Theorem enomnilem 7337

Description: Lemma for enomni 7338. 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 6917	. . . . . . 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 ∈ Omni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) → ∃ℎ ℎ:𝐴–1-1-onto→𝐵)
4		fveq1 5638	. . . . . . . . . 10 ⊢ (𝑓 = (𝑔 ∘ ℎ) → (𝑓‘𝑥) = ((𝑔 ∘ ℎ)‘𝑥))
5	4	eqeq1d 2240	. . . . . . . . 9 ⊢ (𝑓 = (𝑔 ∘ ℎ) → ((𝑓‘𝑥) = ∅ ↔ ((𝑔 ∘ ℎ)‘𝑥) = ∅))
6	5	rexbidv 2533	. . . . . . . 8 ⊢ (𝑓 = (𝑔 ∘ ℎ) → (∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅ ↔ ∃𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = ∅))
7	4	eqeq1d 2240	. . . . . . . . 9 ⊢ (𝑓 = (𝑔 ∘ ℎ) → ((𝑓‘𝑥) = 1o ↔ ((𝑔 ∘ ℎ)‘𝑥) = 1o))
8	7	ralbidv 2532	. . . . . . . 8 ⊢ (𝑓 = (𝑔 ∘ ℎ) → (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o ↔ ∀𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = 1o))
9	6, 8	orbi12d 800	. . . . . . 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 7336	. . . . . . . . 9 ⊢ (𝐴 ∈ Omni → (𝐴 ∈ Omni ↔ ∀𝑓 ∈ (2o ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)))
11	10	ibi 176	. . . . . . . 8 ⊢ (𝐴 ∈ Omni → ∀𝑓 ∈ (2o ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o))
12	11	ad3antlr 493	. . . . . . 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 110	. . . . . . . . . . 11 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Omni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) → 𝑔 ∈ (2o ↑𝑚 𝐵))
14		2onn 6689	. . . . . . . . . . . . 13 ⊢ 2o ∈ ω
15		relen 6913	. . . . . . . . . . . . . 14 ⊢ Rel ≈
16	15	brrelex2i 4770	. . . . . . . . . . . . 13 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ∈ V)
17		elmapg 6830	. . . . . . . . . . . . 13 ⊢ ((2o ∈ ω ∧ 𝐵 ∈ V) → (𝑔 ∈ (2o ↑𝑚 𝐵) ↔ 𝑔:𝐵⟶2o))
18	14, 16, 17	sylancr 414	. . . . . . . . . . . 12 ⊢ (𝐴 ≈ 𝐵 → (𝑔 ∈ (2o ↑𝑚 𝐵) ↔ 𝑔:𝐵⟶2o))
19	18	ad2antrr 488	. . . . . . . . . . 11 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Omni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) → (𝑔 ∈ (2o ↑𝑚 𝐵) ↔ 𝑔:𝐵⟶2o))
20	13, 19	mpbid 147	. . . . . . . . . 10 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Omni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) → 𝑔:𝐵⟶2o)
21	20	adantr 276	. . . . . . . . 9 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Omni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) → 𝑔:𝐵⟶2o)
22		f1of 5583	. . . . . . . . . 10 ⊢ (ℎ:𝐴–1-1-onto→𝐵 → ℎ:𝐴⟶𝐵)
23	22	adantl 277	. . . . . . . . 9 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Omni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) → ℎ:𝐴⟶𝐵)
24		fco 5500	. . . . . . . . 9 ⊢ ((𝑔:𝐵⟶2o ∧ ℎ:𝐴⟶𝐵) → (𝑔 ∘ ℎ):𝐴⟶2o)
25	21, 23, 24	syl2anc 411	. . . . . . . 8 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Omni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) → (𝑔 ∘ ℎ):𝐴⟶2o)
26		simpllr 536	. . . . . . . . 9 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Omni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) → 𝐴 ∈ Omni)
27		elmapg 6830	. . . . . . . . 9 ⊢ ((2o ∈ ω ∧ 𝐴 ∈ Omni) → ((𝑔 ∘ ℎ) ∈ (2o ↑𝑚 𝐴) ↔ (𝑔 ∘ ℎ):𝐴⟶2o))
28	14, 26, 27	sylancr 414	. . . . . . . 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 167	. . . . . . 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 2915	. . . . . 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 5584	. . . . . . . . . . . 12 ⊢ (ℎ:𝐴–1-1-onto→𝐵 → ℎ Fn 𝐴)
32	31	ad2antlr 489	. . . . . . . . . . 11 ⊢ (((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Omni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ 𝑥 ∈ 𝐴) → ℎ Fn 𝐴)
33		fvco2 5715	. . . . . . . . . . 11 ⊢ ((ℎ Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝑔 ∘ ℎ)‘𝑥) = (𝑔‘(ℎ‘𝑥)))
34	32, 33	sylancom 420	. . . . . . . . . 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 2240	. . . . . . . . 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	ffvelcdmda 5782	. . . . . . . . . 10 ⊢ (((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Omni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ 𝑥 ∈ 𝐴) → (ℎ‘𝑥) ∈ 𝐵)
37		simpr 110	. . . . . . . . . . . 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 5643	. . . . . . . . . . 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 2240	. . . . . . . . . 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 2914	. . . . . . . . 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 150	. . . . . . . 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 2650	. . . . . . 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 493	. . . . . . . . . . 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 5596	. . . . . . . . . . . . . 14 ⊢ (ℎ:𝐴–1-1-onto→𝐵 → ◡ℎ:𝐵–1-1-onto→𝐴)
45		f1of 5583	. . . . . . . . . . . . . 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 493	. . . . . . . . . . . 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 110	. . . . . . . . . . . 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	ffvelcdmd 5783	. . . . . . . . . . 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 5715	. . . . . . . . . . 11 ⊢ ((ℎ Fn 𝐴 ∧ (◡ℎ‘𝑦) ∈ 𝐴) → ((𝑔 ∘ ℎ)‘(◡ℎ‘𝑦)) = (𝑔‘(ℎ‘(◡ℎ‘𝑦))))
51	43, 49, 50	syl2anc 411	. . . . . . . . . 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 5639	. . . . . . . . . . . 12 ⊢ (𝑥 = (◡ℎ‘𝑦) → ((𝑔 ∘ ℎ)‘𝑥) = ((𝑔 ∘ ℎ)‘(◡ℎ‘𝑦)))
53	52	eqeq1d 2240	. . . . . . . . . . 11 ⊢ (𝑥 = (◡ℎ‘𝑦) → (((𝑔 ∘ ℎ)‘𝑥) = 1o ↔ ((𝑔 ∘ ℎ)‘(◡ℎ‘𝑦)) = 1o))
54		simplr 529	. . . . . . . . . . 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 2915	. . . . . . . . . 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 536	. . . . . . . . . . 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 5919	. . . . . . . . . . . 12 ⊢ ((ℎ:𝐴–1-1-onto→𝐵 ∧ 𝑦 ∈ 𝐵) → (ℎ‘(◡ℎ‘𝑦)) = 𝑦)
58	57	fveq2d 5643	. . . . . . . . . . 11 ⊢ ((ℎ:𝐴–1-1-onto→𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑔‘(ℎ‘(◡ℎ‘𝑦))) = (𝑔‘𝑦))
59	56, 58	sylancom 420	. . . . . . . . . 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 2273	. . . . . . . . 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 2605	. . . . . . . 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 115	. . . . . . 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 793	. . . . . 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 1947	. . . 4 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Omni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) → (∃𝑦 ∈ 𝐵 (𝑔‘𝑦) = ∅ ∨ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) = 1o))
66	65	ralrimiva 2605	. . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Omni) → ∀𝑔 ∈ (2o ↑𝑚 𝐵)(∃𝑦 ∈ 𝐵 (𝑔‘𝑦) = ∅ ∨ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) = 1o))
67		isomnimap 7336	. . . . 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 276	. . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Omni) → (𝐵 ∈ Omni ↔ ∀𝑔 ∈ (2o ↑𝑚 𝐵)(∃𝑦 ∈ 𝐵 (𝑔‘𝑦) = ∅ ∨ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) = 1o)))
70	66, 69	mpbird 167	. 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Omni) → 𝐵 ∈ Omni)
71	70	ex 115	1 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ Omni → 𝐵 ∈ Omni))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 715   = wceq 1397  ∃wex 1540   ∈ wcel 2202  ∀wral 2510  ∃wrex 2511  Vcvv 2802  ∅c0 3494   class class class wbr 4088  ωcom 4688  ^◡ccnv 4724   ∘ ccom 4729   Fn wfn 5321  ⟶wf 5322  –1-1-onto→wf1o 5325  ‘cfv 5326  (class class class)co 6018  1_oc1o 6575  2_oc2o 6576   ↑_𝑚 cmap 6817   ≈ cen 6907  Omnicomni 7333

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 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-id 4390  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1o 6582  df-2o 6583  df-map 6819  df-en 6910  df-omni 7334

This theorem is referenced by:  enomni  7338