msubco - Mathbox for Mario Carneiro

	Mathbox for Mario Carneiro		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > msubco		Structured version Visualization version GIF version

Theorem msubco 35713

Description: The composition of two substitutions is a substitution. (Contributed by Mario Carneiro, 18-Jul-2016.)

**Hypothesis**
Ref	Expression
msubco.s	⊢ 𝑆 = (mSubst‘𝑇)

**Assertion**
Ref	Expression
msubco	⊢ ((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) → (𝐹 ∘ 𝐺) ∈ ran 𝑆)

Proof of Theorem msubco Dummy variables 𝑓 𝑔 ℎ 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2736	. . . . 5 ⊢ (mEx‘𝑇) = (mEx‘𝑇)
2		eqid 2736	. . . . 5 ⊢ (mRSubst‘𝑇) = (mRSubst‘𝑇)
3		msubco.s	. . . . 5 ⊢ 𝑆 = (mSubst‘𝑇)
4	1, 2, 3	elmsubrn 35710	. . . 4 ⊢ ran 𝑆 = ran (𝑓 ∈ ran (mRSubst‘𝑇) ↦ (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑥), (𝑓‘(2^nd ‘𝑥))⟩))
5	4	eleq2i 2828	. . 3 ⊢ (𝐹 ∈ ran 𝑆 ↔ 𝐹 ∈ ran (𝑓 ∈ ran (mRSubst‘𝑇) ↦ (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑥), (𝑓‘(2^nd ‘𝑥))⟩)))
6		eqid 2736	. . . 4 ⊢ (𝑓 ∈ ran (mRSubst‘𝑇) ↦ (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑥), (𝑓‘(2^nd ‘𝑥))⟩)) = (𝑓 ∈ ran (mRSubst‘𝑇) ↦ (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑥), (𝑓‘(2^nd ‘𝑥))⟩))
7		fvex 6853	. . . . 5 ⊢ (mEx‘𝑇) ∈ V
8	7	mptex 7178	. . . 4 ⊢ (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑥), (𝑓‘(2^nd ‘𝑥))⟩) ∈ V
9	6, 8	elrnmpti 5917	. . 3 ⊢ (𝐹 ∈ ran (𝑓 ∈ ran (mRSubst‘𝑇) ↦ (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑥), (𝑓‘(2^nd ‘𝑥))⟩)) ↔ ∃𝑓 ∈ ran (mRSubst‘𝑇)𝐹 = (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑥), (𝑓‘(2^nd ‘𝑥))⟩))
10	5, 9	bitri 275	. 2 ⊢ (𝐹 ∈ ran 𝑆 ↔ ∃𝑓 ∈ ran (mRSubst‘𝑇)𝐹 = (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑥), (𝑓‘(2^nd ‘𝑥))⟩))
11	1, 2, 3	elmsubrn 35710	. . . 4 ⊢ ran 𝑆 = ran (𝑔 ∈ ran (mRSubst‘𝑇) ↦ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑦), (𝑔‘(2^nd ‘𝑦))⟩))
12	11	eleq2i 2828	. . 3 ⊢ (𝐺 ∈ ran 𝑆 ↔ 𝐺 ∈ ran (𝑔 ∈ ran (mRSubst‘𝑇) ↦ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑦), (𝑔‘(2^nd ‘𝑦))⟩)))
13		eqid 2736	. . . 4 ⊢ (𝑔 ∈ ran (mRSubst‘𝑇) ↦ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑦), (𝑔‘(2^nd ‘𝑦))⟩)) = (𝑔 ∈ ran (mRSubst‘𝑇) ↦ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑦), (𝑔‘(2^nd ‘𝑦))⟩))
14	7	mptex 7178	. . . 4 ⊢ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑦), (𝑔‘(2^nd ‘𝑦))⟩) ∈ V
15	13, 14	elrnmpti 5917	. . 3 ⊢ (𝐺 ∈ ran (𝑔 ∈ ran (mRSubst‘𝑇) ↦ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑦), (𝑔‘(2^nd ‘𝑦))⟩)) ↔ ∃𝑔 ∈ ran (mRSubst‘𝑇)𝐺 = (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑦), (𝑔‘(2^nd ‘𝑦))⟩))
16	12, 15	bitri 275	. 2 ⊢ (𝐺 ∈ ran 𝑆 ↔ ∃𝑔 ∈ ran (mRSubst‘𝑇)𝐺 = (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑦), (𝑔‘(2^nd ‘𝑦))⟩))
17		reeanv 3209	. . 3 ⊢ (∃𝑓 ∈ ran (mRSubst‘𝑇)∃𝑔 ∈ ran (mRSubst‘𝑇)(𝐹 = (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑥), (𝑓‘(2^nd ‘𝑥))⟩) ∧ 𝐺 = (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑦), (𝑔‘(2^nd ‘𝑦))⟩)) ↔ (∃𝑓 ∈ ran (mRSubst‘𝑇)𝐹 = (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑥), (𝑓‘(2^nd ‘𝑥))⟩) ∧ ∃𝑔 ∈ ran (mRSubst‘𝑇)𝐺 = (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑦), (𝑔‘(2^nd ‘𝑦))⟩)))
18		simpr 484	. . . . . . . . . . . 12 ⊢ (((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) ∧ 𝑦 ∈ (mEx‘𝑇)) → 𝑦 ∈ (mEx‘𝑇))
19		eqid 2736	. . . . . . . . . . . . 13 ⊢ (mTC‘𝑇) = (mTC‘𝑇)
20		eqid 2736	. . . . . . . . . . . . 13 ⊢ (mREx‘𝑇) = (mREx‘𝑇)
21	19, 1, 20	mexval 35684	. . . . . . . . . . . 12 ⊢ (mEx‘𝑇) = ((mTC‘𝑇) × (mREx‘𝑇))
22	18, 21	eleqtrdi 2846	. . . . . . . . . . 11 ⊢ (((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) ∧ 𝑦 ∈ (mEx‘𝑇)) → 𝑦 ∈ ((mTC‘𝑇) × (mREx‘𝑇)))
23		xp1st 7974	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ((mTC‘𝑇) × (mREx‘𝑇)) → (1^st ‘𝑦) ∈ (mTC‘𝑇))
24	22, 23	syl 17	. . . . . . . . . 10 ⊢ (((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) ∧ 𝑦 ∈ (mEx‘𝑇)) → (1^st ‘𝑦) ∈ (mTC‘𝑇))
25	2, 20	mrsubf 35699	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ ran (mRSubst‘𝑇) → 𝑔:(mREx‘𝑇)⟶(mREx‘𝑇))
26	25	ad2antlr 728	. . . . . . . . . . 11 ⊢ (((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) ∧ 𝑦 ∈ (mEx‘𝑇)) → 𝑔:(mREx‘𝑇)⟶(mREx‘𝑇))
27		xp2nd 7975	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ((mTC‘𝑇) × (mREx‘𝑇)) → (2^nd ‘𝑦) ∈ (mREx‘𝑇))
28	22, 27	syl 17	. . . . . . . . . . 11 ⊢ (((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) ∧ 𝑦 ∈ (mEx‘𝑇)) → (2^nd ‘𝑦) ∈ (mREx‘𝑇))
29	26, 28	ffvelcdmd 7037	. . . . . . . . . 10 ⊢ (((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) ∧ 𝑦 ∈ (mEx‘𝑇)) → (𝑔‘(2^nd ‘𝑦)) ∈ (mREx‘𝑇))
30		opelxpi 5668	. . . . . . . . . 10 ⊢ (((1^st ‘𝑦) ∈ (mTC‘𝑇) ∧ (𝑔‘(2^nd ‘𝑦)) ∈ (mREx‘𝑇)) → ⟨(1^st ‘𝑦), (𝑔‘(2^nd ‘𝑦))⟩ ∈ ((mTC‘𝑇) × (mREx‘𝑇)))
31	24, 29, 30	syl2anc 585	. . . . . . . . 9 ⊢ (((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) ∧ 𝑦 ∈ (mEx‘𝑇)) → ⟨(1^st ‘𝑦), (𝑔‘(2^nd ‘𝑦))⟩ ∈ ((mTC‘𝑇) × (mREx‘𝑇)))
32	31, 21	eleqtrrdi 2847	. . . . . . . 8 ⊢ (((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) ∧ 𝑦 ∈ (mEx‘𝑇)) → ⟨(1^st ‘𝑦), (𝑔‘(2^nd ‘𝑦))⟩ ∈ (mEx‘𝑇))
33		eqidd 2737	. . . . . . . 8 ⊢ ((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) → (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑦), (𝑔‘(2^nd ‘𝑦))⟩) = (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑦), (𝑔‘(2^nd ‘𝑦))⟩))
34		eqidd 2737	. . . . . . . 8 ⊢ ((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) → (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑥), (𝑓‘(2^nd ‘𝑥))⟩) = (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑥), (𝑓‘(2^nd ‘𝑥))⟩))
35		fvex 6853	. . . . . . . . . 10 ⊢ (1^st ‘𝑦) ∈ V
36		fvex 6853	. . . . . . . . . 10 ⊢ (𝑔‘(2^nd ‘𝑦)) ∈ V
37	35, 36	op1std 7952	. . . . . . . . 9 ⊢ (𝑥 = ⟨(1^st ‘𝑦), (𝑔‘(2^nd ‘𝑦))⟩ → (1^st ‘𝑥) = (1^st ‘𝑦))
38	35, 36	op2ndd 7953	. . . . . . . . . 10 ⊢ (𝑥 = ⟨(1^st ‘𝑦), (𝑔‘(2^nd ‘𝑦))⟩ → (2^nd ‘𝑥) = (𝑔‘(2^nd ‘𝑦)))
39	38	fveq2d 6844	. . . . . . . . 9 ⊢ (𝑥 = ⟨(1^st ‘𝑦), (𝑔‘(2^nd ‘𝑦))⟩ → (𝑓‘(2^nd ‘𝑥)) = (𝑓‘(𝑔‘(2^nd ‘𝑦))))
40	37, 39	opeq12d 4824	. . . . . . . 8 ⊢ (𝑥 = ⟨(1^st ‘𝑦), (𝑔‘(2^nd ‘𝑦))⟩ → ⟨(1^st ‘𝑥), (𝑓‘(2^nd ‘𝑥))⟩ = ⟨(1^st ‘𝑦), (𝑓‘(𝑔‘(2^nd ‘𝑦)))⟩)
41	32, 33, 34, 40	fmptco 7082	. . . . . . 7 ⊢ ((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) → ((𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑥), (𝑓‘(2^nd ‘𝑥))⟩) ∘ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑦), (𝑔‘(2^nd ‘𝑦))⟩)) = (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑦), (𝑓‘(𝑔‘(2^nd ‘𝑦)))⟩))
42		fvco3 6939	. . . . . . . . . 10 ⊢ ((𝑔:(mREx‘𝑇)⟶(mREx‘𝑇) ∧ (2^nd ‘𝑦) ∈ (mREx‘𝑇)) → ((𝑓 ∘ 𝑔)‘(2^nd ‘𝑦)) = (𝑓‘(𝑔‘(2^nd ‘𝑦))))
43	26, 28, 42	syl2anc 585	. . . . . . . . 9 ⊢ (((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) ∧ 𝑦 ∈ (mEx‘𝑇)) → ((𝑓 ∘ 𝑔)‘(2^nd ‘𝑦)) = (𝑓‘(𝑔‘(2^nd ‘𝑦))))
44	43	opeq2d 4823	. . . . . . . 8 ⊢ (((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) ∧ 𝑦 ∈ (mEx‘𝑇)) → ⟨(1^st ‘𝑦), ((𝑓 ∘ 𝑔)‘(2^nd ‘𝑦))⟩ = ⟨(1^st ‘𝑦), (𝑓‘(𝑔‘(2^nd ‘𝑦)))⟩)
45	44	mpteq2dva 5178	. . . . . . 7 ⊢ ((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) → (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑦), ((𝑓 ∘ 𝑔)‘(2^nd ‘𝑦))⟩) = (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑦), (𝑓‘(𝑔‘(2^nd ‘𝑦)))⟩))
46	41, 45	eqtr4d 2774	. . . . . 6 ⊢ ((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) → ((𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑥), (𝑓‘(2^nd ‘𝑥))⟩) ∘ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑦), (𝑔‘(2^nd ‘𝑦))⟩)) = (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑦), ((𝑓 ∘ 𝑔)‘(2^nd ‘𝑦))⟩))
47	2	mrsubco 35703	. . . . . . . 8 ⊢ ((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) → (𝑓 ∘ 𝑔) ∈ ran (mRSubst‘𝑇))
48	7	mptex 7178	. . . . . . . 8 ⊢ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑦), ((𝑓 ∘ 𝑔)‘(2^nd ‘𝑦))⟩) ∈ V
49		eqid 2736	. . . . . . . . 9 ⊢ (ℎ ∈ ran (mRSubst‘𝑇) ↦ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑦), (ℎ‘(2^nd ‘𝑦))⟩)) = (ℎ ∈ ran (mRSubst‘𝑇) ↦ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑦), (ℎ‘(2^nd ‘𝑦))⟩))
50		fveq1 6839	. . . . . . . . . . 11 ⊢ (ℎ = (𝑓 ∘ 𝑔) → (ℎ‘(2^nd ‘𝑦)) = ((𝑓 ∘ 𝑔)‘(2^nd ‘𝑦)))
51	50	opeq2d 4823	. . . . . . . . . 10 ⊢ (ℎ = (𝑓 ∘ 𝑔) → ⟨(1^st ‘𝑦), (ℎ‘(2^nd ‘𝑦))⟩ = ⟨(1^st ‘𝑦), ((𝑓 ∘ 𝑔)‘(2^nd ‘𝑦))⟩)
52	51	mpteq2dv 5179	. . . . . . . . 9 ⊢ (ℎ = (𝑓 ∘ 𝑔) → (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑦), (ℎ‘(2^nd ‘𝑦))⟩) = (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑦), ((𝑓 ∘ 𝑔)‘(2^nd ‘𝑦))⟩))
53	49, 52	elrnmpt1s 5914	. . . . . . . 8 ⊢ (((𝑓 ∘ 𝑔) ∈ ran (mRSubst‘𝑇) ∧ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑦), ((𝑓 ∘ 𝑔)‘(2^nd ‘𝑦))⟩) ∈ V) → (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑦), ((𝑓 ∘ 𝑔)‘(2^nd ‘𝑦))⟩) ∈ ran (ℎ ∈ ran (mRSubst‘𝑇) ↦ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑦), (ℎ‘(2^nd ‘𝑦))⟩)))
54	47, 48, 53	sylancl 587	. . . . . . 7 ⊢ ((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) → (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑦), ((𝑓 ∘ 𝑔)‘(2^nd ‘𝑦))⟩) ∈ ran (ℎ ∈ ran (mRSubst‘𝑇) ↦ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑦), (ℎ‘(2^nd ‘𝑦))⟩)))
55	1, 2, 3	elmsubrn 35710	. . . . . . 7 ⊢ ran 𝑆 = ran (ℎ ∈ ran (mRSubst‘𝑇) ↦ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑦), (ℎ‘(2^nd ‘𝑦))⟩))
56	54, 55	eleqtrrdi 2847	. . . . . 6 ⊢ ((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) → (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑦), ((𝑓 ∘ 𝑔)‘(2^nd ‘𝑦))⟩) ∈ ran 𝑆)
57	46, 56	eqeltrd 2836	. . . . 5 ⊢ ((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) → ((𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑥), (𝑓‘(2^nd ‘𝑥))⟩) ∘ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑦), (𝑔‘(2^nd ‘𝑦))⟩)) ∈ ran 𝑆)
58		coeq1 5812	. . . . . . 7 ⊢ (𝐹 = (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑥), (𝑓‘(2^nd ‘𝑥))⟩) → (𝐹 ∘ 𝐺) = ((𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑥), (𝑓‘(2^nd ‘𝑥))⟩) ∘ 𝐺))
59		coeq2 5813	. . . . . . 7 ⊢ (𝐺 = (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑦), (𝑔‘(2^nd ‘𝑦))⟩) → ((𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑥), (𝑓‘(2^nd ‘𝑥))⟩) ∘ 𝐺) = ((𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑥), (𝑓‘(2^nd ‘𝑥))⟩) ∘ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑦), (𝑔‘(2^nd ‘𝑦))⟩)))
60	58, 59	sylan9eq 2791	. . . . . 6 ⊢ ((𝐹 = (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑥), (𝑓‘(2^nd ‘𝑥))⟩) ∧ 𝐺 = (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑦), (𝑔‘(2^nd ‘𝑦))⟩)) → (𝐹 ∘ 𝐺) = ((𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑥), (𝑓‘(2^nd ‘𝑥))⟩) ∘ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑦), (𝑔‘(2^nd ‘𝑦))⟩)))
61	60	eleq1d 2821	. . . . 5 ⊢ ((𝐹 = (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑥), (𝑓‘(2^nd ‘𝑥))⟩) ∧ 𝐺 = (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑦), (𝑔‘(2^nd ‘𝑦))⟩)) → ((𝐹 ∘ 𝐺) ∈ ran 𝑆 ↔ ((𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑥), (𝑓‘(2^nd ‘𝑥))⟩) ∘ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑦), (𝑔‘(2^nd ‘𝑦))⟩)) ∈ ran 𝑆))
62	57, 61	syl5ibrcom 247	. . . 4 ⊢ ((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) → ((𝐹 = (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑥), (𝑓‘(2^nd ‘𝑥))⟩) ∧ 𝐺 = (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑦), (𝑔‘(2^nd ‘𝑦))⟩)) → (𝐹 ∘ 𝐺) ∈ ran 𝑆))
63	62	rexlimivv 3179	. . 3 ⊢ (∃𝑓 ∈ ran (mRSubst‘𝑇)∃𝑔 ∈ ran (mRSubst‘𝑇)(𝐹 = (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑥), (𝑓‘(2^nd ‘𝑥))⟩) ∧ 𝐺 = (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑦), (𝑔‘(2^nd ‘𝑦))⟩)) → (𝐹 ∘ 𝐺) ∈ ran 𝑆)
64	17, 63	sylbir 235	. 2 ⊢ ((∃𝑓 ∈ ran (mRSubst‘𝑇)𝐹 = (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑥), (𝑓‘(2^nd ‘𝑥))⟩) ∧ ∃𝑔 ∈ ran (mRSubst‘𝑇)𝐺 = (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑦), (𝑔‘(2^nd ‘𝑦))⟩)) → (𝐹 ∘ 𝐺) ∈ ran 𝑆)
65	10, 16, 64	syl2anb 599	1 ⊢ ((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) → (𝐹 ∘ 𝐺) ∈ ran 𝑆)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ∃wrex 3061 Vcvv 3429 ⟨cop 4573 ↦ cmpt 5166 × cxp 5629 ran crn 5632 ∘ ccom 5635 ⟶wf 6494 ‘cfv 6498 1^st c1st 7940 2^nd c2nd 7941 mTCcmtc 35646 mRExcmrex 35648 mExcmex 35649 mRSubstcmrsub 35652 mSubstcmsub 35653

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-1st 7942 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-er 8643 df-map 8775 df-pm 8776 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-nn 12175 df-2 12244 df-n0 12438 df-xnn0 12511 df-z 12525 df-uz 12789 df-fz 13462 df-fzo 13609 df-seq 13964 df-hash 14293 df-word 14476 df-lsw 14525 df-concat 14533 df-s1 14559 df-substr 14604 df-pfx 14634 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-0g 17404 df-gsum 17405 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-mhm 18751 df-submnd 18752 df-frmd 18817 df-vrmd 18818 df-mrex 35668 df-mex 35669 df-mrsub 35672 df-msub 35673

This theorem is referenced by: mclsppslem 35765

W3C validator