Theorem acndom2 10137

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 8990	. 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 6197	. . . . . . . . . . 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 6779	. . . . . . . . . . . 12 ⊢ (𝑓:𝑋–1-1→𝑌 → 𝑓:𝑋⟶𝑌)
6		frn 6718	. . . . . . . . . . . 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 3919	. . . . . . . . . 10 ⊢ ((((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ 𝑥 ∈ 𝐴) → (𝑓 “ (𝑔‘𝑥)) ⊆ 𝑌)
9		elmapi 8873	. . . . . . . . . . . . . . . . . 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 7085	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ 𝑥 ∈ 𝐴) → (𝑔‘𝑥) ∈ (𝒫 𝑋 ∖ {∅}))
12	11	eldifad 4051	. . . . . . . . . . . . . . 15 ⊢ ((((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ 𝑥 ∈ 𝐴) → (𝑔‘𝑥) ∈ 𝒫 𝑋)
13	12	elpwid 4565	. . . . . . . . . . . . . 14 ⊢ ((((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ 𝑥 ∈ 𝐴) → (𝑔‘𝑥) ⊆ 𝑋)
14		f1dm 6785	. . . . . . . . . . . . . . 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 3945	. . . . . . . . . . . . 13 ⊢ ((((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ 𝑥 ∈ 𝐴) → (𝑔‘𝑥) ⊆ dom 𝑓)
17		sseqin2 4120	. . . . . . . . . . . . 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 4749	. . . . . . . . . . . . 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 3022	. . . . . . . . . . 11 ⊢ ((((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) ∧ 𝑥 ∈ 𝐴) → (dom 𝑓 ∩ (𝑔‘𝑥)) ≠ ∅)
22		imadisj 6076	. . . . . . . . . . . 12 ⊢ ((𝑓 “ (𝑔‘𝑥)) = ∅ ↔ (dom 𝑓 ∩ (𝑔‘𝑥)) = ∅)
23	22	necon3bii 3007	. . . . . . . . . . 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 3154	. . . . . . . 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 10129	. . . . . . . 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 10125	. . . . . . . . 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 6837	. . . . . . . . . . . . . . 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 3906	. . . . . . . . . . . . . 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 7286	. . . . . . . . . . . . . 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 2860	. . . . . . . . . . . 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 6838	. . . . . . . . . . . . . . 15 ⊢ (𝑓:𝑋–1-1-onto→ran 𝑓 → ◡𝑓:ran 𝑓–1-1-onto→𝑋)
40		f1of 6825	. . . . . . . . . . . . . . 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 7086	. . . . . . . . . . . . 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 7268	. . . . . . . . . . . . 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 3174	. . . . . . . . 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 10131	. . . . . . . 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 1967	. . . . . 6 ⊢ (((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) ∧ 𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)) → ∃ℎ∀𝑥 ∈ 𝐴 (ℎ‘𝑥) ∈ (𝑔‘𝑥))
53	52	ralrimiva 3154	. . . . 5 ⊢ ((𝑓:𝑋–1-1→𝑌 ∧ 𝑌 ∈ AC 𝐴) → ∀𝑔 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)∃ℎ∀𝑥 ∈ 𝐴 (ℎ‘𝑥) ∈ (𝑔‘𝑥))
54		vex 3454	. . . . . . . 8 ⊢ 𝑓 ∈ V
55	54	dmex 7922	. . . . . . 7 ⊢ dom 𝑓 ∈ V
56	14, 55	eqeltrrdi 2869	. . . . . 6 ⊢ (𝑓:𝑋–1-1→𝑌 → 𝑋 ∈ V)
57		isacn 10127	. . . . . 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 1962	. 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 1569  ∃wex 1811   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076  Vcvv 3450   ⊆ wss 3895   ∖ cdif 4045   ∩ cin 4089  ∅c0 4278  𝒫 cpw 4556  {csn 4584   class class class wbr 5102  ^◡ccnv 5649  dom cdm 5650  ran crn 5651   “ cima 5653  ⟶wf 6535  –1-1→wf1 6536  –1-1-onto→wf1o 6538  ‘cfv 6539  (class class class)co 7421   ↑_m cmap 8851   ≼ cdom 8975  AC wacn 10023

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7752

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-ov 7424  df-oprab 7425  df-mpo 7426  df-1st 8002  df-2nd 8003  df-map 8853  df-dom 8979  df-acn 10027

This theorem is used by:  acnen2  10138  dfac13  10225  iundomg  10629  iunctb  10663