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Theorem msubco 35759

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 2739	. . . . 5 ⊢ (mEx‘𝑇) = (mEx‘𝑇)
2		eqid 2739	. . . . 5 ⊢ (mRSubst‘𝑇) = (mRSubst‘𝑇)
3		msubco.s	. . . . 5 ⊢ 𝑆 = (mSubst‘𝑇)
4	1, 2, 3	elmsubrn 35756	. . . 4 ⊢ ran 𝑆 = ran (𝑓 ∈ ran (mRSubst‘𝑇) ↦ (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑥), (𝑓‘(2^nd ‘𝑥))⟩))
5	4	eleq2i 2831	. . 3 ⊢ (𝐹 ∈ ran 𝑆 ↔ 𝐹 ∈ ran (𝑓 ∈ ran (mRSubst‘𝑇) ↦ (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑥), (𝑓‘(2^nd ‘𝑥))⟩)))
6		eqid 2739	. . . 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 6840	. . . . 5 ⊢ (mEx‘𝑇) ∈ V
8	7	mptex 7167	. . . 4 ⊢ (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑥), (𝑓‘(2^nd ‘𝑥))⟩) ∈ V
9	6, 8	elrnmpti 5904	. . 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 276	. 2 ⊢ (𝐹 ∈ ran 𝑆 ↔ ∃𝑓 ∈ ran (mRSubst‘𝑇)𝐹 = (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑥), (𝑓‘(2^nd ‘𝑥))⟩))
11	1, 2, 3	elmsubrn 35756	. . . 4 ⊢ ran 𝑆 = ran (𝑔 ∈ ran (mRSubst‘𝑇) ↦ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑦), (𝑔‘(2^nd ‘𝑦))⟩))
12	11	eleq2i 2831	. . 3 ⊢ (𝐺 ∈ ran 𝑆 ↔ 𝐺 ∈ ran (𝑔 ∈ ran (mRSubst‘𝑇) ↦ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑦), (𝑔‘(2^nd ‘𝑦))⟩)))
13		eqid 2739	. . . 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 7167	. . . 4 ⊢ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑦), (𝑔‘(2^nd ‘𝑦))⟩) ∈ V
15	13, 14	elrnmpti 5904	. . 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 276	. 2 ⊢ (𝐺 ∈ ran 𝑆 ↔ ∃𝑔 ∈ ran (mRSubst‘𝑇)𝐺 = (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑦), (𝑔‘(2^nd ‘𝑦))⟩))
17		reeanv 3211	. . 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 485	. . . . . . . . . . . 12 ⊢ (((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) ∧ 𝑦 ∈ (mEx‘𝑇)) → 𝑦 ∈ (mEx‘𝑇))
19		eqid 2739	. . . . . . . . . . . . 13 ⊢ (mTC‘𝑇) = (mTC‘𝑇)
20		eqid 2739	. . . . . . . . . . . . 13 ⊢ (mREx‘𝑇) = (mREx‘𝑇)
21	19, 1, 20	mexval 35730	. . . . . . . . . . . 12 ⊢ (mEx‘𝑇) = ((mTC‘𝑇) × (mREx‘𝑇))
22	18, 21	eleqtrdi 2849	. . . . . . . . . . 11 ⊢ (((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) ∧ 𝑦 ∈ (mEx‘𝑇)) → 𝑦 ∈ ((mTC‘𝑇) × (mREx‘𝑇)))
23		xp1st 7963	. . . . . . . . . . 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 35745	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ ran (mRSubst‘𝑇) → 𝑔:(mREx‘𝑇)⟶(mREx‘𝑇))
26	25	ad2antlr 733	. . . . . . . . . . 11 ⊢ (((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) ∧ 𝑦 ∈ (mEx‘𝑇)) → 𝑔:(mREx‘𝑇)⟶(mREx‘𝑇))
27		xp2nd 7964	. . . . . . . . . . . 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 7026	. . . . . . . . . 10 ⊢ (((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) ∧ 𝑦 ∈ (mEx‘𝑇)) → (𝑔‘(2^nd ‘𝑦)) ∈ (mREx‘𝑇))
30		opelxpi 5655	. . . . . . . . . 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 590	. . . . . . . . 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 2850	. . . . . . . 8 ⊢ (((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) ∧ 𝑦 ∈ (mEx‘𝑇)) → ⟨(1^st ‘𝑦), (𝑔‘(2^nd ‘𝑦))⟩ ∈ (mEx‘𝑇))
33		eqidd 2740	. . . . . . . 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 2740	. . . . . . . 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 6840	. . . . . . . . . 10 ⊢ (1^st ‘𝑦) ∈ V
36		fvex 6840	. . . . . . . . . 10 ⊢ (𝑔‘(2^nd ‘𝑦)) ∈ V
37	35, 36	op1std 7941	. . . . . . . . 9 ⊢ (𝑥 = ⟨(1^st ‘𝑦), (𝑔‘(2^nd ‘𝑦))⟩ → (1^st ‘𝑥) = (1^st ‘𝑦))
38	35, 36	op2ndd 7942	. . . . . . . . . 10 ⊢ (𝑥 = ⟨(1^st ‘𝑦), (𝑔‘(2^nd ‘𝑦))⟩ → (2^nd ‘𝑥) = (𝑔‘(2^nd ‘𝑦)))
39	38	fveq2d 6831	. . . . . . . . 9 ⊢ (𝑥 = ⟨(1^st ‘𝑦), (𝑔‘(2^nd ‘𝑦))⟩ → (𝑓‘(2^nd ‘𝑥)) = (𝑓‘(𝑔‘(2^nd ‘𝑦))))
40	37, 39	opeq12d 4812	. . . . . . . 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 7071	. . . . . . 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 6927	. . . . . . . . . 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 590	. . . . . . . . 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 4811	. . . . . . . 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 5165	. . . . . . 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 2777	. . . . . 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 35749	. . . . . . . 8 ⊢ ((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) → (𝑓 ∘ 𝑔) ∈ ran (mRSubst‘𝑇))
48	7	mptex 7167	. . . . . . . 8 ⊢ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑦), ((𝑓 ∘ 𝑔)‘(2^nd ‘𝑦))⟩) ∈ V
49		eqid 2739	. . . . . . . . 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 6826	. . . . . . . . . . 11 ⊢ (ℎ = (𝑓 ∘ 𝑔) → (ℎ‘(2^nd ‘𝑦)) = ((𝑓 ∘ 𝑔)‘(2^nd ‘𝑦)))
51	50	opeq2d 4811	. . . . . . . . . 10 ⊢ (ℎ = (𝑓 ∘ 𝑔) → ⟨(1^st ‘𝑦), (ℎ‘(2^nd ‘𝑦))⟩ = ⟨(1^st ‘𝑦), ((𝑓 ∘ 𝑔)‘(2^nd ‘𝑦))⟩)
52	51	mpteq2dv 5166	. . . . . . . . 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 5901	. . . . . . . 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 592	. . . . . . 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 35756	. . . . . . 7 ⊢ ran 𝑆 = ran (ℎ ∈ ran (mRSubst‘𝑇) ↦ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑦), (ℎ‘(2^nd ‘𝑦))⟩))
56	54, 55	eleqtrrdi 2850	. . . . . 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 2839	. . . . 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 5799	. . . . . . 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 5800	. . . . . . 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 2794	. . . . . 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 2824	. . . . 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 248	. . . 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 3181	. . 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 236	. 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 604	1 ⊢ ((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) → (𝐹 ∘ 𝐺) ∈ ran 𝑆)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1547 ∈ wcel 2119 ∃wrex 3063 Vcvv 3431 ⟨cop 4561 ↦ cmpt 5153 × cxp 5616 ran crn 5619 ∘ ccom 5622 ⟶wf 6481 ‘cfv 6485 1^st c1st 7929 2^nd c2nd 7930 mTCcmtc 35692 mRExcmrex 35694 mExcmex 35695 mRSubstcmrsub 35698 mSubstcmsub 35699

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4839 df-int 4878 df-iun 4923 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-er 8633 df-map 8765 df-pm 8766 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-card 9854 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12166 df-2 12235 df-n0 12429 df-xnn0 12502 df-z 12516 df-uz 12780 df-fz 13453 df-fzo 13600 df-seq 13955 df-hash 14284 df-word 14467 df-lsw 14516 df-concat 14524 df-s1 14550 df-substr 14595 df-pfx 14625 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-0g 17395 df-gsum 17396 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-mhm 18742 df-submnd 18743 df-frmd 18808 df-vrmd 18809 df-mrex 35714 df-mex 35715 df-mrsub 35718 df-msub 35719

This theorem is referenced by: mclsppslem 35811

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