Theorem enmkvlem 7153

Description: Lemma for enmkv 7154. One direction of the biconditional. (Contributed by Jim Kingdon, 25-Jun-2024.)

Assertion

Ref	Expression
enmkvlem	⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ Markov → 𝐵 ∈ Markov))

Proof of Theorem enmkvlem

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

Step	Hyp	Ref	Expression
1		bren 6741	. . . . . . 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 ∈ Markov) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) → ∃ℎ ℎ:𝐴–1-1-onto→𝐵)
4		f1ofn 5458	. . . . . . . . . . . 12 ⊢ (ℎ:𝐴–1-1-onto→𝐵 → ℎ Fn 𝐴)
5	4	ad3antlr 493	. . . . . . . . . . 11 ⊢ ((((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Markov) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ ∀𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = 1o) ∧ 𝑦 ∈ 𝐵) → ℎ Fn 𝐴)
6		f1ocnv 5470	. . . . . . . . . . . . . 14 ⊢ (ℎ:𝐴–1-1-onto→𝐵 → ◡ℎ:𝐵–1-1-onto→𝐴)
7	6	ad3antlr 493	. . . . . . . . . . . . 13 ⊢ ((((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Markov) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ ∀𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = 1o) ∧ 𝑦 ∈ 𝐵) → ◡ℎ:𝐵–1-1-onto→𝐴)
8		f1of 5457	. . . . . . . . . . . . 13 ⊢ (◡ℎ:𝐵–1-1-onto→𝐴 → ◡ℎ:𝐵⟶𝐴)
9	7, 8	syl 14	. . . . . . . . . . . 12 ⊢ ((((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Markov) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ ∀𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = 1o) ∧ 𝑦 ∈ 𝐵) → ◡ℎ:𝐵⟶𝐴)
10		simpr 110	. . . . . . . . . . . 12 ⊢ ((((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Markov) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ ∀𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = 1o) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
11	9, 10	ffvelcdmd 5648	. . . . . . . . . . 11 ⊢ ((((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Markov) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ ∀𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = 1o) ∧ 𝑦 ∈ 𝐵) → (◡ℎ‘𝑦) ∈ 𝐴)
12		fvco2 5581	. . . . . . . . . . 11 ⊢ ((ℎ Fn 𝐴 ∧ (◡ℎ‘𝑦) ∈ 𝐴) → ((𝑔 ∘ ℎ)‘(◡ℎ‘𝑦)) = (𝑔‘(ℎ‘(◡ℎ‘𝑦))))
13	5, 11, 12	syl2anc 411	. . . . . . . . . 10 ⊢ ((((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Markov) ∧ 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))))
14		fveqeq2 5520	. . . . . . . . . . 11 ⊢ (𝑥 = (◡ℎ‘𝑦) → (((𝑔 ∘ ℎ)‘𝑥) = 1o ↔ ((𝑔 ∘ ℎ)‘(◡ℎ‘𝑦)) = 1o))
15		simplr 528	. . . . . . . . . . 11 ⊢ ((((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Markov) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ ∀𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = 1o) ∧ 𝑦 ∈ 𝐵) → ∀𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = 1o)
16	14, 15, 11	rspcdva 2846	. . . . . . . . . 10 ⊢ ((((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Markov) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ ∀𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = 1o) ∧ 𝑦 ∈ 𝐵) → ((𝑔 ∘ ℎ)‘(◡ℎ‘𝑦)) = 1o)
17		simpllr 534	. . . . . . . . . . . 12 ⊢ ((((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Markov) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ ∀𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = 1o) ∧ 𝑦 ∈ 𝐵) → ℎ:𝐴–1-1-onto→𝐵)
18		f1ocnvfv2 5773	. . . . . . . . . . . 12 ⊢ ((ℎ:𝐴–1-1-onto→𝐵 ∧ 𝑦 ∈ 𝐵) → (ℎ‘(◡ℎ‘𝑦)) = 𝑦)
19	17, 18	sylancom 420	. . . . . . . . . . 11 ⊢ ((((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Markov) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ ∀𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = 1o) ∧ 𝑦 ∈ 𝐵) → (ℎ‘(◡ℎ‘𝑦)) = 𝑦)
20	19	fveq2d 5515	. . . . . . . . . 10 ⊢ ((((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Markov) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ ∀𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = 1o) ∧ 𝑦 ∈ 𝐵) → (𝑔‘(ℎ‘(◡ℎ‘𝑦))) = (𝑔‘𝑦))
21	13, 16, 20	3eqtr3rd 2219	. . . . . . . . 9 ⊢ ((((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Markov) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ ∀𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = 1o) ∧ 𝑦 ∈ 𝐵) → (𝑔‘𝑦) = 1o)
22	21	ralrimiva 2550	. . . . . . . 8 ⊢ (((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Markov) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ ∀𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = 1o) → ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) = 1o)
23	22	ex 115	. . . . . . 7 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Markov) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) → (∀𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = 1o → ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) = 1o))
24	23	con3d 631	. . . . . 6 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Markov) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) → (¬ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) = 1o → ¬ ∀𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = 1o))
25		fveq1 5510	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑔 ∘ ℎ) → (𝑓‘𝑥) = ((𝑔 ∘ ℎ)‘𝑥))
26	25	eqeq1d 2186	. . . . . . . . . 10 ⊢ (𝑓 = (𝑔 ∘ ℎ) → ((𝑓‘𝑥) = 1o ↔ ((𝑔 ∘ ℎ)‘𝑥) = 1o))
27	26	ralbidv 2477	. . . . . . . . 9 ⊢ (𝑓 = (𝑔 ∘ ℎ) → (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o ↔ ∀𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = 1o))
28	27	notbid 667	. . . . . . . 8 ⊢ (𝑓 = (𝑔 ∘ ℎ) → (¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o ↔ ¬ ∀𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = 1o))
29	25	eqeq1d 2186	. . . . . . . . 9 ⊢ (𝑓 = (𝑔 ∘ ℎ) → ((𝑓‘𝑥) = ∅ ↔ ((𝑔 ∘ ℎ)‘𝑥) = ∅))
30	29	rexbidv 2478	. . . . . . . 8 ⊢ (𝑓 = (𝑔 ∘ ℎ) → (∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅ ↔ ∃𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = ∅))
31	28, 30	imbi12d 234	. . . . . . 7 ⊢ (𝑓 = (𝑔 ∘ ℎ) → ((¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅) ↔ (¬ ∀𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = ∅)))
32		ismkvmap 7146	. . . . . . . . 9 ⊢ (𝐴 ∈ Markov → (𝐴 ∈ Markov ↔ ∀𝑓 ∈ (2o ↑𝑚 𝐴)(¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅)))
33	32	ibi 176	. . . . . . . 8 ⊢ (𝐴 ∈ Markov → ∀𝑓 ∈ (2o ↑𝑚 𝐴)(¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅))
34	33	ad3antlr 493	. . . . . . 7 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Markov) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) → ∀𝑓 ∈ (2o ↑𝑚 𝐴)(¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅))
35		simpr 110	. . . . . . . . . . 11 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Markov) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) → 𝑔 ∈ (2o ↑𝑚 𝐵))
36		2onn 6516	. . . . . . . . . . . . 13 ⊢ 2o ∈ ω
37		relen 6738	. . . . . . . . . . . . . 14 ⊢ Rel ≈
38	37	brrelex2i 4667	. . . . . . . . . . . . 13 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ∈ V)
39		elmapg 6655	. . . . . . . . . . . . 13 ⊢ ((2o ∈ ω ∧ 𝐵 ∈ V) → (𝑔 ∈ (2o ↑𝑚 𝐵) ↔ 𝑔:𝐵⟶2o))
40	36, 38, 39	sylancr 414	. . . . . . . . . . . 12 ⊢ (𝐴 ≈ 𝐵 → (𝑔 ∈ (2o ↑𝑚 𝐵) ↔ 𝑔:𝐵⟶2o))
41	40	ad2antrr 488	. . . . . . . . . . 11 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Markov) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) → (𝑔 ∈ (2o ↑𝑚 𝐵) ↔ 𝑔:𝐵⟶2o))
42	35, 41	mpbid 147	. . . . . . . . . 10 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Markov) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) → 𝑔:𝐵⟶2o)
43	42	adantr 276	. . . . . . . . 9 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Markov) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) → 𝑔:𝐵⟶2o)
44		f1of 5457	. . . . . . . . . 10 ⊢ (ℎ:𝐴–1-1-onto→𝐵 → ℎ:𝐴⟶𝐵)
45	44	adantl 277	. . . . . . . . 9 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Markov) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) → ℎ:𝐴⟶𝐵)
46		fco 5377	. . . . . . . . 9 ⊢ ((𝑔:𝐵⟶2o ∧ ℎ:𝐴⟶𝐵) → (𝑔 ∘ ℎ):𝐴⟶2o)
47	43, 45, 46	syl2anc 411	. . . . . . . 8 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Markov) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) → (𝑔 ∘ ℎ):𝐴⟶2o)
48		simpllr 534	. . . . . . . . 9 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Markov) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) → 𝐴 ∈ Markov)
49		elmapg 6655	. . . . . . . . 9 ⊢ ((2o ∈ ω ∧ 𝐴 ∈ Markov) → ((𝑔 ∘ ℎ) ∈ (2o ↑𝑚 𝐴) ↔ (𝑔 ∘ ℎ):𝐴⟶2o))
50	36, 48, 49	sylancr 414	. . . . . . . 8 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Markov) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) → ((𝑔 ∘ ℎ) ∈ (2o ↑𝑚 𝐴) ↔ (𝑔 ∘ ℎ):𝐴⟶2o))
51	47, 50	mpbird 167	. . . . . . 7 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Markov) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) → (𝑔 ∘ ℎ) ∈ (2o ↑𝑚 𝐴))
52	31, 34, 51	rspcdva 2846	. . . . . 6 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Markov) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) → (¬ ∀𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = ∅))
53	4	ad2antlr 489	. . . . . . . . . 10 ⊢ (((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Markov) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ 𝑥 ∈ 𝐴) → ℎ Fn 𝐴)
54		fvco2 5581	. . . . . . . . . 10 ⊢ ((ℎ Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝑔 ∘ ℎ)‘𝑥) = (𝑔‘(ℎ‘𝑥)))
55	53, 54	sylancom 420	. . . . . . . . 9 ⊢ (((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Markov) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ 𝑥 ∈ 𝐴) → ((𝑔 ∘ ℎ)‘𝑥) = (𝑔‘(ℎ‘𝑥)))
56	55	eqeq1d 2186	. . . . . . . 8 ⊢ (((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Markov) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ 𝑥 ∈ 𝐴) → (((𝑔 ∘ ℎ)‘𝑥) = ∅ ↔ (𝑔‘(ℎ‘𝑥)) = ∅))
57	45	ffvelcdmda 5647	. . . . . . . . 9 ⊢ (((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Markov) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ 𝑥 ∈ 𝐴) → (ℎ‘𝑥) ∈ 𝐵)
58		simpr 110	. . . . . . . . . 10 ⊢ ((((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Markov) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 = (ℎ‘𝑥)) → 𝑦 = (ℎ‘𝑥))
59	58	fveqeq2d 5519	. . . . . . . . 9 ⊢ ((((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Markov) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 = (ℎ‘𝑥)) → ((𝑔‘𝑦) = ∅ ↔ (𝑔‘(ℎ‘𝑥)) = ∅))
60	57, 59	rspcedv 2845	. . . . . . . 8 ⊢ (((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Markov) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ 𝑥 ∈ 𝐴) → ((𝑔‘(ℎ‘𝑥)) = ∅ → ∃𝑦 ∈ 𝐵 (𝑔‘𝑦) = ∅))
61	56, 60	sylbid 150	. . . . . . 7 ⊢ (((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Markov) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ 𝑥 ∈ 𝐴) → (((𝑔 ∘ ℎ)‘𝑥) = ∅ → ∃𝑦 ∈ 𝐵 (𝑔‘𝑦) = ∅))
62	61	rexlimdva 2594	. . . . . 6 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Markov) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) → (∃𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = ∅ → ∃𝑦 ∈ 𝐵 (𝑔‘𝑦) = ∅))
63	24, 52, 62	3syld 57	. . . . 5 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Markov) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) → (¬ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) = 1o → ∃𝑦 ∈ 𝐵 (𝑔‘𝑦) = ∅))
64	3, 63	exlimddv 1898	. . . 4 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Markov) ∧ 𝑔 ∈ (2o ↑𝑚 𝐵)) → (¬ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) = 1o → ∃𝑦 ∈ 𝐵 (𝑔‘𝑦) = ∅))
65	64	ralrimiva 2550	. . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Markov) → ∀𝑔 ∈ (2o ↑𝑚 𝐵)(¬ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) = 1o → ∃𝑦 ∈ 𝐵 (𝑔‘𝑦) = ∅))
66		ismkvmap 7146	. . . . 5 ⊢ (𝐵 ∈ V → (𝐵 ∈ Markov ↔ ∀𝑔 ∈ (2o ↑𝑚 𝐵)(¬ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) = 1o → ∃𝑦 ∈ 𝐵 (𝑔‘𝑦) = ∅)))
67	38, 66	syl 14	. . . 4 ⊢ (𝐴 ≈ 𝐵 → (𝐵 ∈ Markov ↔ ∀𝑔 ∈ (2o ↑𝑚 𝐵)(¬ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) = 1o → ∃𝑦 ∈ 𝐵 (𝑔‘𝑦) = ∅)))
68	67	adantr 276	. . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Markov) → (𝐵 ∈ Markov ↔ ∀𝑔 ∈ (2o ↑𝑚 𝐵)(¬ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) = 1o → ∃𝑦 ∈ 𝐵 (𝑔‘𝑦) = ∅)))
69	65, 68	mpbird 167	. 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Markov) → 𝐵 ∈ Markov)
70	69	ex 115	1 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ Markov → 𝐵 ∈ Markov))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353  ∃wex 1492   ∈ wcel 2148  ∀wral 2455  ∃wrex 2456  Vcvv 2737  ∅c0 3422   class class class wbr 4000  ωcom 4586  ^◡ccnv 4622   ∘ ccom 4627   Fn wfn 5207  ⟶wf 5208  –1-1-onto→wf1o 5211  ‘cfv 5212  (class class class)co 5869  1_oc1o 6404  2_oc2o 6405   ↑_𝑚 cmap 6642   ≈ cen 6732  Markovcmarkov 7143

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-id 4290  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1o 6411  df-2o 6412  df-map 6644  df-en 6735  df-markov 7144

This theorem is referenced by:  enmkv  7154