Theorem msubvrs 31457

Description: The set of variables in a substitution is the union, indexed by the variables in the original expression, of the variables in the substitution to that variable. (Contributed by Mario Carneiro, 18-Jul-2016.)

Hypotheses

Ref	Expression
msubvrs.s	|- S = (mSubst ` T )
msubvrs.e	|- E = (mEx ` T )
msubvrs.v	|- V = (mVars ` T )
msubvrs.h	|- H = (mVH ` T )

Assertion

Ref	Expression
msubvrs	|- ( ( T e. mFS /\ F e. ran S /\ X e. E ) -> ( V ` ( F ` X ) ) = U_ x e. ( V ` X ) ( V ` ( F ` ( H ` x ) ) ) )

Distinct variable groups:   x, E   x, F   x, T   x, X   x, V

Allowed substitution hints:   S( x)   H( x)

Proof of Theorem msubvrs

Dummy variables e f are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		msubvrs.e	. . . . . 6 |- E = (mEx ` T )
2		eqid 2622	. . . . . 6 |- (mRSubst ` T ) = (mRSubst ` T )
3		msubvrs.s	. . . . . 6 |- S = (mSubst ` T )
4	1, 2, 3	elmsubrn 31425	. . . . 5 |- ran S = ran ( f e. ran (mRSubst ` T ) |-> ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) )
5	4	eleq2i 2693	. . . 4 |- ( F e. ran S <-> F e. ran ( f e. ran (mRSubst ` T ) |-> ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) ) )
6		eqid 2622	. . . . 5 |- ( f e. ran (mRSubst ` T ) |-> ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) ) = ( f e. ran (mRSubst ` T ) |-> ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) )
7		fvex 6201	. . . . . . 7 |- (mEx ` T ) e. _V
8	1, 7	eqeltri 2697	. . . . . 6 |- E e. _V
9	8	mptex 6486	. . . . 5 |- ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) e. _V
10	6, 9	elrnmpti 5376	. . . 4 |- ( F e. ran ( f e. ran (mRSubst ` T ) |-> ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) ) <-> E. f e. ran (mRSubst ` T ) F = ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) )
11	5, 10	bitri 264	. . 3 |- ( F e. ran S <-> E. f e. ran (mRSubst ` T ) F = ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) )
12		simp2 1062	. . . . . . . . 9 |- ( ( T e. mFS /\ f e. ran (mRSubst ` T ) /\ X e. E ) -> f e. ran (mRSubst ` T ) )
13		simp3 1063	. . . . . . . . . . 11 |- ( ( T e. mFS /\ f e. ran (mRSubst ` T ) /\ X e. E ) -> X e. E )
14		eqid 2622	. . . . . . . . . . . 12 |- (mTC ` T ) = (mTC ` T )
15		eqid 2622	. . . . . . . . . . . 12 |- (mREx ` T ) = (mREx ` T )
16	14, 1, 15	mexval 31399	. . . . . . . . . . 11 |- E = ( (mTC ` T ) X. (mREx ` T ) )
17	13, 16	syl6eleq 2711	. . . . . . . . . 10 |- ( ( T e. mFS /\ f e. ran (mRSubst ` T ) /\ X e. E ) -> X e. ( (mTC ` T ) X. (mREx ` T ) ) )
18		xp2nd 7199	. . . . . . . . . 10 |- ( X e. ( (mTC ` T ) X. (mREx ` T ) ) -> ( 2nd ` X ) e. (mREx ` T ) )
19	17, 18	syl 17	. . . . . . . . 9 |- ( ( T e. mFS /\ f e. ran (mRSubst ` T ) /\ X e. E ) -> ( 2nd ` X ) e. (mREx ` T ) )
20		eqid 2622	. . . . . . . . . 10 |- (mVR ` T ) = (mVR ` T )
21	2, 20, 15	mrsubvrs 31419	. . . . . . . . 9 |- ( ( f e. ran (mRSubst ` T ) /\ ( 2nd ` X ) e. (mREx ` T ) ) -> ( ran ( f ` ( 2nd ` X ) ) i^i (mVR ` T ) ) = U_ x e. ( ran ( 2nd ` X ) i^i (mVR ` T ) ) ( ran ( f ` <" x "> ) i^i (mVR ` T ) ) )
22	12, 19, 21	syl2anc 693	. . . . . . . 8 |- ( ( T e. mFS /\ f e. ran (mRSubst ` T ) /\ X e. E ) -> ( ran ( f ` ( 2nd ` X ) ) i^i (mVR ` T ) ) = U_ x e. ( ran ( 2nd ` X ) i^i (mVR ` T ) ) ( ran ( f ` <" x "> ) i^i (mVR ` T ) ) )
23		fveq2 6191	. . . . . . . . . . . . 13 |- ( e = X -> ( 1st ` e ) = ( 1st ` X ) )
24		fveq2 6191	. . . . . . . . . . . . . 14 |- ( e = X -> ( 2nd ` e ) = ( 2nd ` X ) )
25	24	fveq2d 6195	. . . . . . . . . . . . 13 |- ( e = X -> ( f ` ( 2nd ` e ) ) = ( f ` ( 2nd ` X ) ) )
26	23, 25	opeq12d 4410	. . . . . . . . . . . 12 |- ( e = X -> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. = <. ( 1st ` X ) , ( f ` ( 2nd ` X ) ) >. )
27		eqid 2622	. . . . . . . . . . . 12 |- ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) = ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. )
28		opex 4932	. . . . . . . . . . . 12 |- <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. e. _V
29	26, 27, 28	fvmpt3i 6287	. . . . . . . . . . 11 |- ( X e. E -> ( ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) ` X ) = <. ( 1st ` X ) , ( f ` ( 2nd ` X ) ) >. )
30	13, 29	syl 17	. . . . . . . . . 10 |- ( ( T e. mFS /\ f e. ran (mRSubst ` T ) /\ X e. E ) -> ( ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) ` X ) = <. ( 1st ` X ) , ( f ` ( 2nd ` X ) ) >. )
31	30	fveq2d 6195	. . . . . . . . 9 |- ( ( T e. mFS /\ f e. ran (mRSubst ` T ) /\ X e. E ) -> ( V ` ( ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) ` X ) ) = ( V ` <. ( 1st ` X ) , ( f ` ( 2nd ` X ) ) >. ) )
32		xp1st 7198	. . . . . . . . . . . . 13 |- ( X e. ( (mTC ` T ) X. (mREx ` T ) ) -> ( 1st ` X ) e. (mTC ` T ) )
33	17, 32	syl 17	. . . . . . . . . . . 12 |- ( ( T e. mFS /\ f e. ran (mRSubst ` T ) /\ X e. E ) -> ( 1st ` X ) e. (mTC ` T ) )
34	2, 15	mrsubf 31414	. . . . . . . . . . . . . 14 |- ( f e. ran (mRSubst ` T ) -> f : (mREx ` T ) --> (mREx ` T ) )
35	12, 34	syl 17	. . . . . . . . . . . . 13 |- ( ( T e. mFS /\ f e. ran (mRSubst ` T ) /\ X e. E ) -> f : (mREx ` T ) --> (mREx ` T ) )
36	18, 16	eleq2s 2719	. . . . . . . . . . . . . 14 |- ( X e. E -> ( 2nd ` X ) e. (mREx ` T ) )
37	13, 36	syl 17	. . . . . . . . . . . . 13 |- ( ( T e. mFS /\ f e. ran (mRSubst ` T ) /\ X e. E ) -> ( 2nd ` X ) e. (mREx ` T ) )
38	35, 37	ffvelrnd 6360	. . . . . . . . . . . 12 |- ( ( T e. mFS /\ f e. ran (mRSubst ` T ) /\ X e. E ) -> ( f ` ( 2nd ` X ) ) e. (mREx ` T ) )
39		opelxpi 5148	. . . . . . . . . . . 12 |- ( ( ( 1st ` X ) e. (mTC ` T ) /\ ( f ` ( 2nd ` X ) ) e. (mREx ` T ) ) -> <. ( 1st ` X ) , ( f ` ( 2nd ` X ) ) >. e. ( (mTC ` T ) X. (mREx ` T ) ) )
40	33, 38, 39	syl2anc 693	. . . . . . . . . . 11 |- ( ( T e. mFS /\ f e. ran (mRSubst ` T ) /\ X e. E ) -> <. ( 1st ` X ) , ( f ` ( 2nd ` X ) ) >. e. ( (mTC ` T ) X. (mREx ` T ) ) )
41	40, 16	syl6eleqr 2712	. . . . . . . . . 10 |- ( ( T e. mFS /\ f e. ran (mRSubst ` T ) /\ X e. E ) -> <. ( 1st ` X ) , ( f ` ( 2nd ` X ) ) >. e. E )
42		msubvrs.v	. . . . . . . . . . 11 |- V = (mVars ` T )
43	20, 1, 42	mvrsval 31402	. . . . . . . . . 10 |- ( <. ( 1st ` X ) , ( f ` ( 2nd ` X ) ) >. e. E -> ( V ` <. ( 1st ` X ) , ( f ` ( 2nd ` X ) ) >. ) = ( ran ( 2nd ` <. ( 1st ` X ) , ( f ` ( 2nd ` X ) ) >. ) i^i (mVR ` T ) ) )
44	41, 43	syl 17	. . . . . . . . 9 |- ( ( T e. mFS /\ f e. ran (mRSubst ` T ) /\ X e. E ) -> ( V ` <. ( 1st ` X ) , ( f ` ( 2nd ` X ) ) >. ) = ( ran ( 2nd ` <. ( 1st ` X ) , ( f ` ( 2nd ` X ) ) >. ) i^i (mVR ` T ) ) )
45		fvex 6201	. . . . . . . . . . . . 13 |- ( 1st ` X ) e. _V
46		fvex 6201	. . . . . . . . . . . . 13 |- ( f ` ( 2nd ` X ) ) e. _V
47	45, 46	op2nd 7177	. . . . . . . . . . . 12 |- ( 2nd ` <. ( 1st ` X ) , ( f ` ( 2nd ` X ) ) >. ) = ( f ` ( 2nd ` X ) )
48	47	a1i 11	. . . . . . . . . . 11 |- ( ( T e. mFS /\ f e. ran (mRSubst ` T ) /\ X e. E ) -> ( 2nd ` <. ( 1st ` X ) , ( f ` ( 2nd ` X ) ) >. ) = ( f ` ( 2nd ` X ) ) )
49	48	rneqd 5353	. . . . . . . . . 10 |- ( ( T e. mFS /\ f e. ran (mRSubst ` T ) /\ X e. E ) -> ran ( 2nd ` <. ( 1st ` X ) , ( f ` ( 2nd ` X ) ) >. ) = ran ( f ` ( 2nd ` X ) ) )
50	49	ineq1d 3813	. . . . . . . . 9 |- ( ( T e. mFS /\ f e. ran (mRSubst ` T ) /\ X e. E ) -> ( ran ( 2nd ` <. ( 1st ` X ) , ( f ` ( 2nd ` X ) ) >. ) i^i (mVR ` T ) ) = ( ran ( f ` ( 2nd ` X ) ) i^i (mVR ` T ) ) )
51	31, 44, 50	3eqtrd 2660	. . . . . . . 8 |- ( ( T e. mFS /\ f e. ran (mRSubst ` T ) /\ X e. E ) -> ( V ` ( ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) ` X ) ) = ( ran ( f ` ( 2nd ` X ) ) i^i (mVR ` T ) ) )
52	20, 1, 42	mvrsval 31402	. . . . . . . . . . 11 |- ( X e. E -> ( V ` X ) = ( ran ( 2nd ` X ) i^i (mVR ` T ) ) )
53	13, 52	syl 17	. . . . . . . . . 10 |- ( ( T e. mFS /\ f e. ran (mRSubst ` T ) /\ X e. E ) -> ( V ` X ) = ( ran ( 2nd ` X ) i^i (mVR ` T ) ) )
54	53	iuneq1d 4545	. . . . . . . . 9 |- ( ( T e. mFS /\ f e. ran (mRSubst ` T ) /\ X e. E ) -> U_ x e. ( V ` X ) ( V ` ( ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) ` ( H ` x ) ) ) = U_ x e. ( ran ( 2nd ` X ) i^i (mVR ` T ) ) ( V ` ( ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) ` ( H ` x ) ) ) )
55		msubvrs.h	. . . . . . . . . . . . . . . . 17 |- H = (mVH ` T )
56	20, 1, 55	mvhf 31455	. . . . . . . . . . . . . . . 16 |- ( T e. mFS -> H : (mVR ` T ) --> E )
57	56	3ad2ant1 1082	. . . . . . . . . . . . . . 15 |- ( ( T e. mFS /\ f e. ran (mRSubst ` T ) /\ X e. E ) -> H : (mVR ` T ) --> E )
58		inss2 3834	. . . . . . . . . . . . . . . 16 |- ( ran ( 2nd ` X ) i^i (mVR ` T ) ) C_ (mVR ` T )
59	58	sseli 3599	. . . . . . . . . . . . . . 15 |- ( x e. ( ran ( 2nd ` X ) i^i (mVR ` T ) ) -> x e. (mVR ` T ) )
60		ffvelrn 6357	. . . . . . . . . . . . . . 15 |- ( ( H : (mVR ` T ) --> E /\ x e. (mVR ` T ) ) -> ( H ` x ) e. E )
61	57, 59, 60	syl2an 494	. . . . . . . . . . . . . 14 |- ( ( ( T e. mFS /\ f e. ran (mRSubst ` T ) /\ X e. E ) /\ x e. ( ran ( 2nd ` X ) i^i (mVR ` T ) ) ) -> ( H ` x ) e. E )
62		fveq2 6191	. . . . . . . . . . . . . . . 16 |- ( e = ( H ` x ) -> ( 1st ` e ) = ( 1st ` ( H ` x ) ) )
63		fveq2 6191	. . . . . . . . . . . . . . . . 17 |- ( e = ( H ` x ) -> ( 2nd ` e ) = ( 2nd ` ( H ` x ) ) )
64	63	fveq2d 6195	. . . . . . . . . . . . . . . 16 |- ( e = ( H ` x ) -> ( f ` ( 2nd ` e ) ) = ( f ` ( 2nd ` ( H ` x ) ) ) )
65	62, 64	opeq12d 4410	. . . . . . . . . . . . . . 15 |- ( e = ( H ` x ) -> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. = <. ( 1st ` ( H ` x ) ) , ( f ` ( 2nd ` ( H ` x ) ) ) >. )
66	65, 27, 28	fvmpt3i 6287	. . . . . . . . . . . . . 14 |- ( ( H ` x ) e. E -> ( ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) ` ( H ` x ) ) = <. ( 1st ` ( H ` x ) ) , ( f ` ( 2nd ` ( H ` x ) ) ) >. )
67	61, 66	syl 17	. . . . . . . . . . . . 13 |- ( ( ( T e. mFS /\ f e. ran (mRSubst ` T ) /\ X e. E ) /\ x e. ( ran ( 2nd ` X ) i^i (mVR ` T ) ) ) -> ( ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) ` ( H ` x ) ) = <. ( 1st ` ( H ` x ) ) , ( f ` ( 2nd ` ( H ` x ) ) ) >. )
68	59	adantl 482	. . . . . . . . . . . . . . . 16 |- ( ( ( T e. mFS /\ f e. ran (mRSubst ` T ) /\ X e. E ) /\ x e. ( ran ( 2nd ` X ) i^i (mVR ` T ) ) ) -> x e. (mVR ` T ) )
69		eqid 2622	. . . . . . . . . . . . . . . . 17 |- (mType ` T ) = (mType ` T )
70	20, 69, 55	mvhval 31431	. . . . . . . . . . . . . . . 16 |- ( x e. (mVR ` T ) -> ( H ` x ) = <. ( (mType ` T ) ` x ) , <" x "> >. )
71	68, 70	syl 17	. . . . . . . . . . . . . . 15 |- ( ( ( T e. mFS /\ f e. ran (mRSubst ` T ) /\ X e. E ) /\ x e. ( ran ( 2nd ` X ) i^i (mVR ` T ) ) ) -> ( H ` x ) = <. ( (mType ` T ) ` x ) , <" x "> >. )
72		fvex 6201	. . . . . . . . . . . . . . . 16 |- ( (mType ` T ) ` x ) e. _V
73		s1cli 13384	. . . . . . . . . . . . . . . . 17 |- <" x "> e. Word _V
74	73	elexi 3213	. . . . . . . . . . . . . . . 16 |- <" x "> e. _V
75	72, 74	op1std 7178	. . . . . . . . . . . . . . 15 |- ( ( H ` x ) = <. ( (mType ` T ) ` x ) , <" x "> >. -> ( 1st ` ( H ` x ) ) = ( (mType ` T ) ` x ) )
76	71, 75	syl 17	. . . . . . . . . . . . . 14 |- ( ( ( T e. mFS /\ f e. ran (mRSubst ` T ) /\ X e. E ) /\ x e. ( ran ( 2nd ` X ) i^i (mVR ` T ) ) ) -> ( 1st ` ( H ` x ) ) = ( (mType ` T ) ` x ) )
77	72, 74	op2ndd 7179	. . . . . . . . . . . . . . . 16 |- ( ( H ` x ) = <. ( (mType ` T ) ` x ) , <" x "> >. -> ( 2nd ` ( H ` x ) ) = <" x "> )
78	71, 77	syl 17	. . . . . . . . . . . . . . 15 |- ( ( ( T e. mFS /\ f e. ran (mRSubst ` T ) /\ X e. E ) /\ x e. ( ran ( 2nd ` X ) i^i (mVR ` T ) ) ) -> ( 2nd ` ( H ` x ) ) = <" x "> )
79	78	fveq2d 6195	. . . . . . . . . . . . . 14 |- ( ( ( T e. mFS /\ f e. ran (mRSubst ` T ) /\ X e. E ) /\ x e. ( ran ( 2nd ` X ) i^i (mVR ` T ) ) ) -> ( f ` ( 2nd ` ( H ` x ) ) ) = ( f ` <" x "> ) )
80	76, 79	opeq12d 4410	. . . . . . . . . . . . 13 |- ( ( ( T e. mFS /\ f e. ran (mRSubst ` T ) /\ X e. E ) /\ x e. ( ran ( 2nd ` X ) i^i (mVR ` T ) ) ) -> <. ( 1st ` ( H ` x ) ) , ( f ` ( 2nd ` ( H ` x ) ) ) >. = <. ( (mType ` T ) ` x ) , ( f ` <" x "> ) >. )
81	67, 80	eqtrd 2656	. . . . . . . . . . . 12 |- ( ( ( T e. mFS /\ f e. ran (mRSubst ` T ) /\ X e. E ) /\ x e. ( ran ( 2nd ` X ) i^i (mVR ` T ) ) ) -> ( ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) ` ( H ` x ) ) = <. ( (mType ` T ) ` x ) , ( f ` <" x "> ) >. )
82	81	fveq2d 6195	. . . . . . . . . . 11 |- ( ( ( T e. mFS /\ f e. ran (mRSubst ` T ) /\ X e. E ) /\ x e. ( ran ( 2nd ` X ) i^i (mVR ` T ) ) ) -> ( V ` ( ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) ` ( H ` x ) ) ) = ( V ` <. ( (mType ` T ) ` x ) , ( f ` <" x "> ) >. ) )
83		simpl1 1064	. . . . . . . . . . . . . . . 16 |- ( ( ( T e. mFS /\ f e. ran (mRSubst ` T ) /\ X e. E ) /\ x e. ( ran ( 2nd ` X ) i^i (mVR ` T ) ) ) -> T e. mFS )
84	20, 14, 69	mtyf2 31448	. . . . . . . . . . . . . . . 16 |- ( T e. mFS -> (mType ` T ) : (mVR ` T ) --> (mTC ` T ) )
85	83, 84	syl 17	. . . . . . . . . . . . . . 15 |- ( ( ( T e. mFS /\ f e. ran (mRSubst ` T ) /\ X e. E ) /\ x e. ( ran ( 2nd ` X ) i^i (mVR ` T ) ) ) -> (mType ` T ) : (mVR ` T ) --> (mTC ` T ) )
86	85, 68	ffvelrnd 6360	. . . . . . . . . . . . . 14 |- ( ( ( T e. mFS /\ f e. ran (mRSubst ` T ) /\ X e. E ) /\ x e. ( ran ( 2nd ` X ) i^i (mVR ` T ) ) ) -> ( (mType ` T ) ` x ) e. (mTC ` T ) )
87	35	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ( T e. mFS /\ f e. ran (mRSubst ` T ) /\ X e. E ) /\ x e. ( ran ( 2nd ` X ) i^i (mVR ` T ) ) ) -> f : (mREx ` T ) --> (mREx ` T ) )
88		elun2 3781	. . . . . . . . . . . . . . . . . 18 |- ( x e. (mVR ` T ) -> x e. ( (mCN ` T ) u. (mVR ` T ) ) )
89	68, 88	syl 17	. . . . . . . . . . . . . . . . 17 |- ( ( ( T e. mFS /\ f e. ran (mRSubst ` T ) /\ X e. E ) /\ x e. ( ran ( 2nd ` X ) i^i (mVR ` T ) ) ) -> x e. ( (mCN ` T ) u. (mVR ` T ) ) )
90	89	s1cld 13383	. . . . . . . . . . . . . . . 16 |- ( ( ( T e. mFS /\ f e. ran (mRSubst ` T ) /\ X e. E ) /\ x e. ( ran ( 2nd ` X ) i^i (mVR ` T ) ) ) -> <" x "> e. Word ( (mCN ` T ) u. (mVR ` T ) ) )
91		eqid 2622	. . . . . . . . . . . . . . . . . 18 |- (mCN ` T ) = (mCN ` T )
92	91, 20, 15	mrexval 31398	. . . . . . . . . . . . . . . . 17 |- ( T e. mFS -> (mREx ` T ) = Word ( (mCN ` T ) u. (mVR ` T ) ) )
93	83, 92	syl 17	. . . . . . . . . . . . . . . 16 |- ( ( ( T e. mFS /\ f e. ran (mRSubst ` T ) /\ X e. E ) /\ x e. ( ran ( 2nd ` X ) i^i (mVR ` T ) ) ) -> (mREx ` T ) = Word ( (mCN ` T ) u. (mVR ` T ) ) )
94	90, 93	eleqtrrd 2704	. . . . . . . . . . . . . . 15 |- ( ( ( T e. mFS /\ f e. ran (mRSubst ` T ) /\ X e. E ) /\ x e. ( ran ( 2nd ` X ) i^i (mVR ` T ) ) ) -> <" x "> e. (mREx ` T ) )
95	87, 94	ffvelrnd 6360	. . . . . . . . . . . . . 14 |- ( ( ( T e. mFS /\ f e. ran (mRSubst ` T ) /\ X e. E ) /\ x e. ( ran ( 2nd ` X ) i^i (mVR ` T ) ) ) -> ( f ` <" x "> ) e. (mREx ` T ) )
96		opelxpi 5148	. . . . . . . . . . . . . 14 |- ( ( ( (mType ` T ) ` x ) e. (mTC ` T ) /\ ( f ` <" x "> ) e. (mREx ` T ) ) -> <. ( (mType ` T ) ` x ) , ( f ` <" x "> ) >. e. ( (mTC ` T ) X. (mREx ` T ) ) )
97	86, 95, 96	syl2anc 693	. . . . . . . . . . . . 13 |- ( ( ( T e. mFS /\ f e. ran (mRSubst ` T ) /\ X e. E ) /\ x e. ( ran ( 2nd ` X ) i^i (mVR ` T ) ) ) -> <. ( (mType ` T ) ` x ) , ( f ` <" x "> ) >. e. ( (mTC ` T ) X. (mREx ` T ) ) )
98	97, 16	syl6eleqr 2712	. . . . . . . . . . . 12 |- ( ( ( T e. mFS /\ f e. ran (mRSubst ` T ) /\ X e. E ) /\ x e. ( ran ( 2nd ` X ) i^i (mVR ` T ) ) ) -> <. ( (mType ` T ) ` x ) , ( f ` <" x "> ) >. e. E )
99	20, 1, 42	mvrsval 31402	. . . . . . . . . . . 12 |- ( <. ( (mType ` T ) ` x ) , ( f ` <" x "> ) >. e. E -> ( V ` <. ( (mType ` T ) ` x ) , ( f ` <" x "> ) >. ) = ( ran ( 2nd ` <. ( (mType ` T ) ` x ) , ( f ` <" x "> ) >. ) i^i (mVR ` T ) ) )
100	98, 99	syl 17	. . . . . . . . . . 11 |- ( ( ( T e. mFS /\ f e. ran (mRSubst ` T ) /\ X e. E ) /\ x e. ( ran ( 2nd ` X ) i^i (mVR ` T ) ) ) -> ( V ` <. ( (mType ` T ) ` x ) , ( f ` <" x "> ) >. ) = ( ran ( 2nd ` <. ( (mType ` T ) ` x ) , ( f ` <" x "> ) >. ) i^i (mVR ` T ) ) )
101		fvex 6201	. . . . . . . . . . . . . . 15 |- ( f ` <" x "> ) e. _V
102	72, 101	op2nd 7177	. . . . . . . . . . . . . 14 |- ( 2nd ` <. ( (mType ` T ) ` x ) , ( f ` <" x "> ) >. ) = ( f ` <" x "> )
103	102	a1i 11	. . . . . . . . . . . . 13 |- ( ( ( T e. mFS /\ f e. ran (mRSubst ` T ) /\ X e. E ) /\ x e. ( ran ( 2nd ` X ) i^i (mVR ` T ) ) ) -> ( 2nd ` <. ( (mType ` T ) ` x ) , ( f ` <" x "> ) >. ) = ( f ` <" x "> ) )
104	103	rneqd 5353	. . . . . . . . . . . 12 |- ( ( ( T e. mFS /\ f e. ran (mRSubst ` T ) /\ X e. E ) /\ x e. ( ran ( 2nd ` X ) i^i (mVR ` T ) ) ) -> ran ( 2nd ` <. ( (mType ` T ) ` x ) , ( f ` <" x "> ) >. ) = ran ( f ` <" x "> ) )
105	104	ineq1d 3813	. . . . . . . . . . 11 |- ( ( ( T e. mFS /\ f e. ran (mRSubst ` T ) /\ X e. E ) /\ x e. ( ran ( 2nd ` X ) i^i (mVR ` T ) ) ) -> ( ran ( 2nd ` <. ( (mType ` T ) ` x ) , ( f ` <" x "> ) >. ) i^i (mVR ` T ) ) = ( ran ( f ` <" x "> ) i^i (mVR ` T ) ) )
106	82, 100, 105	3eqtrd 2660	. . . . . . . . . 10 |- ( ( ( T e. mFS /\ f e. ran (mRSubst ` T ) /\ X e. E ) /\ x e. ( ran ( 2nd ` X ) i^i (mVR ` T ) ) ) -> ( V ` ( ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) ` ( H ` x ) ) ) = ( ran ( f ` <" x "> ) i^i (mVR ` T ) ) )
107	106	iuneq2dv 4542	. . . . . . . . 9 |- ( ( T e. mFS /\ f e. ran (mRSubst ` T ) /\ X e. E ) -> U_ x e. ( ran ( 2nd ` X ) i^i (mVR ` T ) ) ( V ` ( ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) ` ( H ` x ) ) ) = U_ x e. ( ran ( 2nd ` X ) i^i (mVR ` T ) ) ( ran ( f ` <" x "> ) i^i (mVR ` T ) ) )
108	54, 107	eqtrd 2656	. . . . . . . 8 |- ( ( T e. mFS /\ f e. ran (mRSubst ` T ) /\ X e. E ) -> U_ x e. ( V ` X ) ( V ` ( ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) ` ( H ` x ) ) ) = U_ x e. ( ran ( 2nd ` X ) i^i (mVR ` T ) ) ( ran ( f ` <" x "> ) i^i (mVR ` T ) ) )
109	22, 51, 108	3eqtr4d 2666	. . . . . . 7 |- ( ( T e. mFS /\ f e. ran (mRSubst ` T ) /\ X e. E ) -> ( V ` ( ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) ` X ) ) = U_ x e. ( V ` X ) ( V ` ( ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) ` ( H ` x ) ) ) )
110		fveq1 6190	. . . . . . . . 9 |- ( F = ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) -> ( F ` X ) = ( ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) ` X ) )
111	110	fveq2d 6195	. . . . . . . 8 |- ( F = ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) -> ( V ` ( F ` X ) ) = ( V ` ( ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) ` X ) ) )
112		fveq1 6190	. . . . . . . . . 10 |- ( F = ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) -> ( F ` ( H ` x ) ) = ( ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) ` ( H ` x ) ) )
113	112	fveq2d 6195	. . . . . . . . 9 |- ( F = ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) -> ( V ` ( F ` ( H ` x ) ) ) = ( V ` ( ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) ` ( H ` x ) ) ) )
114	113	iuneq2d 4547	. . . . . . . 8 |- ( F = ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) -> U_ x e. ( V ` X ) ( V ` ( F ` ( H ` x ) ) ) = U_ x e. ( V ` X ) ( V ` ( ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) ` ( H ` x ) ) ) )
115	111, 114	eqeq12d 2637	. . . . . . 7 |- ( F = ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) -> ( ( V ` ( F ` X ) ) = U_ x e. ( V ` X ) ( V ` ( F ` ( H ` x ) ) ) <-> ( V ` ( ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) ` X ) ) = U_ x e. ( V ` X ) ( V ` ( ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) ` ( H ` x ) ) ) ) )
116	109, 115	syl5ibrcom 237	. . . . . 6 |- ( ( T e. mFS /\ f e. ran (mRSubst ` T ) /\ X e. E ) -> ( F = ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) -> ( V ` ( F ` X ) ) = U_ x e. ( V ` X ) ( V ` ( F ` ( H ` x ) ) ) ) )
117	116	3expia 1267	. . . . 5 |- ( ( T e. mFS /\ f e. ran (mRSubst ` T ) ) -> ( X e. E -> ( F = ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) -> ( V ` ( F ` X ) ) = U_ x e. ( V ` X ) ( V ` ( F ` ( H ` x ) ) ) ) ) )
118	117	com23 86	. . . 4 |- ( ( T e. mFS /\ f e. ran (mRSubst ` T ) ) -> ( F = ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) -> ( X e. E -> ( V ` ( F ` X ) ) = U_ x e. ( V ` X ) ( V ` ( F ` ( H ` x ) ) ) ) ) )
119	118	rexlimdva 3031	. . 3 |- ( T e. mFS -> ( E. f e. ran (mRSubst ` T ) F = ( e e. E |-> <. ( 1st ` e ) , ( f ` ( 2nd ` e ) ) >. ) -> ( X e. E -> ( V ` ( F ` X ) ) = U_ x e. ( V ` X ) ( V ` ( F ` ( H ` x ) ) ) ) ) )
120	11, 119	syl5bi 232	. 2 |- ( T e. mFS -> ( F e. ran S -> ( X e. E -> ( V ` ( F ` X ) ) = U_ x e. ( V ` X ) ( V ` ( F ` ( H ` x ) ) ) ) ) )
121	120	3imp 1256	1 |- ( ( T e. mFS /\ F e. ran S /\ X e. E ) -> ( V ` ( F ` X ) ) = U_ x e. ( V ` X ) ( V ` ( F ` ( H ` x ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   _Vcvv 3200   u. cun 3572   i^i cin 3573   <.cop 4183   U_ciun 4520   |-> cmpt 4729   X. cxp 5112   ran crn 5115   -->wf 5884   ` cfv 5888   1stc1st 7166   2ndc2nd 7167  Word cword 13291   <"cs1 13294  mCNcmcn 31357  mVRcmvar 31358  mTypecmty 31359  mTCcmtc 31361  mRExcmrex 31363  mExcmex 31364  mVarscmvrs 31366  mRSubstcmrsub 31367  mSubstcmsub 31368  mVHcmvh 31369  mFScmfs 31373

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-word 13299  df-lsw 13300  df-concat 13301  df-s1 13302  df-substr 13303  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-0g 16102  df-gsum 16103  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-frmd 17386  df-mrex 31383  df-mex 31384  df-mvrs 31386  df-mrsub 31387  df-msub 31388  df-mvh 31389  df-mfs 31393

This theorem is referenced by:  mclsppslem  31480