msubco - Mathbox for Mario Carneiro

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Theorem msubco 32780

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 2823	. . . . 5 ⊢ (mEx‘𝑇) = (mEx‘𝑇)
2		eqid 2823	. . . . 5 ⊢ (mRSubst‘𝑇) = (mRSubst‘𝑇)
3		msubco.s	. . . . 5 ⊢ 𝑆 = (mSubst‘𝑇)
4	1, 2, 3	elmsubrn 32777	. . . 4 ⊢ ran 𝑆 = ran (𝑓 ∈ ran (mRSubst‘𝑇) ↦ (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑥), (𝑓‘(2^nd ‘𝑥))⟩))
5	4	eleq2i 2906	. . 3 ⊢ (𝐹 ∈ ran 𝑆 ↔ 𝐹 ∈ ran (𝑓 ∈ ran (mRSubst‘𝑇) ↦ (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑥), (𝑓‘(2^nd ‘𝑥))⟩)))
6		eqid 2823	. . . 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 6685	. . . . 5 ⊢ (mEx‘𝑇) ∈ V
8	7	mptex 6988	. . . 4 ⊢ (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑥), (𝑓‘(2^nd ‘𝑥))⟩) ∈ V
9	6, 8	elrnmpti 5834	. . 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 277	. 2 ⊢ (𝐹 ∈ ran 𝑆 ↔ ∃𝑓 ∈ ran (mRSubst‘𝑇)𝐹 = (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑥), (𝑓‘(2^nd ‘𝑥))⟩))
11	1, 2, 3	elmsubrn 32777	. . . 4 ⊢ ran 𝑆 = ran (𝑔 ∈ ran (mRSubst‘𝑇) ↦ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑦), (𝑔‘(2^nd ‘𝑦))⟩))
12	11	eleq2i 2906	. . 3 ⊢ (𝐺 ∈ ran 𝑆 ↔ 𝐺 ∈ ran (𝑔 ∈ ran (mRSubst‘𝑇) ↦ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑦), (𝑔‘(2^nd ‘𝑦))⟩)))
13		eqid 2823	. . . 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 6988	. . . 4 ⊢ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑦), (𝑔‘(2^nd ‘𝑦))⟩) ∈ V
15	13, 14	elrnmpti 5834	. . 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 277	. 2 ⊢ (𝐺 ∈ ran 𝑆 ↔ ∃𝑔 ∈ ran (mRSubst‘𝑇)𝐺 = (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑦), (𝑔‘(2^nd ‘𝑦))⟩))
17		reeanv 3369	. . 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 487	. . . . . . . . . . . 12 ⊢ (((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) ∧ 𝑦 ∈ (mEx‘𝑇)) → 𝑦 ∈ (mEx‘𝑇))
19		eqid 2823	. . . . . . . . . . . . 13 ⊢ (mTC‘𝑇) = (mTC‘𝑇)
20		eqid 2823	. . . . . . . . . . . . 13 ⊢ (mREx‘𝑇) = (mREx‘𝑇)
21	19, 1, 20	mexval 32751	. . . . . . . . . . . 12 ⊢ (mEx‘𝑇) = ((mTC‘𝑇) × (mREx‘𝑇))
22	18, 21	eleqtrdi 2925	. . . . . . . . . . 11 ⊢ (((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) ∧ 𝑦 ∈ (mEx‘𝑇)) → 𝑦 ∈ ((mTC‘𝑇) × (mREx‘𝑇)))
23		xp1st 7723	. . . . . . . . . . 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 32766	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ ran (mRSubst‘𝑇) → 𝑔:(mREx‘𝑇)⟶(mREx‘𝑇))
26	25	ad2antlr 725	. . . . . . . . . . 11 ⊢ (((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) ∧ 𝑦 ∈ (mEx‘𝑇)) → 𝑔:(mREx‘𝑇)⟶(mREx‘𝑇))
27		xp2nd 7724	. . . . . . . . . . . 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 6854	. . . . . . . . . 10 ⊢ (((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) ∧ 𝑦 ∈ (mEx‘𝑇)) → (𝑔‘(2^nd ‘𝑦)) ∈ (mREx‘𝑇))
30		opelxpi 5594	. . . . . . . . . 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 586	. . . . . . . . 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 2926	. . . . . . . 8 ⊢ (((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) ∧ 𝑦 ∈ (mEx‘𝑇)) → ⟨(1^st ‘𝑦), (𝑔‘(2^nd ‘𝑦))⟩ ∈ (mEx‘𝑇))
33		eqidd 2824	. . . . . . . 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 2824	. . . . . . . 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 6685	. . . . . . . . . 10 ⊢ (1^st ‘𝑦) ∈ V
36		fvex 6685	. . . . . . . . . 10 ⊢ (𝑔‘(2^nd ‘𝑦)) ∈ V
37	35, 36	op1std 7701	. . . . . . . . 9 ⊢ (𝑥 = ⟨(1^st ‘𝑦), (𝑔‘(2^nd ‘𝑦))⟩ → (1^st ‘𝑥) = (1^st ‘𝑦))
38	35, 36	op2ndd 7702	. . . . . . . . . 10 ⊢ (𝑥 = ⟨(1^st ‘𝑦), (𝑔‘(2^nd ‘𝑦))⟩ → (2^nd ‘𝑥) = (𝑔‘(2^nd ‘𝑦)))
39	38	fveq2d 6676	. . . . . . . . 9 ⊢ (𝑥 = ⟨(1^st ‘𝑦), (𝑔‘(2^nd ‘𝑦))⟩ → (𝑓‘(2^nd ‘𝑥)) = (𝑓‘(𝑔‘(2^nd ‘𝑦))))
40	37, 39	opeq12d 4813	. . . . . . . 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 6893	. . . . . . 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 6762	. . . . . . . . . 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 586	. . . . . . . . 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 4812	. . . . . . . 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 5163	. . . . . . 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 2861	. . . . . 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 32770	. . . . . . . 8 ⊢ ((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) → (𝑓 ∘ 𝑔) ∈ ran (mRSubst‘𝑇))
48	7	mptex 6988	. . . . . . . 8 ⊢ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑦), ((𝑓 ∘ 𝑔)‘(2^nd ‘𝑦))⟩) ∈ V
49		eqid 2823	. . . . . . . . 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 6671	. . . . . . . . . . 11 ⊢ (ℎ = (𝑓 ∘ 𝑔) → (ℎ‘(2^nd ‘𝑦)) = ((𝑓 ∘ 𝑔)‘(2^nd ‘𝑦)))
51	50	opeq2d 4812	. . . . . . . . . 10 ⊢ (ℎ = (𝑓 ∘ 𝑔) → ⟨(1^st ‘𝑦), (ℎ‘(2^nd ‘𝑦))⟩ = ⟨(1^st ‘𝑦), ((𝑓 ∘ 𝑔)‘(2^nd ‘𝑦))⟩)
52	51	mpteq2dv 5164	. . . . . . . . 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 5831	. . . . . . . 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 588	. . . . . . 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 32777	. . . . . . 7 ⊢ ran 𝑆 = ran (ℎ ∈ ran (mRSubst‘𝑇) ↦ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑦), (ℎ‘(2^nd ‘𝑦))⟩))
56	54, 55	eleqtrrdi 2926	. . . . . 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 2915	. . . . 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 5730	. . . . . . 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 5731	. . . . . . 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 2878	. . . . . 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 2899	. . . . 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 249	. . . 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 3294	. . 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 237	. 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 398 = wceq 1537 ∈ wcel 2114 ∃wrex 3141 Vcvv 3496 ⟨cop 4575 ↦ cmpt 5148 × cxp 5555 ran crn 5558 ∘ ccom 5561 ⟶wf 6353 ‘cfv 6357 1^st c1st 7689 2^nd c2nd 7690 mTCcmtc 32713 mRExcmrex 32715 mExcmex 32716 mRSubstcmrsub 32719 mSubstcmsub 32720

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-pm 8411 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-n0 11901 df-xnn0 11971 df-z 11985 df-uz 12247 df-fz 12896 df-fzo 13037 df-seq 13373 df-hash 13694 df-word 13865 df-lsw 13917 df-concat 13925 df-s1 13952 df-substr 14005 df-pfx 14035 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-0g 16717 df-gsum 16718 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-mhm 17958 df-submnd 17959 df-frmd 18016 df-vrmd 18017 df-mrex 32735 df-mex 32736 df-mrsub 32739 df-msub 32740

This theorem is referenced by: mclsppslem 32832

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