Theorem acndom2 10050

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 8958	. 2 ⊢ (𝑋 ≼ 𝑌 → ∃𝑓 𝑓:𝑋–1-1→𝑌)
2		simplr 781	. . . . . . . 8 ⊢ (((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → 𝑌 ∈ AC 𝐴)
3		imassrn 6075	. . . . . . . . . . 11 ⊢ (𝑓 “ (𝑔‘𝑥)) ⊆ ran 𝑓
4		simplll 787	. . . . . . . . . . . 12 ⊢ ((((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ 𝑥 ∈ 𝐴) → 𝑓:𝑋–1-1→𝑌)
5		f1f 6778	. . . . . . . . . . . 12 ⊢ (𝑓:𝑋–1-1→𝑌 → 𝑓:𝑋⟶𝑌)
6		frn 6717	. . . . . . . . . . . 12 ⊢ (𝑓:𝑋⟶𝑌 → ran 𝑓 ⊆ 𝑌)
7	4, 5, 6	3syl 19	. . . . . . . . . . 11 ⊢ ((((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ 𝑥 ∈ 𝐴) → ran 𝑓 ⊆ 𝑌)
8	3, 7	sstrid 3949	. . . . . . . . . 10 ⊢ ((((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ 𝑥 ∈ 𝐴) → (𝑓 “ (𝑔‘𝑥)) ⊆ 𝑌)
9		elmapi 8848	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴) → 𝑔:𝐴⟶(𝒫 𝑋 ∖ {∅}))
10	9	adantl 487	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → 𝑔:𝐴⟶(𝒫 𝑋 ∖ {∅}))
11	10	ffvelcdmda 7083	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ 𝑥 ∈ 𝐴) → (𝑔‘𝑥) ∈ (𝒫 𝑋 ∖ {∅}))
12	11	eldifad 3918	. . . . . . . . . . . . . . 15 ⊢ ((((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ 𝑥 ∈ 𝐴) → (𝑔‘𝑥) ∈ 𝒫 𝑋)
13	12	elpwid 4573	. . . . . . . . . . . . . 14 ⊢ ((((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ 𝑥 ∈ 𝐴) → (𝑔‘𝑥) ⊆ 𝑋)
14		f1dm 6784	. . . . . . . . . . . . . . 15 ⊢ (𝑓:𝑋–1-1→𝑌 → dom 𝑓 = 𝑋)
15	4, 14	syl 18	. . . . . . . . . . . . . 14 ⊢ ((((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ 𝑥 ∈ 𝐴) → dom 𝑓 = 𝑋)
16	13, 15	sseqtrrd 3975	. . . . . . . . . . . . 13 ⊢ ((((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ 𝑥 ∈ 𝐴) → (𝑔‘𝑥) ⊆ dom 𝑓)
17		sseqin2 4176	. . . . . . . . . . . . 13 ⊢ ((𝑔‘𝑥) ⊆ dom 𝑓 ↔ (dom 𝑓 ∩ (𝑔‘𝑥)) = (𝑔‘𝑥))
18	16, 17	sylib 221	. . . . . . . . . . . 12 ⊢ ((((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ 𝑥 ∈ 𝐴) → (dom 𝑓 ∩ (𝑔‘𝑥)) = (𝑔‘𝑥))
19		eldifsni 4760	. . . . . . . . . . . . 13 ⊢ ((𝑔‘𝑥) ∈ (𝒫 𝑋 ∖ {∅}) → (𝑔‘𝑥) ≠ ∅)
20	11, 19	syl 18	. . . . . . . . . . . 12 ⊢ ((((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ 𝑥 ∈ 𝐴) → (𝑔‘𝑥) ≠ ∅)
21	18, 20	eqnetrd 3027	. . . . . . . . . . 11 ⊢ ((((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ 𝑥 ∈ 𝐴) → (dom 𝑓 ∩ (𝑔‘𝑥)) ≠ ∅)
22		imadisj 6084	. . . . . . . . . . . 12 ⊢ ((𝑓 “ (𝑔‘𝑥)) = ∅ ↔ (dom 𝑓 ∩ (𝑔‘𝑥)) = ∅)
23	22	necon3bii 3012	. . . . . . . . . . 11 ⊢ ((𝑓 “ (𝑔‘𝑥)) ≠ ∅ ↔ (dom 𝑓 ∩ (𝑔‘𝑥)) ≠ ∅)
24	21, 23	sylibr 237	. . . . . . . . . 10 ⊢ ((((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ 𝑥 ∈ 𝐴) → (𝑓 “ (𝑔‘𝑥)) ≠ ∅)
25	8, 24	jca 521	. . . . . . . . 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 3159	. . . . . . . 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 10042	. . . . . . . 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 596	. . . . . . 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 10038	. . . . . . . . 9 ⊢ (𝑌 ∈ AC 𝐴 → 𝐴 ∈ V)
30	29	ad3antlr 744	. . . . . . . 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 795	. . . . . . . . . . . . . . 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 6836	. . . . . . . . . . . . . . 15 ⊢ (𝑓:𝑋–1-1→𝑌 → 𝑓:𝑋–1-1-onto→ran 𝑓)
33	31, 32	syl 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:𝑋–1-1-onto→ran 𝑓)
34		simprr 785	. . . . . . . . . . . . . . 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 3936	. . . . . . . . . . . . . 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 7281	. . . . . . . . . . . . . 14 ⊢ ((𝑓:𝑋–1-1-onto→ran 𝑓 ∧ (𝑘‘𝑥) ∈ ran 𝑓) → (𝑓‘(◡𝑓‘(𝑘‘𝑥))) = (𝑘‘𝑥))
37	33, 35, 36	syl2anc 596	. . . . . . . . . . . . 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 2865	. . . . . . . . . . . 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 6837	. . . . . . . . . . . . . . 15 ⊢ (𝑓:𝑋–1-1-onto→ran 𝑓 → ◡𝑓:ran 𝑓–1-1-onto→𝑋)
40		f1of 6824	. . . . . . . . . . . . . . 15 ⊢ (◡𝑓:ran 𝑓–1-1-onto→𝑋 → ◡𝑓:ran 𝑓⟶𝑋)
41	33, 39, 40	3syl 19	. . . . . . . . . . . . . 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 7084	. . . . . . . . . . . . 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 760	. . . . . . . . . . . . 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 7263	. . . . . . . . . . . . 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 1398	. . . . . . . . . . . 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 235	. . . . . . . . . . 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 462	. . . . . . . . . 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 3179	. . . . . . . . 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 460	. . . . . . . 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 10044	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ ∀𝑥 ∈ 𝐴 (◡𝑓‘(𝑘‘𝑥)) ∈ (𝑔‘𝑥)) → ∃ℎ∀𝑥 ∈ 𝐴 (ℎ‘𝑥) ∈ (𝑔‘𝑥))
51	30, 49, 50	syl2anc 596	. . . . . . 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 1968	. . . . . 6 ⊢ (((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → ∃ℎ∀𝑥 ∈ 𝐴 (ℎ‘𝑥) ∈ (𝑔‘𝑥))
53	52	ralrimiva 3159	. . . . 5 ⊢ ((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) → ∀𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)∃ℎ∀𝑥 ∈ 𝐴 (ℎ‘𝑥) ∈ (𝑔‘𝑥))
54		vex 3461	. . . . . . . 8 ⊢ 𝑓 ∈ V
55	54	dmex 7908	. . . . . . 7 ⊢ dom 𝑓 ∈ V
56	14, 55	eqeltrrdi 2874	. . . . . 6 ⊢ (𝑓:𝑋–1-1→𝑌 → 𝑋 ∈ V)
57		isacn 10040	. . . . . 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 608	. . . . 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 260	. . . 4 ⊢ ((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) → 𝑋 ∈ AC 𝐴)
60	59	ex 418	. . 3 ⊢ (𝑓:𝑋–1-1→𝑌 → (𝑌 ∈ AC 𝐴 → 𝑋 ∈ AC 𝐴))
61	60	exlimiv 1963	. 2 ⊢ (∃𝑓 𝑓:𝑋–1-1→𝑌 → (𝑌 ∈ AC 𝐴 → 𝑋 ∈ AC 𝐴))
62	1, 61	syl 18	1 ⊢ (𝑋 ≼ 𝑌 → (𝑌 ∈ AC 𝐴 → 𝑋 ∈ AC 𝐴))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2146   ≠ wne 2960  ∀wral 3081  Vcvv 3457   ∖ cdif 3903   ∩ cin 3905   ⊆ wss 3906  ∅c0 4286  𝒫 cpw 4564  {csn 4591   class class class wbr 5111  ^◡ccnv 5662  dom cdm 5663  ran crn 5664   “ cima 5666  ⟶wf 6536  –1-1→wf1 6537  –1-1-onto→wf1o 6539  ‘cfv 6540  (class class class)co 7416   ↑_m cmap 8826   ≼ cdom 8943  AC wacn 9936

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7419  df-oprab 7420  df-mpo 7421  df-1st 7988  df-2nd 7989  df-map 8828  df-dom 8947  df-acn 9940

This theorem is used by:  acnen2  10051  dfac13  10138  iundomg  10536  iunctb  10570