Theorem mrsubvrs 31419

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
mrsubco.s	|- S = (mRSubst ` T )
mrsubvrs.v	|- V = (mVR ` T )
mrsubvrs.r	|- R = (mREx ` T )

Assertion

Ref	Expression
mrsubvrs	|- ( ( F e. ran S /\ X e. R ) -> ( ran ( F ` X ) i^i V ) = U_ x e. ( ran X i^i V ) ( ran ( F ` <" x "> ) i^i V ) )

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

Allowed substitution hint:   R( x)

Proof of Theorem mrsubvrs

Dummy variables v y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		n0i 3920	. . . . . 6 |- ( F e. ran S -> -. ran S = (/) )
2		mrsubco.s	. . . . . . . . 9 |- S = (mRSubst ` T )
3		fvprc 6185	. . . . . . . . 9 |- ( -. T e. _V -> (mRSubst ` T ) = (/) )
4	2, 3	syl5eq 2668	. . . . . . . 8 |- ( -. T e. _V -> S = (/) )
5	4	rneqd 5353	. . . . . . 7 |- ( -. T e. _V -> ran S = ran (/) )
6		rn0 5377	. . . . . . 7 |- ran (/) = (/)
7	5, 6	syl6eq 2672	. . . . . 6 |- ( -. T e. _V -> ran S = (/) )
8	1, 7	nsyl2 142	. . . . 5 |- ( F e. ran S -> T e. _V )
9		eqid 2622	. . . . . 6 |- (mCN ` T ) = (mCN ` T )
10		mrsubvrs.v	. . . . . 6 |- V = (mVR ` T )
11		mrsubvrs.r	. . . . . 6 |- R = (mREx ` T )
12	9, 10, 11	mrexval 31398	. . . . 5 |- ( T e. _V -> R = Word ( (mCN ` T ) u. V ) )
13	8, 12	syl 17	. . . 4 |- ( F e. ran S -> R = Word ( (mCN ` T ) u. V ) )
14	13	eleq2d 2687	. . 3 |- ( F e. ran S -> ( X e. R <-> X e. Word ( (mCN ` T ) u. V ) ) )
15		fveq2 6191	. . . . . . . . 9 |- ( v = (/) -> ( F ` v ) = ( F ` (/) ) )
16	15	rneqd 5353	. . . . . . . 8 |- ( v = (/) -> ran ( F ` v ) = ran ( F ` (/) ) )
17	16	ineq1d 3813	. . . . . . 7 |- ( v = (/) -> ( ran ( F ` v ) i^i V ) = ( ran ( F ` (/) ) i^i V ) )
18		rneq 5351	. . . . . . . . . . . 12 |- ( v = (/) -> ran v = ran (/) )
19	18, 6	syl6eq 2672	. . . . . . . . . . 11 |- ( v = (/) -> ran v = (/) )
20	19	ineq1d 3813	. . . . . . . . . 10 |- ( v = (/) -> ( ran v i^i V ) = ( (/) i^i V ) )
21		0in 3969	. . . . . . . . . 10 |- ( (/) i^i V ) = (/)
22	20, 21	syl6eq 2672	. . . . . . . . 9 |- ( v = (/) -> ( ran v i^i V ) = (/) )
23	22	iuneq1d 4545	. . . . . . . 8 |- ( v = (/) -> U_ x e. ( ran v i^i V ) ( ran ( F ` <" x "> ) i^i V ) = U_ x e. (/) ( ran ( F ` <" x "> ) i^i V ) )
24		0iun 4577	. . . . . . . 8 |- U_ x e. (/) ( ran ( F ` <" x "> ) i^i V ) = (/)
25	23, 24	syl6eq 2672	. . . . . . 7 |- ( v = (/) -> U_ x e. ( ran v i^i V ) ( ran ( F ` <" x "> ) i^i V ) = (/) )
26	17, 25	eqeq12d 2637	. . . . . 6 |- ( v = (/) -> ( ( ran ( F ` v ) i^i V ) = U_ x e. ( ran v i^i V ) ( ran ( F ` <" x "> ) i^i V ) <-> ( ran ( F ` (/) ) i^i V ) = (/) ) )
27	26	imbi2d 330	. . . . 5 |- ( v = (/) -> ( ( F e. ran S -> ( ran ( F ` v ) i^i V ) = U_ x e. ( ran v i^i V ) ( ran ( F ` <" x "> ) i^i V ) ) <-> ( F e. ran S -> ( ran ( F ` (/) ) i^i V ) = (/) ) ) )
28		fveq2 6191	. . . . . . . . 9 |- ( v = y -> ( F ` v ) = ( F ` y ) )
29	28	rneqd 5353	. . . . . . . 8 |- ( v = y -> ran ( F ` v ) = ran ( F ` y ) )
30	29	ineq1d 3813	. . . . . . 7 |- ( v = y -> ( ran ( F ` v ) i^i V ) = ( ran ( F ` y ) i^i V ) )
31		rneq 5351	. . . . . . . . 9 |- ( v = y -> ran v = ran y )
32	31	ineq1d 3813	. . . . . . . 8 |- ( v = y -> ( ran v i^i V ) = ( ran y i^i V ) )
33	32	iuneq1d 4545	. . . . . . 7 |- ( v = y -> U_ x e. ( ran v i^i V ) ( ran ( F ` <" x "> ) i^i V ) = U_ x e. ( ran y i^i V ) ( ran ( F ` <" x "> ) i^i V ) )
34	30, 33	eqeq12d 2637	. . . . . 6 |- ( v = y -> ( ( ran ( F ` v ) i^i V ) = U_ x e. ( ran v i^i V ) ( ran ( F ` <" x "> ) i^i V ) <-> ( ran ( F ` y ) i^i V ) = U_ x e. ( ran y i^i V ) ( ran ( F ` <" x "> ) i^i V ) ) )
35	34	imbi2d 330	. . . . 5 |- ( v = y -> ( ( F e. ran S -> ( ran ( F ` v ) i^i V ) = U_ x e. ( ran v i^i V ) ( ran ( F ` <" x "> ) i^i V ) ) <-> ( F e. ran S -> ( ran ( F ` y ) i^i V ) = U_ x e. ( ran y i^i V ) ( ran ( F ` <" x "> ) i^i V ) ) ) )
36		fveq2 6191	. . . . . . . . 9 |- ( v = ( y ++ <" z "> ) -> ( F ` v ) = ( F ` ( y ++ <" z "> ) ) )
37	36	rneqd 5353	. . . . . . . 8 |- ( v = ( y ++ <" z "> ) -> ran ( F ` v ) = ran ( F ` ( y ++ <" z "> ) ) )
38	37	ineq1d 3813	. . . . . . 7 |- ( v = ( y ++ <" z "> ) -> ( ran ( F ` v ) i^i V ) = ( ran ( F ` ( y ++ <" z "> ) ) i^i V ) )
39		rneq 5351	. . . . . . . . 9 |- ( v = ( y ++ <" z "> ) -> ran v = ran ( y ++ <" z "> ) )
40	39	ineq1d 3813	. . . . . . . 8 |- ( v = ( y ++ <" z "> ) -> ( ran v i^i V ) = ( ran ( y ++ <" z "> ) i^i V ) )
41	40	iuneq1d 4545	. . . . . . 7 |- ( v = ( y ++ <" z "> ) -> U_ x e. ( ran v i^i V ) ( ran ( F ` <" x "> ) i^i V ) = U_ x e. ( ran ( y ++ <" z "> ) i^i V ) ( ran ( F ` <" x "> ) i^i V ) )
42	38, 41	eqeq12d 2637	. . . . . 6 |- ( v = ( y ++ <" z "> ) -> ( ( ran ( F ` v ) i^i V ) = U_ x e. ( ran v i^i V ) ( ran ( F ` <" x "> ) i^i V ) <-> ( ran ( F ` ( y ++ <" z "> ) ) i^i V ) = U_ x e. ( ran ( y ++ <" z "> ) i^i V ) ( ran ( F ` <" x "> ) i^i V ) ) )
43	42	imbi2d 330	. . . . 5 |- ( v = ( y ++ <" z "> ) -> ( ( F e. ran S -> ( ran ( F ` v ) i^i V ) = U_ x e. ( ran v i^i V ) ( ran ( F ` <" x "> ) i^i V ) ) <-> ( F e. ran S -> ( ran ( F ` ( y ++ <" z "> ) ) i^i V ) = U_ x e. ( ran ( y ++ <" z "> ) i^i V ) ( ran ( F ` <" x "> ) i^i V ) ) ) )
44		fveq2 6191	. . . . . . . . 9 |- ( v = X -> ( F ` v ) = ( F ` X ) )
45	44	rneqd 5353	. . . . . . . 8 |- ( v = X -> ran ( F ` v ) = ran ( F ` X ) )
46	45	ineq1d 3813	. . . . . . 7 |- ( v = X -> ( ran ( F ` v ) i^i V ) = ( ran ( F ` X ) i^i V ) )
47		rneq 5351	. . . . . . . . 9 |- ( v = X -> ran v = ran X )
48	47	ineq1d 3813	. . . . . . . 8 |- ( v = X -> ( ran v i^i V ) = ( ran X i^i V ) )
49	48	iuneq1d 4545	. . . . . . 7 |- ( v = X -> U_ x e. ( ran v i^i V ) ( ran ( F ` <" x "> ) i^i V ) = U_ x e. ( ran X i^i V ) ( ran ( F ` <" x "> ) i^i V ) )
50	46, 49	eqeq12d 2637	. . . . . 6 |- ( v = X -> ( ( ran ( F ` v ) i^i V ) = U_ x e. ( ran v i^i V ) ( ran ( F ` <" x "> ) i^i V ) <-> ( ran ( F ` X ) i^i V ) = U_ x e. ( ran X i^i V ) ( ran ( F ` <" x "> ) i^i V ) ) )
51	50	imbi2d 330	. . . . 5 |- ( v = X -> ( ( F e. ran S -> ( ran ( F ` v ) i^i V ) = U_ x e. ( ran v i^i V ) ( ran ( F ` <" x "> ) i^i V ) ) <-> ( F e. ran S -> ( ran ( F ` X ) i^i V ) = U_ x e. ( ran X i^i V ) ( ran ( F ` <" x "> ) i^i V ) ) ) )
52	2	mrsub0 31413	. . . . . . . . 9 |- ( F e. ran S -> ( F ` (/) ) = (/) )
53	52	rneqd 5353	. . . . . . . 8 |- ( F e. ran S -> ran ( F ` (/) ) = ran (/) )
54	53, 6	syl6eq 2672	. . . . . . 7 |- ( F e. ran S -> ran ( F ` (/) ) = (/) )
55	54	ineq1d 3813	. . . . . 6 |- ( F e. ran S -> ( ran ( F ` (/) ) i^i V ) = ( (/) i^i V ) )
56	55, 21	syl6eq 2672	. . . . 5 |- ( F e. ran S -> ( ran ( F ` (/) ) i^i V ) = (/) )
57		uneq1 3760	. . . . . . . 8 |- ( ( ran ( F ` y ) i^i V ) = U_ x e. ( ran y i^i V ) ( ran ( F ` <" x "> ) i^i V ) -> ( ( ran ( F ` y ) i^i V ) u. ( ran ( F ` <" z "> ) i^i V ) ) = ( U_ x e. ( ran y i^i V ) ( ran ( F ` <" x "> ) i^i V ) u. ( ran ( F ` <" z "> ) i^i V ) ) )
58		simpl 473	. . . . . . . . . . . . . 14 |- ( ( F e. ran S /\ ( y e. Word ( (mCN ` T ) u. V ) /\ z e. ( (mCN ` T ) u. V ) ) ) -> F e. ran S )
59		simprl 794	. . . . . . . . . . . . . . 15 |- ( ( F e. ran S /\ ( y e. Word ( (mCN ` T ) u. V ) /\ z e. ( (mCN ` T ) u. V ) ) ) -> y e. Word ( (mCN ` T ) u. V ) )
60	13	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( F e. ran S /\ ( y e. Word ( (mCN ` T ) u. V ) /\ z e. ( (mCN ` T ) u. V ) ) ) -> R = Word ( (mCN ` T ) u. V ) )
61	59, 60	eleqtrrd 2704	. . . . . . . . . . . . . 14 |- ( ( F e. ran S /\ ( y e. Word ( (mCN ` T ) u. V ) /\ z e. ( (mCN ` T ) u. V ) ) ) -> y e. R )
62		simprr 796	. . . . . . . . . . . . . . . 16 |- ( ( F e. ran S /\ ( y e. Word ( (mCN ` T ) u. V ) /\ z e. ( (mCN ` T ) u. V ) ) ) -> z e. ( (mCN ` T ) u. V ) )
63	62	s1cld 13383	. . . . . . . . . . . . . . 15 |- ( ( F e. ran S /\ ( y e. Word ( (mCN ` T ) u. V ) /\ z e. ( (mCN ` T ) u. V ) ) ) -> <" z "> e. Word ( (mCN ` T ) u. V ) )
64	63, 60	eleqtrrd 2704	. . . . . . . . . . . . . 14 |- ( ( F e. ran S /\ ( y e. Word ( (mCN ` T ) u. V ) /\ z e. ( (mCN ` T ) u. V ) ) ) -> <" z "> e. R )
65	2, 11	mrsubccat 31415	. . . . . . . . . . . . . 14 |- ( ( F e. ran S /\ y e. R /\ <" z "> e. R ) -> ( F ` ( y ++ <" z "> ) ) = ( ( F ` y ) ++ ( F ` <" z "> ) ) )
66	58, 61, 64, 65	syl3anc 1326	. . . . . . . . . . . . 13 |- ( ( F e. ran S /\ ( y e. Word ( (mCN ` T ) u. V ) /\ z e. ( (mCN ` T ) u. V ) ) ) -> ( F ` ( y ++ <" z "> ) ) = ( ( F ` y ) ++ ( F ` <" z "> ) ) )
67	66	rneqd 5353	. . . . . . . . . . . 12 |- ( ( F e. ran S /\ ( y e. Word ( (mCN ` T ) u. V ) /\ z e. ( (mCN ` T ) u. V ) ) ) -> ran ( F ` ( y ++ <" z "> ) ) = ran ( ( F ` y ) ++ ( F ` <" z "> ) ) )
68	2, 11	mrsubf 31414	. . . . . . . . . . . . . . . 16 |- ( F e. ran S -> F : R --> R )
69	68	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( F e. ran S /\ ( y e. Word ( (mCN ` T ) u. V ) /\ z e. ( (mCN ` T ) u. V ) ) ) -> F : R --> R )
70	69, 61	ffvelrnd 6360	. . . . . . . . . . . . . 14 |- ( ( F e. ran S /\ ( y e. Word ( (mCN ` T ) u. V ) /\ z e. ( (mCN ` T ) u. V ) ) ) -> ( F ` y ) e. R )
71	70, 60	eleqtrd 2703	. . . . . . . . . . . . 13 |- ( ( F e. ran S /\ ( y e. Word ( (mCN ` T ) u. V ) /\ z e. ( (mCN ` T ) u. V ) ) ) -> ( F ` y ) e. Word ( (mCN ` T ) u. V ) )
72	69, 64	ffvelrnd 6360	. . . . . . . . . . . . . 14 |- ( ( F e. ran S /\ ( y e. Word ( (mCN ` T ) u. V ) /\ z e. ( (mCN ` T ) u. V ) ) ) -> ( F ` <" z "> ) e. R )
73	72, 60	eleqtrd 2703	. . . . . . . . . . . . 13 |- ( ( F e. ran S /\ ( y e. Word ( (mCN ` T ) u. V ) /\ z e. ( (mCN ` T ) u. V ) ) ) -> ( F ` <" z "> ) e. Word ( (mCN ` T ) u. V ) )
74		ccatrn 13372	. . . . . . . . . . . . 13 |- ( ( ( F ` y ) e. Word ( (mCN ` T ) u. V ) /\ ( F ` <" z "> ) e. Word ( (mCN ` T ) u. V ) ) -> ran ( ( F ` y ) ++ ( F ` <" z "> ) ) = ( ran ( F ` y ) u. ran ( F ` <" z "> ) ) )
75	71, 73, 74	syl2anc 693	. . . . . . . . . . . 12 |- ( ( F e. ran S /\ ( y e. Word ( (mCN ` T ) u. V ) /\ z e. ( (mCN ` T ) u. V ) ) ) -> ran ( ( F ` y ) ++ ( F ` <" z "> ) ) = ( ran ( F ` y ) u. ran ( F ` <" z "> ) ) )
76	67, 75	eqtrd 2656	. . . . . . . . . . 11 |- ( ( F e. ran S /\ ( y e. Word ( (mCN ` T ) u. V ) /\ z e. ( (mCN ` T ) u. V ) ) ) -> ran ( F ` ( y ++ <" z "> ) ) = ( ran ( F ` y ) u. ran ( F ` <" z "> ) ) )
77	76	ineq1d 3813	. . . . . . . . . 10 |- ( ( F e. ran S /\ ( y e. Word ( (mCN ` T ) u. V ) /\ z e. ( (mCN ` T ) u. V ) ) ) -> ( ran ( F ` ( y ++ <" z "> ) ) i^i V ) = ( ( ran ( F ` y ) u. ran ( F ` <" z "> ) ) i^i V ) )
78		indir 3875	. . . . . . . . . 10 |- ( ( ran ( F ` y ) u. ran ( F ` <" z "> ) ) i^i V ) = ( ( ran ( F ` y ) i^i V ) u. ( ran ( F ` <" z "> ) i^i V ) )
79	77, 78	syl6eq 2672	. . . . . . . . 9 |- ( ( F e. ran S /\ ( y e. Word ( (mCN ` T ) u. V ) /\ z e. ( (mCN ` T ) u. V ) ) ) -> ( ran ( F ` ( y ++ <" z "> ) ) i^i V ) = ( ( ran ( F ` y ) i^i V ) u. ( ran ( F ` <" z "> ) i^i V ) ) )
80		ccatrn 13372	. . . . . . . . . . . . . . . 16 |- ( ( y e. Word ( (mCN ` T ) u. V ) /\ <" z "> e. Word ( (mCN ` T ) u. V ) ) -> ran ( y ++ <" z "> ) = ( ran y u. ran <" z "> ) )
81	59, 63, 80	syl2anc 693	. . . . . . . . . . . . . . 15 |- ( ( F e. ran S /\ ( y e. Word ( (mCN ` T ) u. V ) /\ z e. ( (mCN ` T ) u. V ) ) ) -> ran ( y ++ <" z "> ) = ( ran y u. ran <" z "> ) )
82		s1rn 13379	. . . . . . . . . . . . . . . . 17 |- ( z e. ( (mCN ` T ) u. V ) -> ran <" z "> = { z } )
83	82	ad2antll 765	. . . . . . . . . . . . . . . 16 |- ( ( F e. ran S /\ ( y e. Word ( (mCN ` T ) u. V ) /\ z e. ( (mCN ` T ) u. V ) ) ) -> ran <" z "> = { z } )
84	83	uneq2d 3767	. . . . . . . . . . . . . . 15 |- ( ( F e. ran S /\ ( y e. Word ( (mCN ` T ) u. V ) /\ z e. ( (mCN ` T ) u. V ) ) ) -> ( ran y u. ran <" z "> ) = ( ran y u. { z } ) )
85	81, 84	eqtrd 2656	. . . . . . . . . . . . . 14 |- ( ( F e. ran S /\ ( y e. Word ( (mCN ` T ) u. V ) /\ z e. ( (mCN ` T ) u. V ) ) ) -> ran ( y ++ <" z "> ) = ( ran y u. { z } ) )
86	85	ineq1d 3813	. . . . . . . . . . . . 13 |- ( ( F e. ran S /\ ( y e. Word ( (mCN ` T ) u. V ) /\ z e. ( (mCN ` T ) u. V ) ) ) -> ( ran ( y ++ <" z "> ) i^i V ) = ( ( ran y u. { z } ) i^i V ) )
87		indir 3875	. . . . . . . . . . . . 13 |- ( ( ran y u. { z } ) i^i V ) = ( ( ran y i^i V ) u. ( { z } i^i V ) )
88	86, 87	syl6eq 2672	. . . . . . . . . . . 12 |- ( ( F e. ran S /\ ( y e. Word ( (mCN ` T ) u. V ) /\ z e. ( (mCN ` T ) u. V ) ) ) -> ( ran ( y ++ <" z "> ) i^i V ) = ( ( ran y i^i V ) u. ( { z } i^i V ) ) )
89	88	iuneq1d 4545	. . . . . . . . . . 11 |- ( ( F e. ran S /\ ( y e. Word ( (mCN ` T ) u. V ) /\ z e. ( (mCN ` T ) u. V ) ) ) -> U_ x e. ( ran ( y ++ <" z "> ) i^i V ) ( ran ( F ` <" x "> ) i^i V ) = U_ x e. ( ( ran y i^i V ) u. ( { z } i^i V ) ) ( ran ( F ` <" x "> ) i^i V ) )
90		iunxun 4605	. . . . . . . . . . 11 |- U_ x e. ( ( ran y i^i V ) u. ( { z } i^i V ) ) ( ran ( F ` <" x "> ) i^i V ) = ( U_ x e. ( ran y i^i V ) ( ran ( F ` <" x "> ) i^i V ) u. U_ x e. ( { z } i^i V ) ( ran ( F ` <" x "> ) i^i V ) )
91	89, 90	syl6eq 2672	. . . . . . . . . 10 |- ( ( F e. ran S /\ ( y e. Word ( (mCN ` T ) u. V ) /\ z e. ( (mCN ` T ) u. V ) ) ) -> U_ x e. ( ran ( y ++ <" z "> ) i^i V ) ( ran ( F ` <" x "> ) i^i V ) = ( U_ x e. ( ran y i^i V ) ( ran ( F ` <" x "> ) i^i V ) u. U_ x e. ( { z } i^i V ) ( ran ( F ` <" x "> ) i^i V ) ) )
92		simpr 477	. . . . . . . . . . . . . . . 16 |- ( ( ( F e. ran S /\ ( y e. Word ( (mCN ` T ) u. V ) /\ z e. ( (mCN ` T ) u. V ) ) ) /\ z e. V ) -> z e. V )
93	92	snssd 4340	. . . . . . . . . . . . . . 15 |- ( ( ( F e. ran S /\ ( y e. Word ( (mCN ` T ) u. V ) /\ z e. ( (mCN ` T ) u. V ) ) ) /\ z e. V ) -> { z } C_ V )
94		df-ss 3588	. . . . . . . . . . . . . . 15 |- ( { z } C_ V <-> ( { z } i^i V ) = { z } )
95	93, 94	sylib 208	. . . . . . . . . . . . . 14 |- ( ( ( F e. ran S /\ ( y e. Word ( (mCN ` T ) u. V ) /\ z e. ( (mCN ` T ) u. V ) ) ) /\ z e. V ) -> ( { z } i^i V ) = { z } )
96	95	iuneq1d 4545	. . . . . . . . . . . . 13 |- ( ( ( F e. ran S /\ ( y e. Word ( (mCN ` T ) u. V ) /\ z e. ( (mCN ` T ) u. V ) ) ) /\ z e. V ) -> U_ x e. ( { z } i^i V ) ( ran ( F ` <" x "> ) i^i V ) = U_ x e. { z } ( ran ( F ` <" x "> ) i^i V ) )
97		vex 3203	. . . . . . . . . . . . . 14 |- z e. _V
98		s1eq 13380	. . . . . . . . . . . . . . . . 17 |- ( x = z -> <" x "> = <" z "> )
99	98	fveq2d 6195	. . . . . . . . . . . . . . . 16 |- ( x = z -> ( F ` <" x "> ) = ( F ` <" z "> ) )
100	99	rneqd 5353	. . . . . . . . . . . . . . 15 |- ( x = z -> ran ( F ` <" x "> ) = ran ( F ` <" z "> ) )
101	100	ineq1d 3813	. . . . . . . . . . . . . 14 |- ( x = z -> ( ran ( F ` <" x "> ) i^i V ) = ( ran ( F ` <" z "> ) i^i V ) )
102	97, 101	iunxsn 4603	. . . . . . . . . . . . 13 |- U_ x e. { z } ( ran ( F ` <" x "> ) i^i V ) = ( ran ( F ` <" z "> ) i^i V )
103	96, 102	syl6eq 2672	. . . . . . . . . . . 12 |- ( ( ( F e. ran S /\ ( y e. Word ( (mCN ` T ) u. V ) /\ z e. ( (mCN ` T ) u. V ) ) ) /\ z e. V ) -> U_ x e. ( { z } i^i V ) ( ran ( F ` <" x "> ) i^i V ) = ( ran ( F ` <" z "> ) i^i V ) )
104		incom 3805	. . . . . . . . . . . . . . 15 |- ( { z } i^i V ) = ( V i^i { z } )
105		simpr 477	. . . . . . . . . . . . . . . 16 |- ( ( ( F e. ran S /\ ( y e. Word ( (mCN ` T ) u. V ) /\ z e. ( (mCN ` T ) u. V ) ) ) /\ -. z e. V ) -> -. z e. V )
106		disjsn 4246	. . . . . . . . . . . . . . . 16 |- ( ( V i^i { z } ) = (/) <-> -. z e. V )
107	105, 106	sylibr 224	. . . . . . . . . . . . . . 15 |- ( ( ( F e. ran S /\ ( y e. Word ( (mCN ` T ) u. V ) /\ z e. ( (mCN ` T ) u. V ) ) ) /\ -. z e. V ) -> ( V i^i { z } ) = (/) )
108	104, 107	syl5eq 2668	. . . . . . . . . . . . . 14 |- ( ( ( F e. ran S /\ ( y e. Word ( (mCN ` T ) u. V ) /\ z e. ( (mCN ` T ) u. V ) ) ) /\ -. z e. V ) -> ( { z } i^i V ) = (/) )
109	108	iuneq1d 4545	. . . . . . . . . . . . 13 |- ( ( ( F e. ran S /\ ( y e. Word ( (mCN ` T ) u. V ) /\ z e. ( (mCN ` T ) u. V ) ) ) /\ -. z e. V ) -> U_ x e. ( { z } i^i V ) ( ran ( F ` <" x "> ) i^i V ) = U_ x e. (/) ( ran ( F ` <" x "> ) i^i V ) )
110	58	adantr 481	. . . . . . . . . . . . . . . . . 18 |- ( ( ( F e. ran S /\ ( y e. Word ( (mCN ` T ) u. V ) /\ z e. ( (mCN ` T ) u. V ) ) ) /\ -. z e. V ) -> F e. ran S )
111		eldif 3584	. . . . . . . . . . . . . . . . . . . . 21 |- ( z e. ( ( (mCN ` T ) u. V ) \ V ) <-> ( z e. ( (mCN ` T ) u. V ) /\ -. z e. V ) )
112	111	biimpri 218	. . . . . . . . . . . . . . . . . . . 20 |- ( ( z e. ( (mCN ` T ) u. V ) /\ -. z e. V ) -> z e. ( ( (mCN ` T ) u. V ) \ V ) )
113	62, 112	sylan 488	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( F e. ran S /\ ( y e. Word ( (mCN ` T ) u. V ) /\ z e. ( (mCN ` T ) u. V ) ) ) /\ -. z e. V ) -> z e. ( ( (mCN ` T ) u. V ) \ V ) )
114		difun2 4048	. . . . . . . . . . . . . . . . . . 19 |- ( ( (mCN ` T ) u. V ) \ V ) = ( (mCN ` T ) \ V )
115	113, 114	syl6eleq 2711	. . . . . . . . . . . . . . . . . 18 |- ( ( ( F e. ran S /\ ( y e. Word ( (mCN ` T ) u. V ) /\ z e. ( (mCN ` T ) u. V ) ) ) /\ -. z e. V ) -> z e. ( (mCN ` T ) \ V ) )
116	2, 11, 10, 9	mrsubcn 31416	. . . . . . . . . . . . . . . . . 18 |- ( ( F e. ran S /\ z e. ( (mCN ` T ) \ V ) ) -> ( F ` <" z "> ) = <" z "> )
117	110, 115, 116	syl2anc 693	. . . . . . . . . . . . . . . . 17 |- ( ( ( F e. ran S /\ ( y e. Word ( (mCN ` T ) u. V ) /\ z e. ( (mCN ` T ) u. V ) ) ) /\ -. z e. V ) -> ( F ` <" z "> ) = <" z "> )
118	117	rneqd 5353	. . . . . . . . . . . . . . . 16 |- ( ( ( F e. ran S /\ ( y e. Word ( (mCN ` T ) u. V ) /\ z e. ( (mCN ` T ) u. V ) ) ) /\ -. z e. V ) -> ran ( F ` <" z "> ) = ran <" z "> )
119	83	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( ( F e. ran S /\ ( y e. Word ( (mCN ` T ) u. V ) /\ z e. ( (mCN ` T ) u. V ) ) ) /\ -. z e. V ) -> ran <" z "> = { z } )
120	118, 119	eqtrd 2656	. . . . . . . . . . . . . . 15 |- ( ( ( F e. ran S /\ ( y e. Word ( (mCN ` T ) u. V ) /\ z e. ( (mCN ` T ) u. V ) ) ) /\ -. z e. V ) -> ran ( F ` <" z "> ) = { z } )
121	120	ineq1d 3813	. . . . . . . . . . . . . 14 |- ( ( ( F e. ran S /\ ( y e. Word ( (mCN ` T ) u. V ) /\ z e. ( (mCN ` T ) u. V ) ) ) /\ -. z e. V ) -> ( ran ( F ` <" z "> ) i^i V ) = ( { z } i^i V ) )
122	121, 108	eqtrd 2656	. . . . . . . . . . . . 13 |- ( ( ( F e. ran S /\ ( y e. Word ( (mCN ` T ) u. V ) /\ z e. ( (mCN ` T ) u. V ) ) ) /\ -. z e. V ) -> ( ran ( F ` <" z "> ) i^i V ) = (/) )
123	24, 109, 122	3eqtr4a 2682	. . . . . . . . . . . 12 |- ( ( ( F e. ran S /\ ( y e. Word ( (mCN ` T ) u. V ) /\ z e. ( (mCN ` T ) u. V ) ) ) /\ -. z e. V ) -> U_ x e. ( { z } i^i V ) ( ran ( F ` <" x "> ) i^i V ) = ( ran ( F ` <" z "> ) i^i V ) )
124	103, 123	pm2.61dan 832	. . . . . . . . . . 11 |- ( ( F e. ran S /\ ( y e. Word ( (mCN ` T ) u. V ) /\ z e. ( (mCN ` T ) u. V ) ) ) -> U_ x e. ( { z } i^i V ) ( ran ( F ` <" x "> ) i^i V ) = ( ran ( F ` <" z "> ) i^i V ) )
125	124	uneq2d 3767	. . . . . . . . . 10 |- ( ( F e. ran S /\ ( y e. Word ( (mCN ` T ) u. V ) /\ z e. ( (mCN ` T ) u. V ) ) ) -> ( U_ x e. ( ran y i^i V ) ( ran ( F ` <" x "> ) i^i V ) u. U_ x e. ( { z } i^i V ) ( ran ( F ` <" x "> ) i^i V ) ) = ( U_ x e. ( ran y i^i V ) ( ran ( F ` <" x "> ) i^i V ) u. ( ran ( F ` <" z "> ) i^i V ) ) )
126	91, 125	eqtrd 2656	. . . . . . . . 9 |- ( ( F e. ran S /\ ( y e. Word ( (mCN ` T ) u. V ) /\ z e. ( (mCN ` T ) u. V ) ) ) -> U_ x e. ( ran ( y ++ <" z "> ) i^i V ) ( ran ( F ` <" x "> ) i^i V ) = ( U_ x e. ( ran y i^i V ) ( ran ( F ` <" x "> ) i^i V ) u. ( ran ( F ` <" z "> ) i^i V ) ) )
127	79, 126	eqeq12d 2637	. . . . . . . 8 |- ( ( F e. ran S /\ ( y e. Word ( (mCN ` T ) u. V ) /\ z e. ( (mCN ` T ) u. V ) ) ) -> ( ( ran ( F ` ( y ++ <" z "> ) ) i^i V ) = U_ x e. ( ran ( y ++ <" z "> ) i^i V ) ( ran ( F ` <" x "> ) i^i V ) <-> ( ( ran ( F ` y ) i^i V ) u. ( ran ( F ` <" z "> ) i^i V ) ) = ( U_ x e. ( ran y i^i V ) ( ran ( F ` <" x "> ) i^i V ) u. ( ran ( F ` <" z "> ) i^i V ) ) ) )
128	57, 127	syl5ibr 236	. . . . . . 7 |- ( ( F e. ran S /\ ( y e. Word ( (mCN ` T ) u. V ) /\ z e. ( (mCN ` T ) u. V ) ) ) -> ( ( ran ( F ` y ) i^i V ) = U_ x e. ( ran y i^i V ) ( ran ( F ` <" x "> ) i^i V ) -> ( ran ( F ` ( y ++ <" z "> ) ) i^i V ) = U_ x e. ( ran ( y ++ <" z "> ) i^i V ) ( ran ( F ` <" x "> ) i^i V ) ) )
129	128	expcom 451	. . . . . 6 |- ( ( y e. Word ( (mCN ` T ) u. V ) /\ z e. ( (mCN ` T ) u. V ) ) -> ( F e. ran S -> ( ( ran ( F ` y ) i^i V ) = U_ x e. ( ran y i^i V ) ( ran ( F ` <" x "> ) i^i V ) -> ( ran ( F ` ( y ++ <" z "> ) ) i^i V ) = U_ x e. ( ran ( y ++ <" z "> ) i^i V ) ( ran ( F ` <" x "> ) i^i V ) ) ) )
130	129	a2d 29	. . . . 5 |- ( ( y e. Word ( (mCN ` T ) u. V ) /\ z e. ( (mCN ` T ) u. V ) ) -> ( ( F e. ran S -> ( ran ( F ` y ) i^i V ) = U_ x e. ( ran y i^i V ) ( ran ( F ` <" x "> ) i^i V ) ) -> ( F e. ran S -> ( ran ( F ` ( y ++ <" z "> ) ) i^i V ) = U_ x e. ( ran ( y ++ <" z "> ) i^i V ) ( ran ( F ` <" x "> ) i^i V ) ) ) )
131	27, 35, 43, 51, 56, 130	wrdind 13476	. . . 4 |- ( X e. Word ( (mCN ` T ) u. V ) -> ( F e. ran S -> ( ran ( F ` X ) i^i V ) = U_ x e. ( ran X i^i V ) ( ran ( F ` <" x "> ) i^i V ) ) )
132	131	com12 32	. . 3 |- ( F e. ran S -> ( X e. Word ( (mCN ` T ) u. V ) -> ( ran ( F ` X ) i^i V ) = U_ x e. ( ran X i^i V ) ( ran ( F ` <" x "> ) i^i V ) ) )
133	14, 132	sylbid 230	. 2 |- ( F e. ran S -> ( X e. R -> ( ran ( F ` X ) i^i V ) = U_ x e. ( ran X i^i V ) ( ran ( F ` <" x "> ) i^i V ) ) )
134	133	imp 445	1 |- ( ( F e. ran S /\ X e. R ) -> ( ran ( F ` X ) i^i V ) = U_ x e. ( ran X i^i V ) ( ran ( F ` <" x "> ) i^i V ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   \ cdif 3571   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915   {csn 4177   U_ciun 4520   ran crn 5115   -->wf 5884   ` cfv 5888  (class class class)co 6650  Word cword 13291   ++ cconcat 13293   <"cs1 13294  mCNcmcn 31357  mVRcmvar 31358  mRExcmrex 31363  mRSubstcmrsub 31367

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-mrsub 31387

This theorem is referenced by:  msubvrs  31457