. 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 205   ∧ wa 395   = wceq 1539  ∃wex 1783   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063  Vcvv 3422   ∖ cdif 3880   ∩ cin 3882   ⊆ wss 3883  ∅c0 4253  𝒫 cpw 4530  {csn 4558   class class class wbr 5070  ^◡ccnv 5579  dom cdm 5580  ran crn 5581   “ cima 5583  ⟶wf 6414  –1-1→wf1 6415  –1-1-onto→wf1o 6417  ‘cfv 6418  (class class class)co 7255   ↑_m cmap 8573   ≼ cdom 8689  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-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-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  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-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-1st 7804  df-2nd 7805  df-map 8575  df-dom 8693  df-acn 9631

This theorem is referenced by:  acnen2  9742  dfac13  9829  iundomg  10228  iunctb  10261