Theorem enomnilem 7018

Description: Lemma for enomni 7019. 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 6649	. . . . . . 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 ∈ Omni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) → ∃ℎ ℎ:𝐴–1-1-onto→𝐵)
4		fveq1 5428	. . . . . . . . . 10 ⊢ (𝑓 = (𝑔 ∘ ℎ) → (𝑓‘𝑥) = ((𝑔 ∘ ℎ)‘𝑥))
5	4	eqeq1d 2149	. . . . . . . . 9 ⊢ (𝑓 = (𝑔 ∘ ℎ) → ((𝑓‘𝑥) = ∅ ↔ ((𝑔 ∘ ℎ)‘𝑥) = ∅))
6	5	rexbidv 2439	. . . . . . . 8 ⊢ (𝑓 = (𝑔 ∘ ℎ) → (∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅ ↔ ∃𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = ∅))
7	4	eqeq1d 2149	. . . . . . . . 9 ⊢ (𝑓 = (𝑔 ∘ ℎ) → ((𝑓‘𝑥) = 1o ↔ ((𝑔 ∘ ℎ)‘𝑥) = 1o))
8	7	ralbidv 2438	. . . . . . . 8 ⊢ (𝑓 = (𝑔 ∘ ℎ) → (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o ↔ ∀𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = 1o))
9	6, 8	orbi12d 783	. . . . . . 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 7017	. . . . . . . . 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 485	. . . . . . 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 6425	. . . . . . . . . . . . 13 ⊢ 2o ∈ ω
15		relen 6646	. . . . . . . . . . . . . 14 ⊢ Rel ≈
16	15	brrelex2i 4591	. . . . . . . . . . . . 13 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ∈ V)
17		elmapg 6563	. . . . . . . . . . . . 13 ⊢ ((2o ∈ ω ∧ 𝐵 ∈ V) → (𝑔 ∈ (2o ↑𝑚 𝐵) ↔ 𝑔:𝐵⟶2o))
18	14, 16, 17	sylancr 411	. . . . . . . . . . . 12 ⊢ (𝐴 ≈ 𝐵 → (𝑔 ∈ (2o ↑𝑚 𝐵) ↔ 𝑔:𝐵⟶2o))
19	18	ad2antrr 480	. . . . . . . . . . 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 5375	. . . . . . . . . 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 5296	. . . . . . . . 9 ⊢ ((𝑔:𝐵⟶2o ∧ ℎ:𝐴⟶𝐵) → (𝑔 ∘ ℎ):𝐴⟶2o)
25	21, 23, 24	syl2anc 409	. . . . . . . 8 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Omni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) → (𝑔 ∘ ℎ):𝐴⟶2o)
26		simpllr 524	. . . . . . . . 9 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Omni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) → 𝐴 ∈ Omni)
27		elmapg 6563	. . . . . . . . 9 ⊢ ((2o ∈ ω ∧ 𝐴 ∈ Omni) → ((𝑔 ∘ ℎ) ∈ (2o ↑𝑚 𝐴) ↔ (𝑔 ∘ ℎ):𝐴⟶2o))
28	14, 26, 27	sylancr 411	. . . . . . . 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 2798	. . . . . 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 5376	. . . . . . . . . . . 12 ⊢ (ℎ:𝐴–1-1-onto→𝐵 → ℎ Fn 𝐴)
32	31	ad2antlr 481	. . . . . . . . . . 11 ⊢ (((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Omni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ 𝑥 ∈ 𝐴) → ℎ Fn 𝐴)
33		fvco2 5498	. . . . . . . . . . 11 ⊢ ((ℎ Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝑔 ∘ ℎ)‘𝑥) = (𝑔‘(ℎ‘𝑥)))
34	32, 33	sylancom 417	. . . . . . . . . 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 2149	. . . . . . . . 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 5563	. . . . . . . . . 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 5433	. . . . . . . . . . 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 2149	. . . . . . . . . 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 2797	. . . . . . . . 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 2552	. . . . . . 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 485	. . . . . . . . . . 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 5388	. . . . . . . . . . . . . 14 ⊢ (ℎ:𝐴–1-1-onto→𝐵 → ◡ℎ:𝐵–1-1-onto→𝐴)
45		f1of 5375	. . . . . . . . . . . . . 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 485	. . . . . . . . . . . 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 5564	. . . . . . . . . . 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 5498	. . . . . . . . . . 11 ⊢ ((ℎ Fn 𝐴 ∧ (◡ℎ‘𝑦) ∈ 𝐴) → ((𝑔 ∘ ℎ)‘(◡ℎ‘𝑦)) = (𝑔‘(ℎ‘(◡ℎ‘𝑦))))
51	43, 49, 50	syl2anc 409	. . . . . . . . . 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 5429	. . . . . . . . . . . 12 ⊢ (𝑥 = (◡ℎ‘𝑦) → ((𝑔 ∘ ℎ)‘𝑥) = ((𝑔 ∘ ℎ)‘(◡ℎ‘𝑦)))
53	52	eqeq1d 2149	. . . . . . . . . . 11 ⊢ (𝑥 = (◡ℎ‘𝑦) → (((𝑔 ∘ ℎ)‘𝑥) = 1o ↔ ((𝑔 ∘ ℎ)‘(◡ℎ‘𝑦)) = 1o))
54		simplr 520	. . . . . . . . . . 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 2798	. . . . . . . . . 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 524	. . . . . . . . . . 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 5687	. . . . . . . . . . . 12 ⊢ ((ℎ:𝐴–1-1-onto→𝐵 ∧ 𝑦 ∈ 𝐵) → (ℎ‘(◡ℎ‘𝑦)) = 𝑦)
58	57	fveq2d 5433	. . . . . . . . . . 11 ⊢ ((ℎ:𝐴–1-1-onto→𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑔‘(ℎ‘(◡ℎ‘𝑦))) = (𝑔‘𝑦))
59	56, 58	sylancom 417	. . . . . . . . . 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 2182	. . . . . . . . 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 2508	. . . . . . . 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 776	. . . . . 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 1871	. . . 4 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Omni) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) → (∃𝑦 ∈ 𝐵 (𝑔‘𝑦) = ∅ ∨ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) = 1o))
66	65	ralrimiva 2508	. . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Omni) → ∀𝑔 ∈ (2o ↑𝑚 𝐵)(∃𝑦 ∈ 𝐵 (𝑔‘𝑦) = ∅ ∨ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) = 1o))
67		isomnimap 7017	. . . . 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 698   = wceq 1332  ∃wex 1469   ∈ wcel 1481  ∀wral 2417  ∃wrex 2418  Vcvv 2689  ∅c0 3368   class class class wbr 3937  ωcom 4512  ^◡ccnv 4546   ∘ ccom 4551   Fn wfn 5126  ⟶wf 5127  –1-1-onto→wf1o 5130  ‘cfv 5131  (class class class)co 5782  1_oc1o 6314  2_oc2o 6315   ↑_𝑚 cmap 6550   ≈ cen 6640  Omnicomni 7012

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2914  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-br 3938  df-opab 3998  df-id 4223  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1o 6321  df-2o 6322  df-map 6552  df-en 6643  df-omni 7014

This theorem is referenced by:  enomni  7019