Theorem f1od2 6203

Description: Describe an implicit one-to-one onto function of two variables. (Contributed by Thierry Arnoux, 17-Aug-2017.)

Hypotheses

Ref	Expression
f1od2.1	|- F = ( x e. A , y e. B |-> C )
f1od2.2	|- ( ( ph /\ ( x e. A /\ y e. B ) ) -> C e. W )
f1od2.3	|- ( ( ph /\ z e. D ) -> ( I e. X /\ J e. Y ) )
f1od2.4	|- ( ph -> ( ( ( x e. A /\ y e. B ) /\ z = C ) <-> ( z e. D /\ ( x = I /\ y = J ) ) ) )

Assertion

Ref	Expression
f1od2	|- ( ph -> F : ( A X. B ) -1-1-onto-> D )

Distinct variable groups:   x, y, z, A   x, B, y, z   z, C   x, D, y, z   x, I, y   x, J, y   ph, x, y, z

Allowed substitution hints:   C( x, y)   F( x, y, z)   I( z)   J( z)   W( x, y, z)   X( x, y, z)   Y( x, y, z)

Proof of Theorem f1od2

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		f1od2.2	. . . 4 |- ( ( ph /\ ( x e. A /\ y e. B ) ) -> C e. W )
2	1	ralrimivva 2548	. . 3 |- ( ph -> A. x e. A A. y e. B C e. W )
3		f1od2.1	. . . 4 |- F = ( x e. A , y e. B |-> C )
4	3	fnmpo 6170	. . 3 |- ( A. x e. A A. y e. B C e. W -> F Fn ( A X. B ) )
5	2, 4	syl 14	. 2 |- ( ph -> F Fn ( A X. B ) )
6		f1od2.3	. . . . . 6 |- ( ( ph /\ z e. D ) -> ( I e. X /\ J e. Y ) )
7		opelxpi 4636	. . . . . 6 |- ( ( I e. X /\ J e. Y ) -> <. I , J >. e. ( X X. Y ) )
8	6, 7	syl 14	. . . . 5 |- ( ( ph /\ z e. D ) -> <. I , J >. e. ( X X. Y ) )
9	8	ralrimiva 2539	. . . 4 |- ( ph -> A. z e. D <. I , J >. e. ( X X. Y ) )
10		eqid 2165	. . . . 5 |- ( z e. D |-> <. I , J >. ) = ( z e. D |-> <. I , J >. )
11	10	fnmpt 5314	. . . 4 |- ( A. z e. D <. I , J >. e. ( X X. Y ) -> ( z e. D |-> <. I , J >. ) Fn D )
12	9, 11	syl 14	. . 3 |- ( ph -> ( z e. D |-> <. I , J >. ) Fn D )
13		elxp7 6138	. . . . . . . 8 |- ( a e. ( A X. B ) <-> ( a e. ( _V X. _V ) /\ ( ( 1st ` a ) e. A /\ ( 2nd ` a ) e. B ) ) )
14	13	anbi1i 454	. . . . . . 7 |- ( ( a e. ( A X. B ) /\ z = [_ ( 1st ` a ) / x ]_ [_ ( 2nd ` a ) / y ]_ C ) <-> ( ( a e. ( _V X. _V ) /\ ( ( 1st ` a ) e. A /\ ( 2nd ` a ) e. B ) ) /\ z = [_ ( 1st ` a ) / x ]_ [_ ( 2nd ` a ) / y ]_ C ) )
15		anass 399	. . . . . . . . 9 |- ( ( ( a e. ( _V X. _V ) /\ ( ( 1st ` a ) e. A /\ ( 2nd ` a ) e. B ) ) /\ z = [_ ( 1st ` a ) / x ]_ [_ ( 2nd ` a ) / y ]_ C ) <-> ( a e. ( _V X. _V ) /\ ( ( ( 1st ` a ) e. A /\ ( 2nd ` a ) e. B ) /\ z = [_ ( 1st ` a ) / x ]_ [_ ( 2nd ` a ) / y ]_ C ) ) )
16		f1od2.4	. . . . . . . . . . . . 13 |- ( ph -> ( ( ( x e. A /\ y e. B ) /\ z = C ) <-> ( z e. D /\ ( x = I /\ y = J ) ) ) )
17	16	sbcbidv 3009	. . . . . . . . . . . 12 |- ( ph -> ( [. ( 2nd ` a ) / y ]. ( ( x e. A /\ y e. B ) /\ z = C ) <-> [. ( 2nd ` a ) / y ]. ( z e. D /\ ( x = I /\ y = J ) ) ) )
18	17	sbcbidv 3009	. . . . . . . . . . 11 |- ( ph -> ( [. ( 1st ` a ) / x ]. [. ( 2nd ` a ) / y ]. ( ( x e. A /\ y e. B ) /\ z = C ) <-> [. ( 1st ` a ) / x ]. [. ( 2nd ` a ) / y ]. ( z e. D /\ ( x = I /\ y = J ) ) ) )
19		sbcan 2993	. . . . . . . . . . . . . 14 |- ( [. ( 2nd ` a ) / y ]. ( ( x e. A /\ y e. B ) /\ z = C ) <-> ( [. ( 2nd ` a ) / y ]. ( x e. A /\ y e. B ) /\ [. ( 2nd ` a ) / y ]. z = C ) )
20		sbcan 2993	. . . . . . . . . . . . . . . 16 |- ( [. ( 2nd ` a ) / y ]. ( x e. A /\ y e. B ) <-> ( [. ( 2nd ` a ) / y ]. x e. A /\ [. ( 2nd ` a ) / y ]. y e. B ) )
21		vex 2729	. . . . . . . . . . . . . . . . . . 19 |- a e. _V
22		2ndexg 6136	. . . . . . . . . . . . . . . . . . 19 |- ( a e. _V -> ( 2nd ` a ) e. _V )
23	21, 22	ax-mp 5	. . . . . . . . . . . . . . . . . 18 |- ( 2nd ` a ) e. _V
24		sbcg 3020	. . . . . . . . . . . . . . . . . 18 |- ( ( 2nd ` a ) e. _V -> ( [. ( 2nd ` a ) / y ]. x e. A <-> x e. A ) )
25	23, 24	ax-mp 5	. . . . . . . . . . . . . . . . 17 |- ( [. ( 2nd ` a ) / y ]. x e. A <-> x e. A )
26		sbcel1v 3013	. . . . . . . . . . . . . . . . 17 |- ( [. ( 2nd ` a ) / y ]. y e. B <-> ( 2nd ` a ) e. B )
27	25, 26	anbi12i 456	. . . . . . . . . . . . . . . 16 |- ( ( [. ( 2nd ` a ) / y ]. x e. A /\ [. ( 2nd ` a ) / y ]. y e. B ) <-> ( x e. A /\ ( 2nd ` a ) e. B ) )
28	20, 27	bitri 183	. . . . . . . . . . . . . . 15 |- ( [. ( 2nd ` a ) / y ]. ( x e. A /\ y e. B ) <-> ( x e. A /\ ( 2nd ` a ) e. B ) )
29		sbceq2g 3067	. . . . . . . . . . . . . . . 16 |- ( ( 2nd ` a ) e. _V -> ( [. ( 2nd ` a ) / y ]. z = C <-> z = [_ ( 2nd ` a ) / y ]_ C ) )
30	23, 29	ax-mp 5	. . . . . . . . . . . . . . 15 |- ( [. ( 2nd ` a ) / y ]. z = C <-> z = [_ ( 2nd ` a ) / y ]_ C )
31	28, 30	anbi12i 456	. . . . . . . . . . . . . 14 |- ( ( [. ( 2nd ` a ) / y ]. ( x e. A /\ y e. B ) /\ [. ( 2nd ` a ) / y ]. z = C ) <-> ( ( x e. A /\ ( 2nd ` a ) e. B ) /\ z = [_ ( 2nd ` a ) / y ]_ C ) )
32	19, 31	bitri 183	. . . . . . . . . . . . 13 |- ( [. ( 2nd ` a ) / y ]. ( ( x e. A /\ y e. B ) /\ z = C ) <-> ( ( x e. A /\ ( 2nd ` a ) e. B ) /\ z = [_ ( 2nd ` a ) / y ]_ C ) )
33	32	sbcbii 3010	. . . . . . . . . . . 12 |- ( [. ( 1st ` a ) / x ]. [. ( 2nd ` a ) / y ]. ( ( x e. A /\ y e. B ) /\ z = C ) <-> [. ( 1st ` a ) / x ]. ( ( x e. A /\ ( 2nd ` a ) e. B ) /\ z = [_ ( 2nd ` a ) / y ]_ C ) )
34		sbcan 2993	. . . . . . . . . . . 12 |- ( [. ( 1st ` a ) / x ]. ( ( x e. A /\ ( 2nd ` a ) e. B ) /\ z = [_ ( 2nd ` a ) / y ]_ C ) <-> ( [. ( 1st ` a ) / x ]. ( x e. A /\ ( 2nd ` a ) e. B ) /\ [. ( 1st ` a ) / x ]. z = [_ ( 2nd ` a ) / y ]_ C ) )
35		sbcan 2993	. . . . . . . . . . . . . 14 |- ( [. ( 1st ` a ) / x ]. ( x e. A /\ ( 2nd ` a ) e. B ) <-> ( [. ( 1st ` a ) / x ]. x e. A /\ [. ( 1st ` a ) / x ]. ( 2nd ` a ) e. B ) )
36		sbcel1v 3013	. . . . . . . . . . . . . . 15 |- ( [. ( 1st ` a ) / x ]. x e. A <-> ( 1st ` a ) e. A )
37		1stexg 6135	. . . . . . . . . . . . . . . . 17 |- ( a e. _V -> ( 1st ` a ) e. _V )
38	21, 37	ax-mp 5	. . . . . . . . . . . . . . . 16 |- ( 1st ` a ) e. _V
39		sbcg 3020	. . . . . . . . . . . . . . . 16 |- ( ( 1st ` a ) e. _V -> ( [. ( 1st ` a ) / x ]. ( 2nd ` a ) e. B <-> ( 2nd ` a ) e. B ) )
40	38, 39	ax-mp 5	. . . . . . . . . . . . . . 15 |- ( [. ( 1st ` a ) / x ]. ( 2nd ` a ) e. B <-> ( 2nd ` a ) e. B )
41	36, 40	anbi12i 456	. . . . . . . . . . . . . 14 |- ( ( [. ( 1st ` a ) / x ]. x e. A /\ [. ( 1st ` a ) / x ]. ( 2nd ` a ) e. B ) <-> ( ( 1st ` a ) e. A /\ ( 2nd ` a ) e. B ) )
42	35, 41	bitri 183	. . . . . . . . . . . . 13 |- ( [. ( 1st ` a ) / x ]. ( x e. A /\ ( 2nd ` a ) e. B ) <-> ( ( 1st ` a ) e. A /\ ( 2nd ` a ) e. B ) )
43		sbceq2g 3067	. . . . . . . . . . . . . 14 |- ( ( 1st ` a ) e. _V -> ( [. ( 1st ` a ) / x ]. z = [_ ( 2nd ` a ) / y ]_ C <-> z = [_ ( 1st ` a ) / x ]_ [_ ( 2nd ` a ) / y ]_ C ) )
44	38, 43	ax-mp 5	. . . . . . . . . . . . 13 |- ( [. ( 1st ` a ) / x ]. z = [_ ( 2nd ` a ) / y ]_ C <-> z = [_ ( 1st ` a ) / x ]_ [_ ( 2nd ` a ) / y ]_ C )
45	42, 44	anbi12i 456	. . . . . . . . . . . 12 |- ( ( [. ( 1st ` a ) / x ]. ( x e. A /\ ( 2nd ` a ) e. B ) /\ [. ( 1st ` a ) / x ]. z = [_ ( 2nd ` a ) / y ]_ C ) <-> ( ( ( 1st ` a ) e. A /\ ( 2nd ` a ) e. B ) /\ z = [_ ( 1st ` a ) / x ]_ [_ ( 2nd ` a ) / y ]_ C ) )
46	33, 34, 45	3bitri 205	. . . . . . . . . . 11 |- ( [. ( 1st ` a ) / x ]. [. ( 2nd ` a ) / y ]. ( ( x e. A /\ y e. B ) /\ z = C ) <-> ( ( ( 1st ` a ) e. A /\ ( 2nd ` a ) e. B ) /\ z = [_ ( 1st ` a ) / x ]_ [_ ( 2nd ` a ) / y ]_ C ) )
47		sbcan 2993	. . . . . . . . . . . . . 14 |- ( [. ( 2nd ` a ) / y ]. ( z e. D /\ ( x = I /\ y = J ) ) <-> ( [. ( 2nd ` a ) / y ]. z e. D /\ [. ( 2nd ` a ) / y ]. ( x = I /\ y = J ) ) )
48		sbcg 3020	. . . . . . . . . . . . . . . 16 |- ( ( 2nd ` a ) e. _V -> ( [. ( 2nd ` a ) / y ]. z e. D <-> z e. D ) )
49	23, 48	ax-mp 5	. . . . . . . . . . . . . . 15 |- ( [. ( 2nd ` a ) / y ]. z e. D <-> z e. D )
50		sbcan 2993	. . . . . . . . . . . . . . . 16 |- ( [. ( 2nd ` a ) / y ]. ( x = I /\ y = J ) <-> ( [. ( 2nd ` a ) / y ]. x = I /\ [. ( 2nd ` a ) / y ]. y = J ) )
51		sbcg 3020	. . . . . . . . . . . . . . . . . 18 |- ( ( 2nd ` a ) e. _V -> ( [. ( 2nd ` a ) / y ]. x = I <-> x = I ) )
52	23, 51	ax-mp 5	. . . . . . . . . . . . . . . . 17 |- ( [. ( 2nd ` a ) / y ]. x = I <-> x = I )
53		sbceq1g 3065	. . . . . . . . . . . . . . . . . . 19 |- ( ( 2nd ` a ) e. _V -> ( [. ( 2nd ` a ) / y ]. y = J <-> [_ ( 2nd ` a ) / y ]_ y = J ) )
54	23, 53	ax-mp 5	. . . . . . . . . . . . . . . . . 18 |- ( [. ( 2nd ` a ) / y ]. y = J <-> [_ ( 2nd ` a ) / y ]_ y = J )
55		csbvarg 3073	. . . . . . . . . . . . . . . . . . . 20 |- ( ( 2nd ` a ) e. _V -> [_ ( 2nd ` a ) / y ]_ y = ( 2nd ` a ) )
56	23, 55	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 |- [_ ( 2nd ` a ) / y ]_ y = ( 2nd ` a )
57	56	eqeq1i 2173	. . . . . . . . . . . . . . . . . 18 |- ( [_ ( 2nd ` a ) / y ]_ y = J <-> ( 2nd ` a ) = J )
58	54, 57	bitri 183	. . . . . . . . . . . . . . . . 17 |- ( [. ( 2nd ` a ) / y ]. y = J <-> ( 2nd ` a ) = J )
59	52, 58	anbi12i 456	. . . . . . . . . . . . . . . 16 |- ( ( [. ( 2nd ` a ) / y ]. x = I /\ [. ( 2nd ` a ) / y ]. y = J ) <-> ( x = I /\ ( 2nd ` a ) = J ) )
60	50, 59	bitri 183	. . . . . . . . . . . . . . 15 |- ( [. ( 2nd ` a ) / y ]. ( x = I /\ y = J ) <-> ( x = I /\ ( 2nd ` a ) = J ) )
61	49, 60	anbi12i 456	. . . . . . . . . . . . . 14 |- ( ( [. ( 2nd ` a ) / y ]. z e. D /\ [. ( 2nd ` a ) / y ]. ( x = I /\ y = J ) ) <-> ( z e. D /\ ( x = I /\ ( 2nd ` a ) = J ) ) )
62	47, 61	bitri 183	. . . . . . . . . . . . 13 |- ( [. ( 2nd ` a ) / y ]. ( z e. D /\ ( x = I /\ y = J ) ) <-> ( z e. D /\ ( x = I /\ ( 2nd ` a ) = J ) ) )
63	62	sbcbii 3010	. . . . . . . . . . . 12 |- ( [. ( 1st ` a ) / x ]. [. ( 2nd ` a ) / y ]. ( z e. D /\ ( x = I /\ y = J ) ) <-> [. ( 1st ` a ) / x ]. ( z e. D /\ ( x = I /\ ( 2nd ` a ) = J ) ) )
64		sbcan 2993	. . . . . . . . . . . 12 |- ( [. ( 1st ` a ) / x ]. ( z e. D /\ ( x = I /\ ( 2nd ` a ) = J ) ) <-> ( [. ( 1st ` a ) / x ]. z e. D /\ [. ( 1st ` a ) / x ]. ( x = I /\ ( 2nd ` a ) = J ) ) )
65		sbcg 3020	. . . . . . . . . . . . . 14 |- ( ( 1st ` a ) e. _V -> ( [. ( 1st ` a ) / x ]. z e. D <-> z e. D ) )
66	38, 65	ax-mp 5	. . . . . . . . . . . . 13 |- ( [. ( 1st ` a ) / x ]. z e. D <-> z e. D )
67		sbcan 2993	. . . . . . . . . . . . . 14 |- ( [. ( 1st ` a ) / x ]. ( x = I /\ ( 2nd ` a ) = J ) <-> ( [. ( 1st ` a ) / x ]. x = I /\ [. ( 1st ` a ) / x ]. ( 2nd ` a ) = J ) )
68		sbceq1g 3065	. . . . . . . . . . . . . . . . 17 |- ( ( 1st ` a ) e. _V -> ( [. ( 1st ` a ) / x ]. x = I <-> [_ ( 1st ` a ) / x ]_ x = I ) )
69	38, 68	ax-mp 5	. . . . . . . . . . . . . . . 16 |- ( [. ( 1st ` a ) / x ]. x = I <-> [_ ( 1st ` a ) / x ]_ x = I )
70		csbvarg 3073	. . . . . . . . . . . . . . . . . 18 |- ( ( 1st ` a ) e. _V -> [_ ( 1st ` a ) / x ]_ x = ( 1st ` a ) )
71	38, 70	ax-mp 5	. . . . . . . . . . . . . . . . 17 |- [_ ( 1st ` a ) / x ]_ x = ( 1st ` a )
72	71	eqeq1i 2173	. . . . . . . . . . . . . . . 16 |- ( [_ ( 1st ` a ) / x ]_ x = I <-> ( 1st ` a ) = I )
73	69, 72	bitri 183	. . . . . . . . . . . . . . 15 |- ( [. ( 1st ` a ) / x ]. x = I <-> ( 1st ` a ) = I )
74		sbcg 3020	. . . . . . . . . . . . . . . 16 |- ( ( 1st ` a ) e. _V -> ( [. ( 1st ` a ) / x ]. ( 2nd ` a ) = J <-> ( 2nd ` a ) = J ) )
75	38, 74	ax-mp 5	. . . . . . . . . . . . . . 15 |- ( [. ( 1st ` a ) / x ]. ( 2nd ` a ) = J <-> ( 2nd ` a ) = J )
76	73, 75	anbi12i 456	. . . . . . . . . . . . . 14 |- ( ( [. ( 1st ` a ) / x ]. x = I /\ [. ( 1st ` a ) / x ]. ( 2nd ` a ) = J ) <-> ( ( 1st ` a ) = I /\ ( 2nd ` a ) = J ) )
77	67, 76	bitri 183	. . . . . . . . . . . . 13 |- ( [. ( 1st ` a ) / x ]. ( x = I /\ ( 2nd ` a ) = J ) <-> ( ( 1st ` a ) = I /\ ( 2nd ` a ) = J ) )
78	66, 77	anbi12i 456	. . . . . . . . . . . 12 |- ( ( [. ( 1st ` a ) / x ]. z e. D /\ [. ( 1st ` a ) / x ]. ( x = I /\ ( 2nd ` a ) = J ) ) <-> ( z e. D /\ ( ( 1st ` a ) = I /\ ( 2nd ` a ) = J ) ) )
79	63, 64, 78	3bitri 205	. . . . . . . . . . 11 |- ( [. ( 1st ` a ) / x ]. [. ( 2nd ` a ) / y ]. ( z e. D /\ ( x = I /\ y = J ) ) <-> ( z e. D /\ ( ( 1st ` a ) = I /\ ( 2nd ` a ) = J ) ) )
80	18, 46, 79	3bitr3g 221	. . . . . . . . . 10 |- ( ph -> ( ( ( ( 1st ` a ) e. A /\ ( 2nd ` a ) e. B ) /\ z = [_ ( 1st ` a ) / x ]_ [_ ( 2nd ` a ) / y ]_ C ) <-> ( z e. D /\ ( ( 1st ` a ) = I /\ ( 2nd ` a ) = J ) ) ) )
81	80	anbi2d 460	. . . . . . . . 9 |- ( ph -> ( ( a e. ( _V X. _V ) /\ ( ( ( 1st ` a ) e. A /\ ( 2nd ` a ) e. B ) /\ z = [_ ( 1st ` a ) / x ]_ [_ ( 2nd ` a ) / y ]_ C ) ) <-> ( a e. ( _V X. _V ) /\ ( z e. D /\ ( ( 1st ` a ) = I /\ ( 2nd ` a ) = J ) ) ) ) )
82	15, 81	syl5bb 191	. . . . . . . 8 |- ( ph -> ( ( ( a e. ( _V X. _V ) /\ ( ( 1st ` a ) e. A /\ ( 2nd ` a ) e. B ) ) /\ z = [_ ( 1st ` a ) / x ]_ [_ ( 2nd ` a ) / y ]_ C ) <-> ( a e. ( _V X. _V ) /\ ( z e. D /\ ( ( 1st ` a ) = I /\ ( 2nd ` a ) = J ) ) ) ) )
83		xpss 4712	. . . . . . . . . . . 12 |- ( X X. Y ) C_ ( _V X. _V )
84		simprr 522	. . . . . . . . . . . . 13 |- ( ( ph /\ ( z e. D /\ a = <. I , J >. ) ) -> a = <. I , J >. )
85	8	adantrr 471	. . . . . . . . . . . . 13 |- ( ( ph /\ ( z e. D /\ a = <. I , J >. ) ) -> <. I , J >. e. ( X X. Y ) )
86	84, 85	eqeltrd 2243	. . . . . . . . . . . 12 |- ( ( ph /\ ( z e. D /\ a = <. I , J >. ) ) -> a e. ( X X. Y ) )
87	83, 86	sselid 3140	. . . . . . . . . . 11 |- ( ( ph /\ ( z e. D /\ a = <. I , J >. ) ) -> a e. ( _V X. _V ) )
88	87	ex 114	. . . . . . . . . 10 |- ( ph -> ( ( z e. D /\ a = <. I , J >. ) -> a e. ( _V X. _V ) ) )
89	88	pm4.71rd 392	. . . . . . . . 9 |- ( ph -> ( ( z e. D /\ a = <. I , J >. ) <-> ( a e. ( _V X. _V ) /\ ( z e. D /\ a = <. I , J >. ) ) ) )
90		eqop 6145	. . . . . . . . . . 11 |- ( a e. ( _V X. _V ) -> ( a = <. I , J >. <-> ( ( 1st ` a ) = I /\ ( 2nd ` a ) = J ) ) )
91	90	anbi2d 460	. . . . . . . . . 10 |- ( a e. ( _V X. _V ) -> ( ( z e. D /\ a = <. I , J >. ) <-> ( z e. D /\ ( ( 1st ` a ) = I /\ ( 2nd ` a ) = J ) ) ) )
92	91	pm5.32i 450	. . . . . . . . 9 |- ( ( a e. ( _V X. _V ) /\ ( z e. D /\ a = <. I , J >. ) ) <-> ( a e. ( _V X. _V ) /\ ( z e. D /\ ( ( 1st ` a ) = I /\ ( 2nd ` a ) = J ) ) ) )
93	89, 92	bitr2di 196	. . . . . . . 8 |- ( ph -> ( ( a e. ( _V X. _V ) /\ ( z e. D /\ ( ( 1st ` a ) = I /\ ( 2nd ` a ) = J ) ) ) <-> ( z e. D /\ a = <. I , J >. ) ) )
94	82, 93	bitrd 187	. . . . . . 7 |- ( ph -> ( ( ( a e. ( _V X. _V ) /\ ( ( 1st ` a ) e. A /\ ( 2nd ` a ) e. B ) ) /\ z = [_ ( 1st ` a ) / x ]_ [_ ( 2nd ` a ) / y ]_ C ) <-> ( z e. D /\ a = <. I , J >. ) ) )
95	14, 94	syl5bb 191	. . . . . 6 |- ( ph -> ( ( a e. ( A X. B ) /\ z = [_ ( 1st ` a ) / x ]_ [_ ( 2nd ` a ) / y ]_ C ) <-> ( z e. D /\ a = <. I , J >. ) ) )
96	95	opabbidv 4048	. . . . 5 |- ( ph -> { <. z , a >. | ( a e. ( A X. B ) /\ z = [_ ( 1st ` a ) / x ]_ [_ ( 2nd ` a ) / y ]_ C ) } = { <. z , a >. | ( z e. D /\ a = <. I , J >. ) } )
97		df-mpo 5847	. . . . . . . 8 |- ( x e. A , y e. B |-> C ) = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = C ) }
98	3, 97	eqtri 2186	. . . . . . 7 |- F = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = C ) }
99	98	cnveqi 4779	. . . . . 6 |- `' F = `' { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = C ) }
100		nfv 1516	. . . . . . . 8 |- F/ x a e. ( A X. B )
101		nfcsb1v 3078	. . . . . . . . 9 |- F/_ x [_ ( 1st ` a ) / x ]_ [_ ( 2nd ` a ) / y ]_ C
102	101	nfeq2 2320	. . . . . . . 8 |- F/ x z = [_ ( 1st ` a ) / x ]_ [_ ( 2nd ` a ) / y ]_ C
103	100, 102	nfan 1553	. . . . . . 7 |- F/ x ( a e. ( A X. B ) /\ z = [_ ( 1st ` a ) / x ]_ [_ ( 2nd ` a ) / y ]_ C )
104		nfv 1516	. . . . . . . 8 |- F/ y a e. ( A X. B )
105		nfcv 2308	. . . . . . . . . 10 |- F/_ y ( 1st ` a )
106		nfcsb1v 3078	. . . . . . . . . 10 |- F/_ y [_ ( 2nd ` a ) / y ]_ C
107	105, 106	nfcsb 3082	. . . . . . . . 9 |- F/_ y [_ ( 1st ` a ) / x ]_ [_ ( 2nd ` a ) / y ]_ C
108	107	nfeq2 2320	. . . . . . . 8 |- F/ y z = [_ ( 1st ` a ) / x ]_ [_ ( 2nd ` a ) / y ]_ C
109	104, 108	nfan 1553	. . . . . . 7 |- F/ y ( a e. ( A X. B ) /\ z = [_ ( 1st ` a ) / x ]_ [_ ( 2nd ` a ) / y ]_ C )
110		eleq1 2229	. . . . . . . . 9 |- ( a = <. x , y >. -> ( a e. ( A X. B ) <-> <. x , y >. e. ( A X. B ) ) )
111		opelxp 4634	. . . . . . . . 9 |- ( <. x , y >. e. ( A X. B ) <-> ( x e. A /\ y e. B ) )
112	110, 111	bitrdi 195	. . . . . . . 8 |- ( a = <. x , y >. -> ( a e. ( A X. B ) <-> ( x e. A /\ y e. B ) ) )
113		csbopeq1a 6156	. . . . . . . . 9 |- ( a = <. x , y >. -> [_ ( 1st ` a ) / x ]_ [_ ( 2nd ` a ) / y ]_ C = C )
114	113	eqeq2d 2177	. . . . . . . 8 |- ( a = <. x , y >. -> ( z = [_ ( 1st ` a ) / x ]_ [_ ( 2nd ` a ) / y ]_ C <-> z = C ) )
115	112, 114	anbi12d 465	. . . . . . 7 |- ( a = <. x , y >. -> ( ( a e. ( A X. B ) /\ z = [_ ( 1st ` a ) / x ]_ [_ ( 2nd ` a ) / y ]_ C ) <-> ( ( x e. A /\ y e. B ) /\ z = C ) ) )
116		xpss 4712	. . . . . . . . 9 |- ( A X. B ) C_ ( _V X. _V )
117	116	sseli 3138	. . . . . . . 8 |- ( a e. ( A X. B ) -> a e. ( _V X. _V ) )
118	117	adantr 274	. . . . . . 7 |- ( ( a e. ( A X. B ) /\ z = [_ ( 1st ` a ) / x ]_ [_ ( 2nd ` a ) / y ]_ C ) -> a e. ( _V X. _V ) )
119	103, 109, 115, 118	cnvoprab 6202	. . . . . 6 |- `' { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = C ) } = { <. z , a >. | ( a e. ( A X. B ) /\ z = [_ ( 1st ` a ) / x ]_ [_ ( 2nd ` a ) / y ]_ C ) }
120	99, 119	eqtri 2186	. . . . 5 |- `' F = { <. z , a >. | ( a e. ( A X. B ) /\ z = [_ ( 1st ` a ) / x ]_ [_ ( 2nd ` a ) / y ]_ C ) }
121		df-mpt 4045	. . . . 5 |- ( z e. D |-> <. I , J >. ) = { <. z , a >. | ( z e. D /\ a = <. I , J >. ) }
122	96, 120, 121	3eqtr4g 2224	. . . 4 |- ( ph -> `' F = ( z e. D |-> <. I , J >. ) )
123	122	fneq1d 5278	. . 3 |- ( ph -> ( `' F Fn D <-> ( z e. D |-> <. I , J >. ) Fn D ) )
124	12, 123	mpbird 166	. 2 |- ( ph -> `' F Fn D )
125		dff1o4 5440	. 2 |- ( F : ( A X. B ) -1-1-onto-> D <-> ( F Fn ( A X. B ) /\ `' F Fn D ) )
126	5, 124, 125	sylanbrc 414	1 |- ( ph -> F : ( A X. B ) -1-1-onto-> D )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   e. wcel 2136   A.wral 2444   _Vcvv 2726   [.wsbc 2951   [_csb 3045   <.cop 3579   {copab 4042   |-> cmpt 4043   X. cxp 4602   `'ccnv 4603   Fn wfn 5183   -1-1-onto->wf1o 5187   ` cfv 5188   {coprab 5843   e. cmpo 5844   1stc1st 6106   2ndc2nd 6107

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109

This theorem is referenced by:  oddpwdc  12106