msubco - Mathbox for Mario Carneiro

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Theorem msubco 34522

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 2733	. . . . 5 ⊢ (mEx‘𝑇) = (mEx‘𝑇)
2		eqid 2733	. . . . 5 ⊢ (mRSubst‘𝑇) = (mRSubst‘𝑇)
3		msubco.s	. . . . 5 ⊢ 𝑆 = (mSubst‘𝑇)
4	1, 2, 3	elmsubrn 34519	. . . 4 ⊢ ran 𝑆 = ran (𝑓 ∈ ran (mRSubst‘𝑇) ↦ (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑥), (𝑓‘(2^nd ‘𝑥))⟩))
5	4	eleq2i 2826	. . 3 ⊢ (𝐹 ∈ ran 𝑆 ↔ 𝐹 ∈ ran (𝑓 ∈ ran (mRSubst‘𝑇) ↦ (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑥), (𝑓‘(2^nd ‘𝑥))⟩)))
6		eqid 2733	. . . 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 6905	. . . . 5 ⊢ (mEx‘𝑇) ∈ V
8	7	mptex 7225	. . . 4 ⊢ (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑥), (𝑓‘(2^nd ‘𝑥))⟩) ∈ V
9	6, 8	elrnmpti 5960	. . 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 34519	. . . 4 ⊢ ran 𝑆 = ran (𝑔 ∈ ran (mRSubst‘𝑇) ↦ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑦), (𝑔‘(2^nd ‘𝑦))⟩))
12	11	eleq2i 2826	. . 3 ⊢ (𝐺 ∈ ran 𝑆 ↔ 𝐺 ∈ ran (𝑔 ∈ ran (mRSubst‘𝑇) ↦ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑦), (𝑔‘(2^nd ‘𝑦))⟩)))
13		eqid 2733	. . . 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 7225	. . . 4 ⊢ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑦), (𝑔‘(2^nd ‘𝑦))⟩) ∈ V
15	13, 14	elrnmpti 5960	. . 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 3227	. . 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 486	. . . . . . . . . . . 12 ⊢ (((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) ∧ 𝑦 ∈ (mEx‘𝑇)) → 𝑦 ∈ (mEx‘𝑇))
19		eqid 2733	. . . . . . . . . . . . 13 ⊢ (mTC‘𝑇) = (mTC‘𝑇)
20		eqid 2733	. . . . . . . . . . . . 13 ⊢ (mREx‘𝑇) = (mREx‘𝑇)
21	19, 1, 20	mexval 34493	. . . . . . . . . . . 12 ⊢ (mEx‘𝑇) = ((mTC‘𝑇) × (mREx‘𝑇))
22	18, 21	eleqtrdi 2844	. . . . . . . . . . 11 ⊢ (((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) ∧ 𝑦 ∈ (mEx‘𝑇)) → 𝑦 ∈ ((mTC‘𝑇) × (mREx‘𝑇)))
23		xp1st 8007	. . . . . . . . . . 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 34508	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ ran (mRSubst‘𝑇) → 𝑔:(mREx‘𝑇)⟶(mREx‘𝑇))
26	25	ad2antlr 726	. . . . . . . . . . 11 ⊢ (((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) ∧ 𝑦 ∈ (mEx‘𝑇)) → 𝑔:(mREx‘𝑇)⟶(mREx‘𝑇))
27		xp2nd 8008	. . . . . . . . . . . 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 7088	. . . . . . . . . 10 ⊢ (((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) ∧ 𝑦 ∈ (mEx‘𝑇)) → (𝑔‘(2^nd ‘𝑦)) ∈ (mREx‘𝑇))
30		opelxpi 5714	. . . . . . . . . 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 2845	. . . . . . . 8 ⊢ (((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) ∧ 𝑦 ∈ (mEx‘𝑇)) → ⟨(1^st ‘𝑦), (𝑔‘(2^nd ‘𝑦))⟩ ∈ (mEx‘𝑇))
33		eqidd 2734	. . . . . . . 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 2734	. . . . . . . 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 6905	. . . . . . . . . 10 ⊢ (1^st ‘𝑦) ∈ V
36		fvex 6905	. . . . . . . . . 10 ⊢ (𝑔‘(2^nd ‘𝑦)) ∈ V
37	35, 36	op1std 7985	. . . . . . . . 9 ⊢ (𝑥 = ⟨(1^st ‘𝑦), (𝑔‘(2^nd ‘𝑦))⟩ → (1^st ‘𝑥) = (1^st ‘𝑦))
38	35, 36	op2ndd 7986	. . . . . . . . . 10 ⊢ (𝑥 = ⟨(1^st ‘𝑦), (𝑔‘(2^nd ‘𝑦))⟩ → (2^nd ‘𝑥) = (𝑔‘(2^nd ‘𝑦)))
39	38	fveq2d 6896	. . . . . . . . 9 ⊢ (𝑥 = ⟨(1^st ‘𝑦), (𝑔‘(2^nd ‘𝑦))⟩ → (𝑓‘(2^nd ‘𝑥)) = (𝑓‘(𝑔‘(2^nd ‘𝑦))))
40	37, 39	opeq12d 4882	. . . . . . . 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 7127	. . . . . . 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 6991	. . . . . . . . . 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 4881	. . . . . . . 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 5249	. . . . . . 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 2776	. . . . . 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 34512	. . . . . . . 8 ⊢ ((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) → (𝑓 ∘ 𝑔) ∈ ran (mRSubst‘𝑇))
48	7	mptex 7225	. . . . . . . 8 ⊢ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑦), ((𝑓 ∘ 𝑔)‘(2^nd ‘𝑦))⟩) ∈ V
49		eqid 2733	. . . . . . . . 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 6891	. . . . . . . . . . 11 ⊢ (ℎ = (𝑓 ∘ 𝑔) → (ℎ‘(2^nd ‘𝑦)) = ((𝑓 ∘ 𝑔)‘(2^nd ‘𝑦)))
51	50	opeq2d 4881	. . . . . . . . . 10 ⊢ (ℎ = (𝑓 ∘ 𝑔) → ⟨(1^st ‘𝑦), (ℎ‘(2^nd ‘𝑦))⟩ = ⟨(1^st ‘𝑦), ((𝑓 ∘ 𝑔)‘(2^nd ‘𝑦))⟩)
52	51	mpteq2dv 5251	. . . . . . . . 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 5957	. . . . . . . 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 34519	. . . . . . 7 ⊢ ran 𝑆 = ran (ℎ ∈ ran (mRSubst‘𝑇) ↦ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1^st ‘𝑦), (ℎ‘(2^nd ‘𝑦))⟩))
56	54, 55	eleqtrrdi 2845	. . . . . 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 2834	. . . . 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 5858	. . . . . . 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 5859	. . . . . . 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 2793	. . . . . 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 2819	. . . . 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 246	. . . 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 3200	. . 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 234	. 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 397 = wceq 1542 ∈ wcel 2107 ∃wrex 3071 Vcvv 3475 ⟨cop 4635 ↦ cmpt 5232 × cxp 5675 ran crn 5678 ∘ ccom 5681 ⟶wf 6540 ‘cfv 6544 1^st c1st 7973 2^nd c2nd 7974 mTCcmtc 34455 mRExcmrex 34457 mExcmex 34458 mRSubstcmrsub 34461 mSubstcmsub 34462

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-map 8822 df-pm 8823 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-2 12275 df-n0 12473 df-xnn0 12545 df-z 12559 df-uz 12823 df-fz 13485 df-fzo 13628 df-seq 13967 df-hash 14291 df-word 14465 df-lsw 14513 df-concat 14521 df-s1 14546 df-substr 14591 df-pfx 14621 df-struct 17080 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ress 17174 df-plusg 17210 df-0g 17387 df-gsum 17388 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-mhm 18671 df-submnd 18672 df-frmd 18730 df-vrmd 18731 df-mrex 34477 df-mex 34478 df-mrsub 34481 df-msub 34482

This theorem is referenced by: mclsppslem 34574

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