msubco - Mathbox for Mario Carneiro

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Theorem msubco 30682

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 2610	. . . . 5 ⊢ (mEx‘𝑇) = (mEx‘𝑇)
2		eqid 2610	. . . . 5 ⊢ (mRSubst‘𝑇) = (mRSubst‘𝑇)
3		msubco.s	. . . . 5 ⊢ 𝑆 = (mSubst‘𝑇)
4	1, 2, 3	elmsubrn 30679	. . . 4 ⊢ ran 𝑆 = ran (𝑓 ∈ ran (mRSubst‘𝑇) ↦ (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑥), (𝑓‘(2^nd ‘𝑥))⟩))
5	4	eleq2i 2680	. . 3 ⊢ (𝐹 ∈ ran 𝑆 ↔ 𝐹 ∈ ran (𝑓 ∈ ran (mRSubst‘𝑇) ↦ (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑥), (𝑓‘(2^nd ‘𝑥))⟩)))
6		eqid 2610	. . . 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 6113	. . . . 5 ⊢ (mEx‘𝑇) ∈ V
8	7	mptex 6390	. . . 4 ⊢ (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑥), (𝑓‘(2^nd ‘𝑥))⟩) ∈ V
9	6, 8	elrnmpti 5297	. . 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 263	. 2 ⊢ (𝐹 ∈ ran 𝑆 ↔ ∃𝑓 ∈ ran (mRSubst‘𝑇)𝐹 = (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑥), (𝑓‘(2^nd ‘𝑥))⟩))
11	1, 2, 3	elmsubrn 30679	. . . 4 ⊢ ran 𝑆 = ran (𝑔 ∈ ran (mRSubst‘𝑇) ↦ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑦), (𝑔‘(2^nd ‘𝑦))⟩))
12	11	eleq2i 2680	. . 3 ⊢ (𝐺 ∈ ran 𝑆 ↔ 𝐺 ∈ ran (𝑔 ∈ ran (mRSubst‘𝑇) ↦ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑦), (𝑔‘(2^nd ‘𝑦))⟩)))
13		eqid 2610	. . . 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 6390	. . . 4 ⊢ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑦), (𝑔‘(2^nd ‘𝑦))⟩) ∈ V
15	13, 14	elrnmpti 5297	. . 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 263	. 2 ⊢ (𝐺 ∈ ran 𝑆 ↔ ∃𝑔 ∈ ran (mRSubst‘𝑇)𝐺 = (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑦), (𝑔‘(2^nd ‘𝑦))⟩))
17		reeanv 3086	. . 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 476	. . . . . . . . . . . 12 ⊢ (((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) ∧ 𝑦 ∈ (mEx‘𝑇)) → 𝑦 ∈ (mEx‘𝑇))
19		eqid 2610	. . . . . . . . . . . . 13 ⊢ (mTC‘𝑇) = (mTC‘𝑇)
20		eqid 2610	. . . . . . . . . . . . 13 ⊢ (mREx‘𝑇) = (mREx‘𝑇)
21	19, 1, 20	mexval 30653	. . . . . . . . . . . 12 ⊢ (mEx‘𝑇) = ((mTC‘𝑇) × (mREx‘𝑇))
22	18, 21	syl6eleq 2698	. . . . . . . . . . 11 ⊢ (((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) ∧ 𝑦 ∈ (mEx‘𝑇)) → 𝑦 ∈ ((mTC‘𝑇) × (mREx‘𝑇)))
23		xp1st 7089	. . . . . . . . . . 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 30668	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ ran (mRSubst‘𝑇) → 𝑔:(mREx‘𝑇)⟶(mREx‘𝑇))
26	25	ad2antlr 759	. . . . . . . . . . 11 ⊢ (((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) ∧ 𝑦 ∈ (mEx‘𝑇)) → 𝑔:(mREx‘𝑇)⟶(mREx‘𝑇))
27		xp2nd 7090	. . . . . . . . . . . 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	ffvelrnd 6268	. . . . . . . . . 10 ⊢ (((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) ∧ 𝑦 ∈ (mEx‘𝑇)) → (𝑔‘(2^nd ‘𝑦)) ∈ (mREx‘𝑇))
30		opelxpi 5072	. . . . . . . . . 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 691	. . . . . . . . 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	syl6eleqr 2699	. . . . . . . 8 ⊢ (((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) ∧ 𝑦 ∈ (mEx‘𝑇)) → ⟨(1^st ‘𝑦), (𝑔‘(2^nd ‘𝑦))⟩ ∈ (mEx‘𝑇))
33		eqidd 2611	. . . . . . . 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 2611	. . . . . . . 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 6113	. . . . . . . . . 10 ⊢ (1^st ‘𝑦) ∈ V
36		fvex 6113	. . . . . . . . . 10 ⊢ (𝑔‘(2^nd ‘𝑦)) ∈ V
37	35, 36	op1std 7069	. . . . . . . . 9 ⊢ (𝑥 = ⟨(1^st ‘𝑦), (𝑔‘(2^nd ‘𝑦))⟩ → (1^st ‘𝑥) = (1^st ‘𝑦))
38	35, 36	op2ndd 7070	. . . . . . . . . 10 ⊢ (𝑥 = ⟨(1^st ‘𝑦), (𝑔‘(2^nd ‘𝑦))⟩ → (2^nd ‘𝑥) = (𝑔‘(2^nd ‘𝑦)))
39	38	fveq2d 6107	. . . . . . . . 9 ⊢ (𝑥 = ⟨(1^st ‘𝑦), (𝑔‘(2^nd ‘𝑦))⟩ → (𝑓‘(2^nd ‘𝑥)) = (𝑓‘(𝑔‘(2^nd ‘𝑦))))
40	37, 39	opeq12d 4348	. . . . . . . 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 6303	. . . . . . 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 6185	. . . . . . . . . 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 691	. . . . . . . . 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 4347	. . . . . . . 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 4672	. . . . . . 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 2647	. . . . . 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 30672	. . . . . . . 8 ⊢ ((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) → (𝑓 ∘ 𝑔) ∈ ran (mRSubst‘𝑇))
48	7	mptex 6390	. . . . . . . 8 ⊢ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑦), ((𝑓 ∘ 𝑔)‘(2^nd ‘𝑦))⟩) ∈ V
49		eqid 2610	. . . . . . . . 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 6102	. . . . . . . . . . 11 ⊢ (ℎ = (𝑓 ∘ 𝑔) → (ℎ‘(2^nd ‘𝑦)) = ((𝑓 ∘ 𝑔)‘(2^nd ‘𝑦)))
51	50	opeq2d 4347	. . . . . . . . . 10 ⊢ (ℎ = (𝑓 ∘ 𝑔) → ⟨(1^st ‘𝑦), (ℎ‘(2^nd ‘𝑦))⟩ = ⟨(1^st ‘𝑦), ((𝑓 ∘ 𝑔)‘(2^nd ‘𝑦))⟩)
52	51	mpteq2dv 4673	. . . . . . . . 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 5294	. . . . . . . 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 693	. . . . . . 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 30679	. . . . . . 7 ⊢ ran 𝑆 = ran (ℎ ∈ ran (mRSubst‘𝑇) ↦ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑦), (ℎ‘(2^nd ‘𝑦))⟩))
56	54, 55	syl6eleqr 2699	. . . . . 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 2688	. . . . 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 5201	. . . . . . 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 5202	. . . . . . 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 2664	. . . . . 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 2672	. . . . 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 236	. . . 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 3018	. . 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 224	. 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 495	1 ⊢ ((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) → (𝐹 ∘ 𝐺) ∈ ran 𝑆)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 Vcvv 3173 ⟨cop 4131 ↦ cmpt 4643 × cxp 5036 ran crn 5039 ∘ ccom 5042 ⟶wf 5800 ‘cfv 5804 1^st c1st 7057 2^nd c2nd 7058 mTCcmtc 30615 mRExcmrex 30617 mExcmex 30618 mRSubstcmrsub 30621 mSubstcmsub 30622

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-n0 11170 df-xnn0 11241 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-seq 12664 df-hash 12980 df-word 13154 df-lsw 13155 df-concat 13156 df-s1 13157 df-substr 13158 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-0g 15925 df-gsum 15926 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mhm 17158 df-submnd 17159 df-frmd 17209 df-vrmd 17210 df-mrex 30637 df-mex 30638 df-mrsub 30641 df-msub 30642

This theorem is referenced by: mclsppslem 30734

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