Theorem fnwelem 7292

Description: Lemma for fnwe 7293. (Contributed by Mario Carneiro, 10-Mar-2013.) (Revised by Mario Carneiro, 18-Nov-2014.)

Hypotheses

Ref	Expression
fnwe.1	|- T = { <. x , y >. | ( ( x e. A /\ y e. A ) /\ ( ( F ` x ) R ( F ` y ) \/ ( ( F ` x ) = ( F ` y ) /\ x S y ) ) ) }
fnwe.2	|- ( ph -> F : A --> B )
fnwe.3	|- ( ph -> R We B )
fnwe.4	|- ( ph -> S We A )
fnwe.5	|- ( ph -> ( F " w ) e. _V )
fnwelem.6	|- Q = { <. u , v >. | ( ( u e. ( B X. A ) /\ v e. ( B X. A ) ) /\ ( ( 1st ` u ) R ( 1st ` v ) \/ ( ( 1st ` u ) = ( 1st ` v ) /\ ( 2nd ` u ) S ( 2nd ` v ) ) ) ) }
fnwelem.7	|- G = ( z e. A |-> <. ( F ` z ) , z >. )

Assertion

Ref	Expression
fnwelem	|- ( ph -> T We A )

Distinct variable groups:   v, u, w, x, y, z, A   u, B, v, w, x, y, z   w, G, x, y   ph, w, x, z   u, F, v, w, x, y, z   w, Q, x, y   u, R, v, w, x, y   u, S, v, w, x, y   w, T

Allowed substitution hints:   ph( y, v, u)   Q( z, v, u)   R( z)   S( z)   T( x, y, z, v, u)   G( z, v, u)

Proof of Theorem fnwelem

Step	Hyp	Ref	Expression
1		fnwe.2	. . . 4 |- ( ph -> F : A --> B )
2		ffvelrn 6357	. . . . . 6 |- ( ( F : A --> B /\ z e. A ) -> ( F ` z ) e. B )
3		simpr 477	. . . . . 6 |- ( ( F : A --> B /\ z e. A ) -> z e. A )
4	2, 3	opelxpd 5149	. . . . 5 |- ( ( F : A --> B /\ z e. A ) -> <. ( F ` z ) , z >. e. ( B X. A ) )
5		fnwelem.7	. . . . 5 |- G = ( z e. A |-> <. ( F ` z ) , z >. )
6	4, 5	fmptd 6385	. . . 4 |- ( F : A --> B -> G : A --> ( B X. A ) )
7		frn 6053	. . . 4 |- ( G : A --> ( B X. A ) -> ran G C_ ( B X. A ) )
8	1, 6, 7	3syl 18	. . 3 |- ( ph -> ran G C_ ( B X. A ) )
9		fnwe.3	. . . 4 |- ( ph -> R We B )
10		fnwe.4	. . . 4 |- ( ph -> S We A )
11		fnwelem.6	. . . . 5 |- Q = { <. u , v >. | ( ( u e. ( B X. A ) /\ v e. ( B X. A ) ) /\ ( ( 1st ` u ) R ( 1st ` v ) \/ ( ( 1st ` u ) = ( 1st ` v ) /\ ( 2nd ` u ) S ( 2nd ` v ) ) ) ) }
12	11	wexp 7291	. . . 4 |- ( ( R We B /\ S We A ) -> Q We ( B X. A ) )
13	9, 10, 12	syl2anc 693	. . 3 |- ( ph -> Q We ( B X. A ) )
14		wess 5101	. . 3 |- ( ran G C_ ( B X. A ) -> ( Q We ( B X. A ) -> Q We ran G ) )
15	8, 13, 14	sylc 65	. 2 |- ( ph -> Q We ran G )
16		fveq2 6191	. . . . . . . . . . . 12 |- ( z = x -> ( F ` z ) = ( F ` x ) )
17		id 22	. . . . . . . . . . . 12 |- ( z = x -> z = x )
18	16, 17	opeq12d 4410	. . . . . . . . . . 11 |- ( z = x -> <. ( F ` z ) , z >. = <. ( F ` x ) , x >. )
19		opex 4932	. . . . . . . . . . 11 |- <. ( F ` x ) , x >. e. _V
20	18, 5, 19	fvmpt 6282	. . . . . . . . . 10 |- ( x e. A -> ( G ` x ) = <. ( F ` x ) , x >. )
21		fveq2 6191	. . . . . . . . . . . 12 |- ( z = y -> ( F ` z ) = ( F ` y ) )
22		id 22	. . . . . . . . . . . 12 |- ( z = y -> z = y )
23	21, 22	opeq12d 4410	. . . . . . . . . . 11 |- ( z = y -> <. ( F ` z ) , z >. = <. ( F ` y ) , y >. )
24		opex 4932	. . . . . . . . . . 11 |- <. ( F ` y ) , y >. e. _V
25	23, 5, 24	fvmpt 6282	. . . . . . . . . 10 |- ( y e. A -> ( G ` y ) = <. ( F ` y ) , y >. )
26	20, 25	eqeqan12d 2638	. . . . . . . . 9 |- ( ( x e. A /\ y e. A ) -> ( ( G ` x ) = ( G ` y ) <-> <. ( F ` x ) , x >. = <. ( F ` y ) , y >. ) )
27		fvex 6201	. . . . . . . . . . 11 |- ( F ` x ) e. _V
28		vex 3203	. . . . . . . . . . 11 |- x e. _V
29	27, 28	opth 4945	. . . . . . . . . 10 |- ( <. ( F ` x ) , x >. = <. ( F ` y ) , y >. <-> ( ( F ` x ) = ( F ` y ) /\ x = y ) )
30	29	simprbi 480	. . . . . . . . 9 |- ( <. ( F ` x ) , x >. = <. ( F ` y ) , y >. -> x = y )
31	26, 30	syl6bi 243	. . . . . . . 8 |- ( ( x e. A /\ y e. A ) -> ( ( G ` x ) = ( G ` y ) -> x = y ) )
32	31	rgen2a 2977	. . . . . . 7 |- A. x e. A A. y e. A ( ( G ` x ) = ( G ` y ) -> x = y )
33		dff13 6512	. . . . . . 7 |- ( G : A -1-1-> ( B X. A ) <-> ( G : A --> ( B X. A ) /\ A. x e. A A. y e. A ( ( G ` x ) = ( G ` y ) -> x = y ) ) )
34	6, 32, 33	sylanblrc 697	. . . . . 6 |- ( F : A --> B -> G : A -1-1-> ( B X. A ) )
35		f1f1orn 6148	. . . . . 6 |- ( G : A -1-1-> ( B X. A ) -> G : A -1-1-onto-> ran G )
36		f1ocnv 6149	. . . . . 6 |- ( G : A -1-1-onto-> ran G -> `' G : ran G -1-1-onto-> A )
37	1, 34, 35, 36	4syl 19	. . . . 5 |- ( ph -> `' G : ran G -1-1-onto-> A )
38		eqid 2622	. . . . . . 7 |- { <. x , y >. | ( ( x e. A /\ y e. A ) /\ ( `' `' G ` x ) Q ( `' `' G ` y ) ) } = { <. x , y >. | ( ( x e. A /\ y e. A ) /\ ( `' `' G ` x ) Q ( `' `' G ` y ) ) }
39	38	f1oiso2 6602	. . . . . 6 |- ( `' G : ran G -1-1-onto-> A -> `' G Isom Q , { <. x , y >. | ( ( x e. A /\ y e. A ) /\ ( `' `' G ` x ) Q ( `' `' G ` y ) ) } ( ran G , A ) )
40		fnwe.1	. . . . . . . 8 |- T = { <. x , y >. | ( ( x e. A /\ y e. A ) /\ ( ( F ` x ) R ( F ` y ) \/ ( ( F ` x ) = ( F ` y ) /\ x S y ) ) ) }
41		frel 6050	. . . . . . . . . . . . . . . 16 |- ( G : A --> ( B X. A ) -> Rel G )
42		dfrel2 5583	. . . . . . . . . . . . . . . 16 |- ( Rel G <-> `' `' G = G )
43	41, 42	sylib 208	. . . . . . . . . . . . . . 15 |- ( G : A --> ( B X. A ) -> `' `' G = G )
44	43	fveq1d 6193	. . . . . . . . . . . . . 14 |- ( G : A --> ( B X. A ) -> ( `' `' G ` x ) = ( G ` x ) )
45	43	fveq1d 6193	. . . . . . . . . . . . . 14 |- ( G : A --> ( B X. A ) -> ( `' `' G ` y ) = ( G ` y ) )
46	44, 45	breq12d 4666	. . . . . . . . . . . . 13 |- ( G : A --> ( B X. A ) -> ( ( `' `' G ` x ) Q ( `' `' G ` y ) <-> ( G ` x ) Q ( G ` y ) ) )
47	6, 46	syl 17	. . . . . . . . . . . 12 |- ( F : A --> B -> ( ( `' `' G ` x ) Q ( `' `' G ` y ) <-> ( G ` x ) Q ( G ` y ) ) )
48	47	adantr 481	. . . . . . . . . . 11 |- ( ( F : A --> B /\ ( x e. A /\ y e. A ) ) -> ( ( `' `' G ` x ) Q ( `' `' G ` y ) <-> ( G ` x ) Q ( G ` y ) ) )
49	20, 25	breqan12d 4669	. . . . . . . . . . . 12 |- ( ( x e. A /\ y e. A ) -> ( ( G ` x ) Q ( G ` y ) <-> <. ( F ` x ) , x >. Q <. ( F ` y ) , y >. ) )
50	49	adantl 482	. . . . . . . . . . 11 |- ( ( F : A --> B /\ ( x e. A /\ y e. A ) ) -> ( ( G ` x ) Q ( G ` y ) <-> <. ( F ` x ) , x >. Q <. ( F ` y ) , y >. ) )
51		ffvelrn 6357	. . . . . . . . . . . . . . 15 |- ( ( F : A --> B /\ x e. A ) -> ( F ` x ) e. B )
52		simpr 477	. . . . . . . . . . . . . . 15 |- ( ( F : A --> B /\ x e. A ) -> x e. A )
53	51, 52	jca 554	. . . . . . . . . . . . . 14 |- ( ( F : A --> B /\ x e. A ) -> ( ( F ` x ) e. B /\ x e. A ) )
54		ffvelrn 6357	. . . . . . . . . . . . . . 15 |- ( ( F : A --> B /\ y e. A ) -> ( F ` y ) e. B )
55		simpr 477	. . . . . . . . . . . . . . 15 |- ( ( F : A --> B /\ y e. A ) -> y e. A )
56	54, 55	jca 554	. . . . . . . . . . . . . 14 |- ( ( F : A --> B /\ y e. A ) -> ( ( F ` y ) e. B /\ y e. A ) )
57	53, 56	anim12dan 882	. . . . . . . . . . . . 13 |- ( ( F : A --> B /\ ( x e. A /\ y e. A ) ) -> ( ( ( F ` x ) e. B /\ x e. A ) /\ ( ( F ` y ) e. B /\ y e. A ) ) )
58	57	biantrurd 529	. . . . . . . . . . . 12 |- ( ( F : A --> B /\ ( x e. A /\ y e. A ) ) -> ( ( ( F ` x ) R ( F ` y ) \/ ( ( F ` x ) = ( F ` y ) /\ x S y ) ) <-> ( ( ( ( F ` x ) e. B /\ x e. A ) /\ ( ( F ` y ) e. B /\ y e. A ) ) /\ ( ( F ` x ) R ( F ` y ) \/ ( ( F ` x ) = ( F ` y ) /\ x S y ) ) ) ) )
59		eleq1 2689	. . . . . . . . . . . . . . . 16 |- ( u = <. ( F ` x ) , x >. -> ( u e. ( B X. A ) <-> <. ( F ` x ) , x >. e. ( B X. A ) ) )
60		opelxp 5146	. . . . . . . . . . . . . . . 16 |- ( <. ( F ` x ) , x >. e. ( B X. A ) <-> ( ( F ` x ) e. B /\ x e. A ) )
61	59, 60	syl6bb 276	. . . . . . . . . . . . . . 15 |- ( u = <. ( F ` x ) , x >. -> ( u e. ( B X. A ) <-> ( ( F ` x ) e. B /\ x e. A ) ) )
62	61	anbi1d 741	. . . . . . . . . . . . . 14 |- ( u = <. ( F ` x ) , x >. -> ( ( u e. ( B X. A ) /\ v e. ( B X. A ) ) <-> ( ( ( F ` x ) e. B /\ x e. A ) /\ v e. ( B X. A ) ) ) )
63	27, 28	op1std 7178	. . . . . . . . . . . . . . . 16 |- ( u = <. ( F ` x ) , x >. -> ( 1st ` u ) = ( F ` x ) )
64	63	breq1d 4663	. . . . . . . . . . . . . . 15 |- ( u = <. ( F ` x ) , x >. -> ( ( 1st ` u ) R ( 1st ` v ) <-> ( F ` x ) R ( 1st ` v ) ) )
65	63	eqeq1d 2624	. . . . . . . . . . . . . . . 16 |- ( u = <. ( F ` x ) , x >. -> ( ( 1st ` u ) = ( 1st ` v ) <-> ( F ` x ) = ( 1st ` v ) ) )
66	27, 28	op2ndd 7179	. . . . . . . . . . . . . . . . 17 |- ( u = <. ( F ` x ) , x >. -> ( 2nd ` u ) = x )
67	66	breq1d 4663	. . . . . . . . . . . . . . . 16 |- ( u = <. ( F ` x ) , x >. -> ( ( 2nd ` u ) S ( 2nd ` v ) <-> x S ( 2nd ` v ) ) )
68	65, 67	anbi12d 747	. . . . . . . . . . . . . . 15 |- ( u = <. ( F ` x ) , x >. -> ( ( ( 1st ` u ) = ( 1st ` v ) /\ ( 2nd ` u ) S ( 2nd ` v ) ) <-> ( ( F ` x ) = ( 1st ` v ) /\ x S ( 2nd ` v ) ) ) )
69	64, 68	orbi12d 746	. . . . . . . . . . . . . 14 |- ( u = <. ( F ` x ) , x >. -> ( ( ( 1st ` u ) R ( 1st ` v ) \/ ( ( 1st ` u ) = ( 1st ` v ) /\ ( 2nd ` u ) S ( 2nd ` v ) ) ) <-> ( ( F ` x ) R ( 1st ` v ) \/ ( ( F ` x ) = ( 1st ` v ) /\ x S ( 2nd ` v ) ) ) ) )
70	62, 69	anbi12d 747	. . . . . . . . . . . . 13 |- ( u = <. ( F ` x ) , x >. -> ( ( ( u e. ( B X. A ) /\ v e. ( B X. A ) ) /\ ( ( 1st ` u ) R ( 1st ` v ) \/ ( ( 1st ` u ) = ( 1st ` v ) /\ ( 2nd ` u ) S ( 2nd ` v ) ) ) ) <-> ( ( ( ( F ` x ) e. B /\ x e. A ) /\ v e. ( B X. A ) ) /\ ( ( F ` x ) R ( 1st ` v ) \/ ( ( F ` x ) = ( 1st ` v ) /\ x S ( 2nd ` v ) ) ) ) ) )
71		eleq1 2689	. . . . . . . . . . . . . . . 16 |- ( v = <. ( F ` y ) , y >. -> ( v e. ( B X. A ) <-> <. ( F ` y ) , y >. e. ( B X. A ) ) )
72		opelxp 5146	. . . . . . . . . . . . . . . 16 |- ( <. ( F ` y ) , y >. e. ( B X. A ) <-> ( ( F ` y ) e. B /\ y e. A ) )
73	71, 72	syl6bb 276	. . . . . . . . . . . . . . 15 |- ( v = <. ( F ` y ) , y >. -> ( v e. ( B X. A ) <-> ( ( F ` y ) e. B /\ y e. A ) ) )
74	73	anbi2d 740	. . . . . . . . . . . . . 14 |- ( v = <. ( F ` y ) , y >. -> ( ( ( ( F ` x ) e. B /\ x e. A ) /\ v e. ( B X. A ) ) <-> ( ( ( F ` x ) e. B /\ x e. A ) /\ ( ( F ` y ) e. B /\ y e. A ) ) ) )
75		fvex 6201	. . . . . . . . . . . . . . . . 17 |- ( F ` y ) e. _V
76		vex 3203	. . . . . . . . . . . . . . . . 17 |- y e. _V
77	75, 76	op1std 7178	. . . . . . . . . . . . . . . 16 |- ( v = <. ( F ` y ) , y >. -> ( 1st ` v ) = ( F ` y ) )
78	77	breq2d 4665	. . . . . . . . . . . . . . 15 |- ( v = <. ( F ` y ) , y >. -> ( ( F ` x ) R ( 1st ` v ) <-> ( F ` x ) R ( F ` y ) ) )
79	77	eqeq2d 2632	. . . . . . . . . . . . . . . 16 |- ( v = <. ( F ` y ) , y >. -> ( ( F ` x ) = ( 1st ` v ) <-> ( F ` x ) = ( F ` y ) ) )
80	75, 76	op2ndd 7179	. . . . . . . . . . . . . . . . 17 |- ( v = <. ( F ` y ) , y >. -> ( 2nd ` v ) = y )
81	80	breq2d 4665	. . . . . . . . . . . . . . . 16 |- ( v = <. ( F ` y ) , y >. -> ( x S ( 2nd ` v ) <-> x S y ) )
82	79, 81	anbi12d 747	. . . . . . . . . . . . . . 15 |- ( v = <. ( F ` y ) , y >. -> ( ( ( F ` x ) = ( 1st ` v ) /\ x S ( 2nd ` v ) ) <-> ( ( F ` x ) = ( F ` y ) /\ x S y ) ) )
83	78, 82	orbi12d 746	. . . . . . . . . . . . . 14 |- ( v = <. ( F ` y ) , y >. -> ( ( ( F ` x ) R ( 1st ` v ) \/ ( ( F ` x ) = ( 1st ` v ) /\ x S ( 2nd ` v ) ) ) <-> ( ( F ` x ) R ( F ` y ) \/ ( ( F ` x ) = ( F ` y ) /\ x S y ) ) ) )
84	74, 83	anbi12d 747	. . . . . . . . . . . . 13 |- ( v = <. ( F ` y ) , y >. -> ( ( ( ( ( F ` x ) e. B /\ x e. A ) /\ v e. ( B X. A ) ) /\ ( ( F ` x ) R ( 1st ` v ) \/ ( ( F ` x ) = ( 1st ` v ) /\ x S ( 2nd ` v ) ) ) ) <-> ( ( ( ( F ` x ) e. B /\ x e. A ) /\ ( ( F ` y ) e. B /\ y e. A ) ) /\ ( ( F ` x ) R ( F ` y ) \/ ( ( F ` x ) = ( F ` y ) /\ x S y ) ) ) ) )
85	19, 24, 70, 84, 11	brab 4998	. . . . . . . . . . . 12 |- ( <. ( F ` x ) , x >. Q <. ( F ` y ) , y >. <-> ( ( ( ( F ` x ) e. B /\ x e. A ) /\ ( ( F ` y ) e. B /\ y e. A ) ) /\ ( ( F ` x ) R ( F ` y ) \/ ( ( F ` x ) = ( F ` y ) /\ x S y ) ) ) )
86	58, 85	syl6rbbr 279	. . . . . . . . . . 11 |- ( ( F : A --> B /\ ( x e. A /\ y e. A ) ) -> ( <. ( F ` x ) , x >. Q <. ( F ` y ) , y >. <-> ( ( F ` x ) R ( F ` y ) \/ ( ( F ` x ) = ( F ` y ) /\ x S y ) ) ) )
87	48, 50, 86	3bitrrd 295	. . . . . . . . . 10 |- ( ( F : A --> B /\ ( x e. A /\ y e. A ) ) -> ( ( ( F ` x ) R ( F ` y ) \/ ( ( F ` x ) = ( F ` y ) /\ x S y ) ) <-> ( `' `' G ` x ) Q ( `' `' G ` y ) ) )
88	87	pm5.32da 673	. . . . . . . . 9 |- ( F : A --> B -> ( ( ( x e. A /\ y e. A ) /\ ( ( F ` x ) R ( F ` y ) \/ ( ( F ` x ) = ( F ` y ) /\ x S y ) ) ) <-> ( ( x e. A /\ y e. A ) /\ ( `' `' G ` x ) Q ( `' `' G ` y ) ) ) )
89	88	opabbidv 4716	. . . . . . . 8 |- ( F : A --> B -> { <. x , y >. | ( ( x e. A /\ y e. A ) /\ ( ( F ` x ) R ( F ` y ) \/ ( ( F ` x ) = ( F ` y ) /\ x S y ) ) ) } = { <. x , y >. | ( ( x e. A /\ y e. A ) /\ ( `' `' G ` x ) Q ( `' `' G ` y ) ) } )
90	40, 89	syl5eq 2668	. . . . . . 7 |- ( F : A --> B -> T = { <. x , y >. | ( ( x e. A /\ y e. A ) /\ ( `' `' G ` x ) Q ( `' `' G ` y ) ) } )
91		isoeq3 6569	. . . . . . 7 |- ( T = { <. x , y >. | ( ( x e. A /\ y e. A ) /\ ( `' `' G ` x ) Q ( `' `' G ` y ) ) } -> ( `' G Isom Q , T ( ran G , A ) <-> `' G Isom Q , { <. x , y >. | ( ( x e. A /\ y e. A ) /\ ( `' `' G ` x ) Q ( `' `' G ` y ) ) } ( ran G , A ) ) )
92	90, 91	syl 17	. . . . . 6 |- ( F : A --> B -> ( `' G Isom Q , T ( ran G , A ) <-> `' G Isom Q , { <. x , y >. | ( ( x e. A /\ y e. A ) /\ ( `' `' G ` x ) Q ( `' `' G ` y ) ) } ( ran G , A ) ) )
93	39, 92	syl5ibr 236	. . . . 5 |- ( F : A --> B -> ( `' G : ran G -1-1-onto-> A -> `' G Isom Q , T ( ran G , A ) ) )
94	1, 37, 93	sylc 65	. . . 4 |- ( ph -> `' G Isom Q , T ( ran G , A ) )
95		isocnv 6580	. . . 4 |- ( `' G Isom Q , T ( ran G , A ) -> `' `' G Isom T , Q ( A , ran G ) )
96	94, 95	syl 17	. . 3 |- ( ph -> `' `' G Isom T , Q ( A , ran G ) )
97		imacnvcnv 5599	. . . . 5 |- ( `' `' G " w ) = ( G " w )
98		fnwe.5	. . . . . . 7 |- ( ph -> ( F " w ) e. _V )
99		vex 3203	. . . . . . 7 |- w e. _V
100		xpexg 6960	. . . . . . 7 |- ( ( ( F " w ) e. _V /\ w e. _V ) -> ( ( F " w ) X. w ) e. _V )
101	98, 99, 100	sylancl 694	. . . . . 6 |- ( ph -> ( ( F " w ) X. w ) e. _V )
102		imadmres 5627	. . . . . . 7 |- ( G " dom ( G |` w ) ) = ( G " w )
103		dmres 5419	. . . . . . . . . . 11 |- dom ( G |` w ) = ( w i^i dom G )
104	103	elin2 3801	. . . . . . . . . 10 |- ( x e. dom ( G |` w ) <-> ( x e. w /\ x e. dom G ) )
105		simprr 796	. . . . . . . . . . . . 13 |- ( ( ph /\ ( x e. w /\ x e. dom G ) ) -> x e. dom G )
106		f1dm 6105	. . . . . . . . . . . . . . 15 |- ( G : A -1-1-> ( B X. A ) -> dom G = A )
107	1, 34, 106	3syl 18	. . . . . . . . . . . . . 14 |- ( ph -> dom G = A )
108	107	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ ( x e. w /\ x e. dom G ) ) -> dom G = A )
109	105, 108	eleqtrd 2703	. . . . . . . . . . . 12 |- ( ( ph /\ ( x e. w /\ x e. dom G ) ) -> x e. A )
110	109, 20	syl 17	. . . . . . . . . . 11 |- ( ( ph /\ ( x e. w /\ x e. dom G ) ) -> ( G ` x ) = <. ( F ` x ) , x >. )
111	1	ffnd 6046	. . . . . . . . . . . . . . 15 |- ( ph -> F Fn A )
112	111	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( x e. w /\ x e. dom G ) ) -> F Fn A )
113		dmres 5419	. . . . . . . . . . . . . . 15 |- dom ( F |` w ) = ( w i^i dom F )
114		inss2 3834	. . . . . . . . . . . . . . . 16 |- ( w i^i dom F ) C_ dom F
115		fndm 5990	. . . . . . . . . . . . . . . . 17 |- ( F Fn A -> dom F = A )
116	112, 115	syl 17	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ ( x e. w /\ x e. dom G ) ) -> dom F = A )
117	114, 116	syl5sseq 3653	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( x e. w /\ x e. dom G ) ) -> ( w i^i dom F ) C_ A )
118	113, 117	syl5eqss 3649	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( x e. w /\ x e. dom G ) ) -> dom ( F |` w ) C_ A )
119		simprl 794	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( x e. w /\ x e. dom G ) ) -> x e. w )
120	109, 116	eleqtrrd 2704	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( x e. w /\ x e. dom G ) ) -> x e. dom F )
121	113	elin2 3801	. . . . . . . . . . . . . . 15 |- ( x e. dom ( F |` w ) <-> ( x e. w /\ x e. dom F ) )
122	119, 120, 121	sylanbrc 698	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( x e. w /\ x e. dom G ) ) -> x e. dom ( F |` w ) )
123		fnfvima 6496	. . . . . . . . . . . . . 14 |- ( ( F Fn A /\ dom ( F |` w ) C_ A /\ x e. dom ( F |` w ) ) -> ( F ` x ) e. ( F " dom ( F |` w ) ) )
124	112, 118, 122, 123	syl3anc 1326	. . . . . . . . . . . . 13 |- ( ( ph /\ ( x e. w /\ x e. dom G ) ) -> ( F ` x ) e. ( F " dom ( F |` w ) ) )
125		imadmres 5627	. . . . . . . . . . . . 13 |- ( F " dom ( F |` w ) ) = ( F " w )
126	124, 125	syl6eleq 2711	. . . . . . . . . . . 12 |- ( ( ph /\ ( x e. w /\ x e. dom G ) ) -> ( F ` x ) e. ( F " w ) )
127	126, 119	opelxpd 5149	. . . . . . . . . . 11 |- ( ( ph /\ ( x e. w /\ x e. dom G ) ) -> <. ( F ` x ) , x >. e. ( ( F " w ) X. w ) )
128	110, 127	eqeltrd 2701	. . . . . . . . . 10 |- ( ( ph /\ ( x e. w /\ x e. dom G ) ) -> ( G ` x ) e. ( ( F " w ) X. w ) )
129	104, 128	sylan2b 492	. . . . . . . . 9 |- ( ( ph /\ x e. dom ( G |` w ) ) -> ( G ` x ) e. ( ( F " w ) X. w ) )
130	129	ralrimiva 2966	. . . . . . . 8 |- ( ph -> A. x e. dom ( G |` w ) ( G ` x ) e. ( ( F " w ) X. w ) )
131		f1fun 6103	. . . . . . . . . 10 |- ( G : A -1-1-> ( B X. A ) -> Fun G )
132	1, 34, 131	3syl 18	. . . . . . . . 9 |- ( ph -> Fun G )
133		resss 5422	. . . . . . . . . 10 |- ( G |` w ) C_ G
134		dmss 5323	. . . . . . . . . 10 |- ( ( G |` w ) C_ G -> dom ( G |` w ) C_ dom G )
135	133, 134	ax-mp 5	. . . . . . . . 9 |- dom ( G |` w ) C_ dom G
136		funimass4 6247	. . . . . . . . 9 |- ( ( Fun G /\ dom ( G |` w ) C_ dom G ) -> ( ( G " dom ( G |` w ) ) C_ ( ( F " w ) X. w ) <-> A. x e. dom ( G |` w ) ( G ` x ) e. ( ( F " w ) X. w ) ) )
137	132, 135, 136	sylancl 694	. . . . . . . 8 |- ( ph -> ( ( G " dom ( G |` w ) ) C_ ( ( F " w ) X. w ) <-> A. x e. dom ( G |` w ) ( G ` x ) e. ( ( F " w ) X. w ) ) )
138	130, 137	mpbird 247	. . . . . . 7 |- ( ph -> ( G " dom ( G |` w ) ) C_ ( ( F " w ) X. w ) )
139	102, 138	syl5eqssr 3650	. . . . . 6 |- ( ph -> ( G " w ) C_ ( ( F " w ) X. w ) )
140	101, 139	ssexd 4805	. . . . 5 |- ( ph -> ( G " w ) e. _V )
141	97, 140	syl5eqel 2705	. . . 4 |- ( ph -> ( `' `' G " w ) e. _V )
142	141	alrimiv 1855	. . 3 |- ( ph -> A. w ( `' `' G " w ) e. _V )
143		isowe2 6600	. . 3 |- ( ( `' `' G Isom T , Q ( A , ran G ) /\ A. w ( `' `' G " w ) e. _V ) -> ( Q We ran G -> T We A ) )
144	96, 142, 143	syl2anc 693	. 2 |- ( ph -> ( Q We ran G -> T We A ) )
145	15, 144	mpd 15	1 |- ( ph -> T We A )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   A.wal 1481   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   i^i cin 3573   C_ wss 3574   <.cop 4183   class class class wbr 4653   {copab 4712   |-> cmpt 4729   We wwe 5072   X. cxp 5112   `'ccnv 5113   dom cdm 5114   ran crn 5115   |` cres 5116   "cima 5117   Rel wrel 5119   Fun wfun 5882   Fn wfn 5883   -->wf 5884   -1-1->wf1 5885   -1-1-onto->wf1o 5887   ` cfv 5888   Isom wiso 5889   1stc1st 7166   2ndc2nd 7167

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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-1st 7168  df-2nd 7169

This theorem is referenced by:  fnwe  7293