Theorem enomnilem 7010

Description: Lemma for enomni 7011. One direction of the biconditional. (Contributed by Jim Kingdon, 13-Jul-2022.)

Assertion

Ref	Expression
enomnilem	⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ Omni → 𝐵 ∈ Omni))

Proof of Theorem enomnilem

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

Step	Hyp	Ref	Expression
1		bren 6641	. . . . . . 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 479	. . . . 5 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Omni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) → ∃ℎ ℎ:𝐴–1-1-onto→𝐵)
4		fveq1 5420	. . . . . . . . . 10 ⊢ (𝑓 = (𝑔 ∘ ℎ) → (𝑓‘𝑥) = ((𝑔 ∘ ℎ)‘𝑥))
5	4	eqeq1d 2148	. . . . . . . . 9 ⊢ (𝑓 = (𝑔 ∘ ℎ) → ((𝑓‘𝑥) = ∅ ↔ ((𝑔 ∘ ℎ)‘𝑥) = ∅))
6	5	rexbidv 2438	. . . . . . . 8 ⊢ (𝑓 = (𝑔 ∘ ℎ) → (∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅ ↔ ∃𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = ∅))
7	4	eqeq1d 2148	. . . . . . . . 9 ⊢ (𝑓 = (𝑔 ∘ ℎ) → ((𝑓‘𝑥) = 1o ↔ ((𝑔 ∘ ℎ)‘𝑥) = 1o))
8	7	ralbidv 2437	. . . . . . . 8 ⊢ (𝑓 = (𝑔 ∘ ℎ) → (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o ↔ ∀𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = 1o))
9	6, 8	orbi12d 782	. . . . . . 7 ⊢ (𝑓 = (𝑔 ∘ ℎ) → ((∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o) ↔ (∃𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = 1o)))
10		isomnimap 7009	. . . . . . . . 9 ⊢ (𝐴 ∈ Omni → (𝐴 ∈ Omni ↔ ∀𝑓 ∈ (2o ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)))
11	10	ibi 175	. . . . . . . 8 ⊢ (𝐴 ∈ Omni → ∀𝑓 ∈ (2o ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o))
12	11	ad3antlr 484	. . . . . . 7 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Omni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) → ∀𝑓 ∈ (2o ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o))
13		simpr 109	. . . . . . . . . . 11 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Omni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) → 𝑔 ∈ (2o ↑𝑚 𝐵))
14		2onn 6417	. . . . . . . . . . . . 13 ⊢ 2o ∈ ω
15		relen 6638	. . . . . . . . . . . . . 14 ⊢ Rel ≈
16	15	brrelex2i 4583	. . . . . . . . . . . . 13 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ∈ V)
17		elmapg 6555	. . . . . . . . . . . . 13 ⊢ ((2o ∈ ω ∧ 𝐵 ∈ V) → (𝑔 ∈ (2o ↑𝑚 𝐵) ↔ 𝑔:𝐵⟶2o))
18	14, 16, 17	sylancr 410	. . . . . . . . . . . 12 ⊢ (𝐴 ≈ 𝐵 → (𝑔 ∈ (2o ↑𝑚 𝐵) ↔ 𝑔:𝐵⟶2o))
19	18	ad2antrr 479	. . . . . . . . . . 11 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Omni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) → (𝑔 ∈ (2o ↑𝑚 𝐵) ↔ 𝑔:𝐵⟶2o))
20	13, 19	mpbid 146	. . . . . . . . . 10 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Omni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) → 𝑔:𝐵⟶2o)
21	20	adantr 274	. . . . . . . . 9 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Omni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) → 𝑔:𝐵⟶2o)
22		f1of 5367	. . . . . . . . . 10 ⊢ (ℎ:𝐴–1-1-onto→𝐵 → ℎ:𝐴⟶𝐵)
23	22	adantl 275	. . . . . . . . 9 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Omni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) → ℎ:𝐴⟶𝐵)
24		fco 5288	. . . . . . . . 9 ⊢ ((𝑔:𝐵⟶2o ∧ ℎ:𝐴⟶𝐵) → (𝑔 ∘ ℎ):𝐴⟶2o)
25	21, 23, 24	syl2anc 408	. . . . . . . 8 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Omni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) → (𝑔 ∘ ℎ):𝐴⟶2o)
26		simpllr 523	. . . . . . . . 9 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Omni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) → 𝐴 ∈ Omni)
27		elmapg 6555	. . . . . . . . 9 ⊢ ((2o ∈ ω ∧ 𝐴 ∈ Omni) → ((𝑔 ∘ ℎ) ∈ (2o ↑𝑚 𝐴) ↔ (𝑔 ∘ ℎ):𝐴⟶2o))
28	14, 26, 27	sylancr 410	. . . . . . . 8 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Omni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) → ((𝑔 ∘ ℎ) ∈ (2o ↑𝑚 𝐴) ↔ (𝑔 ∘ ℎ):𝐴⟶2o))
29	25, 28	mpbird 166	. . . . . . 7 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Omni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) → (𝑔 ∘ ℎ) ∈ (2o ↑𝑚 𝐴))
30	9, 12, 29	rspcdva 2794	. . . . . 6 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Omni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) → (∃𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = 1o))
31		f1ofn 5368	. . . . . . . . . . . 12 ⊢ (ℎ:𝐴–1-1-onto→𝐵 → ℎ Fn 𝐴)
32	31	ad2antlr 480	. . . . . . . . . . 11 ⊢ (((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Omni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ 𝑥 ∈ 𝐴) → ℎ Fn 𝐴)
33		fvco2 5490	. . . . . . . . . . 11 ⊢ ((ℎ Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝑔 ∘ ℎ)‘𝑥) = (𝑔‘(ℎ‘𝑥)))
34	32, 33	sylancom 416	. . . . . . . . . 10 ⊢ (((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Omni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ 𝑥 ∈ 𝐴) → ((𝑔 ∘ ℎ)‘𝑥) = (𝑔‘(ℎ‘𝑥)))
35	34	eqeq1d 2148	. . . . . . . . 9 ⊢ (((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Omni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ 𝑥 ∈ 𝐴) → (((𝑔 ∘ ℎ)‘𝑥) = ∅ ↔ (𝑔‘(ℎ‘𝑥)) = ∅))
36	23	ffvelrnda 5555	. . . . . . . . . 10 ⊢ (((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Omni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ 𝑥 ∈ 𝐴) → (ℎ‘𝑥) ∈ 𝐵)
37		simpr 109	. . . . . . . . . . . 12 ⊢ ((((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Omni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 = (ℎ‘𝑥)) → 𝑦 = (ℎ‘𝑥))
38	37	fveq2d 5425	. . . . . . . . . . 11 ⊢ ((((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Omni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 = (ℎ‘𝑥)) → (𝑔‘𝑦) = (𝑔‘(ℎ‘𝑥)))
39	38	eqeq1d 2148	. . . . . . . . . 10 ⊢ ((((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Omni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 = (ℎ‘𝑥)) → ((𝑔‘𝑦) = ∅ ↔ (𝑔‘(ℎ‘𝑥)) = ∅))
40	36, 39	rspcedv 2793	. . . . . . . . 9 ⊢ (((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Omni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ 𝑥 ∈ 𝐴) → ((𝑔‘(ℎ‘𝑥)) = ∅ → ∃𝑦 ∈ 𝐵 (𝑔‘𝑦) = ∅))
41	35, 40	sylbid 149	. . . . . . . 8 ⊢ (((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Omni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ 𝑥 ∈ 𝐴) → (((𝑔 ∘ ℎ)‘𝑥) = ∅ → ∃𝑦 ∈ 𝐵 (𝑔‘𝑦) = ∅))
42	41	rexlimdva 2549	. . . . . . 7 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Omni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) → (∃𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = ∅ → ∃𝑦 ∈ 𝐵 (𝑔‘𝑦) = ∅))
43	31	ad3antlr 484	. . . . . . . . . . 11 ⊢ ((((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Omni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ ∀𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = 1o) ∧ 𝑦 ∈ 𝐵) → ℎ Fn 𝐴)
44		f1ocnv 5380	. . . . . . . . . . . . . 14 ⊢ (ℎ:𝐴–1-1-onto→𝐵 → ◡ℎ:𝐵–1-1-onto→𝐴)
45		f1of 5367	. . . . . . . . . . . . . 14 ⊢ (◡ℎ:𝐵–1-1-onto→𝐴 → ◡ℎ:𝐵⟶𝐴)
46	44, 45	syl 14	. . . . . . . . . . . . 13 ⊢ (ℎ:𝐴–1-1-onto→𝐵 → ◡ℎ:𝐵⟶𝐴)
47	46	ad3antlr 484	. . . . . . . . . . . 12 ⊢ ((((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Omni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ ∀𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = 1o) ∧ 𝑦 ∈ 𝐵) → ◡ℎ:𝐵⟶𝐴)
48		simpr 109	. . . . . . . . . . . 12 ⊢ ((((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Omni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ ∀𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = 1o) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
49	47, 48	ffvelrnd 5556	. . . . . . . . . . 11 ⊢ ((((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Omni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ ∀𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = 1o) ∧ 𝑦 ∈ 𝐵) → (◡ℎ‘𝑦) ∈ 𝐴)
50		fvco2 5490	. . . . . . . . . . 11 ⊢ ((ℎ Fn 𝐴 ∧ (◡ℎ‘𝑦) ∈ 𝐴) → ((𝑔 ∘ ℎ)‘(◡ℎ‘𝑦)) = (𝑔‘(ℎ‘(◡ℎ‘𝑦))))
51	43, 49, 50	syl2anc 408	. . . . . . . . . 10 ⊢ ((((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Omni) ∧ 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))))
52		fveq2 5421	. . . . . . . . . . . 12 ⊢ (𝑥 = (◡ℎ‘𝑦) → ((𝑔 ∘ ℎ)‘𝑥) = ((𝑔 ∘ ℎ)‘(◡ℎ‘𝑦)))
53	52	eqeq1d 2148	. . . . . . . . . . 11 ⊢ (𝑥 = (◡ℎ‘𝑦) → (((𝑔 ∘ ℎ)‘𝑥) = 1o ↔ ((𝑔 ∘ ℎ)‘(◡ℎ‘𝑦)) = 1o))
54		simplr 519	. . . . . . . . . . 11 ⊢ ((((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Omni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ ∀𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = 1o) ∧ 𝑦 ∈ 𝐵) → ∀𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = 1o)
55	53, 54, 49	rspcdva 2794	. . . . . . . . . 10 ⊢ ((((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Omni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ ∀𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = 1o) ∧ 𝑦 ∈ 𝐵) → ((𝑔 ∘ ℎ)‘(◡ℎ‘𝑦)) = 1o)
56		simpllr 523	. . . . . . . . . . 11 ⊢ ((((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Omni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ ∀𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = 1o) ∧ 𝑦 ∈ 𝐵) → ℎ:𝐴–1-1-onto→𝐵)
57		f1ocnvfv2 5679	. . . . . . . . . . . 12 ⊢ ((ℎ:𝐴–1-1-onto→𝐵 ∧ 𝑦 ∈ 𝐵) → (ℎ‘(◡ℎ‘𝑦)) = 𝑦)
58	57	fveq2d 5425	. . . . . . . . . . 11 ⊢ ((ℎ:𝐴–1-1-onto→𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑔‘(ℎ‘(◡ℎ‘𝑦))) = (𝑔‘𝑦))
59	56, 58	sylancom 416	. . . . . . . . . 10 ⊢ ((((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Omni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ ∀𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = 1o) ∧ 𝑦 ∈ 𝐵) → (𝑔‘(ℎ‘(◡ℎ‘𝑦))) = (𝑔‘𝑦))
60	51, 55, 59	3eqtr3rd 2181	. . . . . . . . 9 ⊢ ((((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Omni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ ∀𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = 1o) ∧ 𝑦 ∈ 𝐵) → (𝑔‘𝑦) = 1o)
61	60	ralrimiva 2505	. . . . . . . 8 ⊢ (((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Omni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ ∀𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = 1o) → ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) = 1o)
62	61	ex 114	. . . . . . 7 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Omni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) → (∀𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = 1o → ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) = 1o))
63	42, 62	orim12d 775	. . . . . 6 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Omni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) → ((∃𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = 1o) → (∃𝑦 ∈ 𝐵 (𝑔‘𝑦) = ∅ ∨ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) = 1o)))
64	30, 63	mpd 13	. . . . 5 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Omni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) → (∃𝑦 ∈ 𝐵 (𝑔‘𝑦) = ∅ ∨ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) = 1o))
65	3, 64	exlimddv 1870	. . . 4 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Omni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) → (∃𝑦 ∈ 𝐵 (𝑔‘𝑦) = ∅ ∨ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) = 1o))
66	65	ralrimiva 2505	. . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Omni) → ∀𝑔 ∈ (2o ↑𝑚 𝐵)(∃𝑦 ∈ 𝐵 (𝑔‘𝑦) = ∅ ∨ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) = 1o))
67		isomnimap 7009	. . . . 5 ⊢ (𝐵 ∈ V → (𝐵 ∈ Omni ↔ ∀𝑔 ∈ (2o ↑𝑚 𝐵)(∃𝑦 ∈ 𝐵 (𝑔‘𝑦) = ∅ ∨ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) = 1o)))
68	16, 67	syl 14	. . . 4 ⊢ (𝐴 ≈ 𝐵 → (𝐵 ∈ Omni ↔ ∀𝑔 ∈ (2o ↑𝑚 𝐵)(∃𝑦 ∈ 𝐵 (𝑔‘𝑦) = ∅ ∨ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) = 1o)))
69	68	adantr 274	. . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Omni) → (𝐵 ∈ Omni ↔ ∀𝑔 ∈ (2o ↑𝑚 𝐵)(∃𝑦 ∈ 𝐵 (𝑔‘𝑦) = ∅ ∨ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) = 1o)))
70	66, 69	mpbird 166	. 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Omni) → 𝐵 ∈ Omni)
71	70	ex 114	1 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ Omni → 𝐵 ∈ Omni))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 697   = wceq 1331  ∃wex 1468   ∈ wcel 1480  ∀wral 2416  ∃wrex 2417  Vcvv 2686  ∅c0 3363   class class class wbr 3929  ωcom 4504  ^◡ccnv 4538   ∘ ccom 4543   Fn wfn 5118  ⟶wf 5119  –1-1-onto→wf1o 5122  ‘cfv 5123  (class class class)co 5774  1_oc1o 6306  2_oc2o 6307   ↑_𝑚 cmap 6542   ≈ cen 6632  Omnicomni 7004

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-br 3930  df-opab 3990  df-id 4215  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1o 6313  df-2o 6314  df-map 6544  df-en 6635  df-omni 7006

This theorem is referenced by:  enomni  7011