Theorem enwomnilem 7124

Description: Lemma for enwomni 7125. 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 6704	. . . . . . 7 ⊢ (𝐴 ≈ 𝐵 ↔ ∃ℎ ℎ:𝐴–1-1-onto→𝐵)
2	1	biimpi 119	. . . . . 6 ⊢ (𝐴 ≈ 𝐵 → ∃ℎ ℎ:𝐴–1-1-onto→𝐵)
3	2	ad2antrr 480	. . . . 5 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) → ∃ℎ ℎ:𝐴–1-1-onto→𝐵)
4		fveq1 5479	. . . . . . . . . 10 ⊢ (𝑓 = (𝑔 ∘ ℎ) → (𝑓‘𝑥) = ((𝑔 ∘ ℎ)‘𝑥))
5	4	eqeq1d 2173	. . . . . . . . 9 ⊢ (𝑓 = (𝑔 ∘ ℎ) → ((𝑓‘𝑥) = 1o ↔ ((𝑔 ∘ ℎ)‘𝑥) = 1o))
6	5	ralbidv 2464	. . . . . . . 8 ⊢ (𝑓 = (𝑔 ∘ ℎ) → (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o ↔ ∀𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = 1o))
7	6	dcbid 828	. . . . . . 7 ⊢ (𝑓 = (𝑔 ∘ ℎ) → (DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o ↔ DECID ∀𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = 1o))
8		iswomnimap 7121	. . . . . . . . 9 ⊢ (𝐴 ∈ WOmni → (𝐴 ∈ WOmni ↔ ∀𝑓 ∈ (2o ↑𝑚 𝐴)DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o))
9	8	ibi 175	. . . . . . . 8 ⊢ (𝐴 ∈ WOmni → ∀𝑓 ∈ (2o ↑𝑚 𝐴)DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)
10	9	ad3antlr 485	. . . . . . 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 109	. . . . . . . . . . 11 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) → 𝑔 ∈ (2o ↑𝑚 𝐵))
12		2onn 6480	. . . . . . . . . . . . 13 ⊢ 2o ∈ ω
13		relen 6701	. . . . . . . . . . . . . 14 ⊢ Rel ≈
14	13	brrelex2i 4642	. . . . . . . . . . . . 13 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ∈ V)
15		elmapg 6618	. . . . . . . . . . . . 13 ⊢ ((2o ∈ ω ∧ 𝐵 ∈ V) → (𝑔 ∈ (2o ↑𝑚 𝐵) ↔ 𝑔:𝐵⟶2o))
16	12, 14, 15	sylancr 411	. . . . . . . . . . . 12 ⊢ (𝐴 ≈ 𝐵 → (𝑔 ∈ (2o ↑𝑚 𝐵) ↔ 𝑔:𝐵⟶2o))
17	16	ad2antrr 480	. . . . . . . . . . 11 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) → (𝑔 ∈ (2o ↑𝑚 𝐵) ↔ 𝑔:𝐵⟶2o))
18	11, 17	mpbid 146	. . . . . . . . . 10 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) → 𝑔:𝐵⟶2o)
19	18	adantr 274	. . . . . . . . 9 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) → 𝑔:𝐵⟶2o)
20		f1of 5426	. . . . . . . . . 10 ⊢ (ℎ:𝐴–1-1-onto→𝐵 → ℎ:𝐴⟶𝐵)
21	20	adantl 275	. . . . . . . . 9 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) → ℎ:𝐴⟶𝐵)
22		fco 5347	. . . . . . . . 9 ⊢ ((𝑔:𝐵⟶2o ∧ ℎ:𝐴⟶𝐵) → (𝑔 ∘ ℎ):𝐴⟶2o)
23	19, 21, 22	syl2anc 409	. . . . . . . 8 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) → (𝑔 ∘ ℎ):𝐴⟶2o)
24		simpllr 524	. . . . . . . . 9 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) → 𝐴 ∈ WOmni)
25		elmapg 6618	. . . . . . . . 9 ⊢ ((2o ∈ ω ∧ 𝐴 ∈ WOmni) → ((𝑔 ∘ ℎ) ∈ (2o ↑𝑚 𝐴) ↔ (𝑔 ∘ ℎ):𝐴⟶2o))
26	12, 24, 25	sylancr 411	. . . . . . . 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 166	. . . . . . 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 2830	. . . . . 6 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) → DECID ∀𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = 1o)
29		f1ofn 5427	. . . . . . . . . . . 12 ⊢ (ℎ:𝐴–1-1-onto→𝐵 → ℎ Fn 𝐴)
30	29	ad3antlr 485	. . . . . . . . . . 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 5439	. . . . . . . . . . . . . 14 ⊢ (ℎ:𝐴–1-1-onto→𝐵 → ◡ℎ:𝐵–1-1-onto→𝐴)
32		f1of 5426	. . . . . . . . . . . . . 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 485	. . . . . . . . . . . 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 109	. . . . . . . . . . . 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	ffvelrnd 5615	. . . . . . . . . . 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 5549	. . . . . . . . . . 11 ⊢ ((ℎ Fn 𝐴 ∧ (◡ℎ‘𝑦) ∈ 𝐴) → ((𝑔 ∘ ℎ)‘(◡ℎ‘𝑦)) = (𝑔‘(ℎ‘(◡ℎ‘𝑦))))
38	30, 36, 37	syl2anc 409	. . . . . . . . . 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 5489	. . . . . . . . . . 11 ⊢ (𝑥 = (◡ℎ‘𝑦) → (((𝑔 ∘ ℎ)‘𝑥) = 1o ↔ ((𝑔 ∘ ℎ)‘(◡ℎ‘𝑦)) = 1o))
40		simplr 520	. . . . . . . . . . 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 2830	. . . . . . . . . 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 5740	. . . . . . . . . . . 12 ⊢ ((ℎ:𝐴–1-1-onto→𝐵 ∧ 𝑦 ∈ 𝐵) → (ℎ‘(◡ℎ‘𝑦)) = 𝑦)
43	42	fveq2d 5484	. . . . . . . . . . 11 ⊢ ((ℎ:𝐴–1-1-onto→𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑔‘(ℎ‘(◡ℎ‘𝑦))) = (𝑔‘𝑦))
44	43	ad4ant24 508	. . . . . . . . . 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 2206	. . . . . . . . 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 2537	. . . . . . . 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 485	. . . . . . . . . . 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 109	. . . . . . . . . . 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 5549	. . . . . . . . . . 11 ⊢ ((ℎ Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝑔 ∘ ℎ)‘𝑥) = (𝑔‘(ℎ‘𝑥)))
50	47, 48, 49	syl2anc 409	. . . . . . . . . 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 5489	. . . . . . . . . . 11 ⊢ (𝑦 = (ℎ‘𝑥) → ((𝑔‘𝑦) = 1o ↔ (𝑔‘(ℎ‘𝑥)) = 1o))
52		simplr 520	. . . . . . . . . . 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	ffvelrnda 5614	. . . . . . . . . . . 12 ⊢ (((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ 𝑥 ∈ 𝐴) → (ℎ‘𝑥) ∈ 𝐵)
54	53	adantlr 469	. . . . . . . . . . 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 2830	. . . . . . . . . 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 2197	. . . . . . . . 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 2537	. . . . . . . 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 586	. . . . . . 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 828	. . . . . 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 146	. . . . 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 1885	. . . 4 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) → DECID ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) = 1o)
62	61	ralrimiva 2537	. . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ WOmni) → ∀𝑔 ∈ (2o ↑𝑚 𝐵)DECID ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) = 1o)
63		iswomnimap 7121	. . . . 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 274	. . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ WOmni) → (𝐵 ∈ WOmni ↔ ∀𝑔 ∈ (2o ↑𝑚 𝐵)DECID ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) = 1o))
66	62, 65	mpbird 166	. 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ WOmni) → 𝐵 ∈ WOmni)
67	66	ex 114	1 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ WOmni → 𝐵 ∈ WOmni))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  DECID wdc 824   = wceq 1342  ∃wex 1479   ∈ wcel 2135  ∀wral 2442  Vcvv 2721   class class class wbr 3976  ωcom 4561  ^◡ccnv 4597   ∘ ccom 4602   Fn wfn 5177  ⟶wf 5178  –1-1-onto→wf1o 5181  ‘cfv 5182  (class class class)co 5836  1_oc1o 6368  2_oc2o 6369   ↑_𝑚 cmap 6605   ≈ cen 6695  WOmnicwomni 7118

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-sep 4094  ax-nul 4102  ax-pow 4147  ax-pr 4181  ax-un 4405  ax-setind 4508

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-ral 2447  df-rex 2448  df-v 2723  df-sbc 2947  df-dif 3113  df-un 3115  df-in 3117  df-ss 3124  df-nul 3405  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-int 3819  df-br 3977  df-opab 4038  df-id 4265  df-suc 4343  df-iom 4562  df-xp 4604  df-rel 4605  df-cnv 4606  df-co 4607  df-dm 4608  df-rn 4609  df-res 4610  df-ima 4611  df-iota 5147  df-fun 5184  df-fn 5185  df-f 5186  df-f1 5187  df-fo 5188  df-f1o 5189  df-fv 5190  df-ov 5839  df-oprab 5840  df-mpo 5841  df-1o 6375  df-2o 6376  df-map 6607  df-en 6698  df-womni 7119

This theorem is referenced by:  enwomni  7125