Theorem enwomnilem 7373

Description: Lemma for enwomni 7374. One direction of the biconditional. (Contributed by Jim Kingdon, 20-Jun-2024.)

Assertion

Ref	Expression
enwomnilem	⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ WOmni → 𝐵 ∈ WOmni))

Proof of Theorem enwomnilem

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

Step	Hyp	Ref	Expression
1		bren 6922	. . . . . . 7 ⊢ (𝐴 ≈ 𝐵 ↔ ∃ℎ ℎ:𝐴–1-1-onto→𝐵)
2	1	biimpi 120	. . . . . 6 ⊢ (𝐴 ≈ 𝐵 → ∃ℎ ℎ:𝐴–1-1-onto→𝐵)
3	2	ad2antrr 488	. . . . 5 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) → ∃ℎ ℎ:𝐴–1-1-onto→𝐵)
4		fveq1 5641	. . . . . . . . . 10 ⊢ (𝑓 = (𝑔 ∘ ℎ) → (𝑓‘𝑥) = ((𝑔 ∘ ℎ)‘𝑥))
5	4	eqeq1d 2239	. . . . . . . . 9 ⊢ (𝑓 = (𝑔 ∘ ℎ) → ((𝑓‘𝑥) = 1o ↔ ((𝑔 ∘ ℎ)‘𝑥) = 1o))
6	5	ralbidv 2531	. . . . . . . 8 ⊢ (𝑓 = (𝑔 ∘ ℎ) → (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o ↔ ∀𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = 1o))
7	6	dcbid 845	. . . . . . 7 ⊢ (𝑓 = (𝑔 ∘ ℎ) → (DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o ↔ DECID ∀𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = 1o))
8		iswomnimap 7370	. . . . . . . . 9 ⊢ (𝐴 ∈ WOmni → (𝐴 ∈ WOmni ↔ ∀𝑓 ∈ (2o ↑𝑚 𝐴)DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o))
9	8	ibi 176	. . . . . . . 8 ⊢ (𝐴 ∈ WOmni → ∀𝑓 ∈ (2o ↑𝑚 𝐴)DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)
10	9	ad3antlr 493	. . . . . . 7 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) → ∀𝑓 ∈ (2o ↑𝑚 𝐴)DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)
11		simpr 110	. . . . . . . . . . 11 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) → 𝑔 ∈ (2o ↑𝑚 𝐵))
12		2onn 6694	. . . . . . . . . . . . 13 ⊢ 2o ∈ ω
13		relen 6918	. . . . . . . . . . . . . 14 ⊢ Rel ≈
14	13	brrelex2i 4772	. . . . . . . . . . . . 13 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ∈ V)
15		elmapg 6835	. . . . . . . . . . . . 13 ⊢ ((2o ∈ ω ∧ 𝐵 ∈ V) → (𝑔 ∈ (2o ↑𝑚 𝐵) ↔ 𝑔:𝐵⟶2o))
16	12, 14, 15	sylancr 414	. . . . . . . . . . . 12 ⊢ (𝐴 ≈ 𝐵 → (𝑔 ∈ (2o ↑𝑚 𝐵) ↔ 𝑔:𝐵⟶2o))
17	16	ad2antrr 488	. . . . . . . . . . 11 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) → (𝑔 ∈ (2o ↑𝑚 𝐵) ↔ 𝑔:𝐵⟶2o))
18	11, 17	mpbid 147	. . . . . . . . . 10 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) → 𝑔:𝐵⟶2o)
19	18	adantr 276	. . . . . . . . 9 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) → 𝑔:𝐵⟶2o)
20		f1of 5586	. . . . . . . . . 10 ⊢ (ℎ:𝐴–1-1-onto→𝐵 → ℎ:𝐴⟶𝐵)
21	20	adantl 277	. . . . . . . . 9 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) → ℎ:𝐴⟶𝐵)
22		fco 5502	. . . . . . . . 9 ⊢ ((𝑔:𝐵⟶2o ∧ ℎ:𝐴⟶𝐵) → (𝑔 ∘ ℎ):𝐴⟶2o)
23	19, 21, 22	syl2anc 411	. . . . . . . 8 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) → (𝑔 ∘ ℎ):𝐴⟶2o)
24		simpllr 536	. . . . . . . . 9 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) → 𝐴 ∈ WOmni)
25		elmapg 6835	. . . . . . . . 9 ⊢ ((2o ∈ ω ∧ 𝐴 ∈ WOmni) → ((𝑔 ∘ ℎ) ∈ (2o ↑𝑚 𝐴) ↔ (𝑔 ∘ ℎ):𝐴⟶2o))
26	12, 24, 25	sylancr 414	. . . . . . . 8 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) → ((𝑔 ∘ ℎ) ∈ (2o ↑𝑚 𝐴) ↔ (𝑔 ∘ ℎ):𝐴⟶2o))
27	23, 26	mpbird 167	. . . . . . 7 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) → (𝑔 ∘ ℎ) ∈ (2o ↑𝑚 𝐴))
28	7, 10, 27	rspcdva 2914	. . . . . 6 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) → DECID ∀𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = 1o)
29		f1ofn 5587	. . . . . . . . . . . 12 ⊢ (ℎ:𝐴–1-1-onto→𝐵 → ℎ Fn 𝐴)
30	29	ad3antlr 493	. . . . . . . . . . 11 ⊢ ((((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ ∀𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = 1o) ∧ 𝑦 ∈ 𝐵) → ℎ Fn 𝐴)
31		f1ocnv 5599	. . . . . . . . . . . . . 14 ⊢ (ℎ:𝐴–1-1-onto→𝐵 → ◡ℎ:𝐵–1-1-onto→𝐴)
32		f1of 5586	. . . . . . . . . . . . . 14 ⊢ (◡ℎ:𝐵–1-1-onto→𝐴 → ◡ℎ:𝐵⟶𝐴)
33	31, 32	syl 14	. . . . . . . . . . . . 13 ⊢ (ℎ:𝐴–1-1-onto→𝐵 → ◡ℎ:𝐵⟶𝐴)
34	33	ad3antlr 493	. . . . . . . . . . . 12 ⊢ ((((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ ∀𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = 1o) ∧ 𝑦 ∈ 𝐵) → ◡ℎ:𝐵⟶𝐴)
35		simpr 110	. . . . . . . . . . . 12 ⊢ ((((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ ∀𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = 1o) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
36	34, 35	ffvelcdmd 5786	. . . . . . . . . . 11 ⊢ ((((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ ∀𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = 1o) ∧ 𝑦 ∈ 𝐵) → (◡ℎ‘𝑦) ∈ 𝐴)
37		fvco2 5718	. . . . . . . . . . 11 ⊢ ((ℎ Fn 𝐴 ∧ (◡ℎ‘𝑦) ∈ 𝐴) → ((𝑔 ∘ ℎ)‘(◡ℎ‘𝑦)) = (𝑔‘(ℎ‘(◡ℎ‘𝑦))))
38	30, 36, 37	syl2anc 411	. . . . . . . . . 10 ⊢ ((((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ ∀𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = 1o) ∧ 𝑦 ∈ 𝐵) → ((𝑔 ∘ ℎ)‘(◡ℎ‘𝑦)) = (𝑔‘(ℎ‘(◡ℎ‘𝑦))))
39		fveqeq2 5651	. . . . . . . . . . 11 ⊢ (𝑥 = (◡ℎ‘𝑦) → (((𝑔 ∘ ℎ)‘𝑥) = 1o ↔ ((𝑔 ∘ ℎ)‘(◡ℎ‘𝑦)) = 1o))
40		simplr 529	. . . . . . . . . . 11 ⊢ ((((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ ∀𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = 1o) ∧ 𝑦 ∈ 𝐵) → ∀𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = 1o)
41	39, 40, 36	rspcdva 2914	. . . . . . . . . 10 ⊢ ((((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ ∀𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = 1o) ∧ 𝑦 ∈ 𝐵) → ((𝑔 ∘ ℎ)‘(◡ℎ‘𝑦)) = 1o)
42		f1ocnvfv2 5924	. . . . . . . . . . . 12 ⊢ ((ℎ:𝐴–1-1-onto→𝐵 ∧ 𝑦 ∈ 𝐵) → (ℎ‘(◡ℎ‘𝑦)) = 𝑦)
43	42	fveq2d 5646	. . . . . . . . . . 11 ⊢ ((ℎ:𝐴–1-1-onto→𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑔‘(ℎ‘(◡ℎ‘𝑦))) = (𝑔‘𝑦))
44	43	ad4ant24 516	. . . . . . . . . 10 ⊢ ((((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ ∀𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = 1o) ∧ 𝑦 ∈ 𝐵) → (𝑔‘(ℎ‘(◡ℎ‘𝑦))) = (𝑔‘𝑦))
45	38, 41, 44	3eqtr3rd 2272	. . . . . . . . 9 ⊢ ((((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ ∀𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = 1o) ∧ 𝑦 ∈ 𝐵) → (𝑔‘𝑦) = 1o)
46	45	ralrimiva 2604	. . . . . . . 8 ⊢ (((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ ∀𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = 1o) → ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) = 1o)
47	29	ad3antlr 493	. . . . . . . . . . 11 ⊢ ((((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) = 1o) ∧ 𝑥 ∈ 𝐴) → ℎ Fn 𝐴)
48		simpr 110	. . . . . . . . . . 11 ⊢ ((((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) = 1o) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
49		fvco2 5718	. . . . . . . . . . 11 ⊢ ((ℎ Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝑔 ∘ ℎ)‘𝑥) = (𝑔‘(ℎ‘𝑥)))
50	47, 48, 49	syl2anc 411	. . . . . . . . . 10 ⊢ ((((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) = 1o) ∧ 𝑥 ∈ 𝐴) → ((𝑔 ∘ ℎ)‘𝑥) = (𝑔‘(ℎ‘𝑥)))
51		fveqeq2 5651	. . . . . . . . . . 11 ⊢ (𝑦 = (ℎ‘𝑥) → ((𝑔‘𝑦) = 1o ↔ (𝑔‘(ℎ‘𝑥)) = 1o))
52		simplr 529	. . . . . . . . . . 11 ⊢ ((((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) = 1o) ∧ 𝑥 ∈ 𝐴) → ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) = 1o)
53	21	ffvelcdmda 5785	. . . . . . . . . . . 12 ⊢ (((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ 𝑥 ∈ 𝐴) → (ℎ‘𝑥) ∈ 𝐵)
54	53	adantlr 477	. . . . . . . . . . 11 ⊢ ((((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) = 1o) ∧ 𝑥 ∈ 𝐴) → (ℎ‘𝑥) ∈ 𝐵)
55	51, 52, 54	rspcdva 2914	. . . . . . . . . 10 ⊢ ((((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) = 1o) ∧ 𝑥 ∈ 𝐴) → (𝑔‘(ℎ‘𝑥)) = 1o)
56	50, 55	eqtrd 2263	. . . . . . . . 9 ⊢ ((((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) = 1o) ∧ 𝑥 ∈ 𝐴) → ((𝑔 ∘ ℎ)‘𝑥) = 1o)
57	56	ralrimiva 2604	. . . . . . . 8 ⊢ (((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) = 1o) → ∀𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = 1o)
58	46, 57	impbida 600	. . . . . . 7 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) → (∀𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = 1o ↔ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) = 1o))
59	58	dcbid 845	. . . . . 6 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) → (DECID ∀𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = 1o ↔ DECID ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) = 1o))
60	28, 59	mpbid 147	. . . . 5 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) → DECID ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) = 1o)
61	3, 60	exlimddv 1946	. . . 4 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) → DECID ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) = 1o)
62	61	ralrimiva 2604	. . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ WOmni) → ∀𝑔 ∈ (2o ↑𝑚 𝐵)DECID ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) = 1o)
63		iswomnimap 7370	. . . . 5 ⊢ (𝐵 ∈ V → (𝐵 ∈ WOmni ↔ ∀𝑔 ∈ (2o ↑𝑚 𝐵)DECID ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) = 1o))
64	14, 63	syl 14	. . . 4 ⊢ (𝐴 ≈ 𝐵 → (𝐵 ∈ WOmni ↔ ∀𝑔 ∈ (2o ↑𝑚 𝐵)DECID ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) = 1o))
65	64	adantr 276	. . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ WOmni) → (𝐵 ∈ WOmni ↔ ∀𝑔 ∈ (2o ↑𝑚 𝐵)DECID ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) = 1o))
66	62, 65	mpbird 167	. 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ WOmni) → 𝐵 ∈ WOmni)
67	66	ex 115	1 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ WOmni → 𝐵 ∈ WOmni))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  DECID wdc 841   = wceq 1397  ∃wex 1540   ∈ wcel 2201  ∀wral 2509  Vcvv 2801   class class class wbr 4089  ωcom 4690  ^◡ccnv 4726   ∘ ccom 4731   Fn wfn 5323  ⟶wf 5324  –1-1-onto→wf1o 5327  ‘cfv 5328  (class class class)co 6023  1_oc1o 6580  2_oc2o 6581   ↑_𝑚 cmap 6822   ≈ cen 6912  WOmnicwomni 7367

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-sep 4208  ax-nul 4216  ax-pow 4266  ax-pr 4301  ax-un 4532  ax-setind 4637

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-ral 2514  df-rex 2515  df-v 2803  df-sbc 3031  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-nul 3494  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-int 3930  df-br 4090  df-opab 4152  df-id 4392  df-suc 4470  df-iom 4691  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-res 4739  df-ima 4740  df-iota 5288  df-fun 5330  df-fn 5331  df-f 5332  df-f1 5333  df-fo 5334  df-f1o 5335  df-fv 5336  df-ov 6026  df-oprab 6027  df-mpo 6028  df-1o 6587  df-2o 6588  df-map 6824  df-en 6915  df-womni 7368

This theorem is referenced by:  enwomni  7374