Theorem f1od2 6230

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 2559	. . 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 6197	. . 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 4655	. . . . . 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 2550	. . . 4 |- ( ph -> A. z e. D <. I , J >. e. ( X X. Y ) )
10		eqid 2177	. . . . 5 |- ( z e. D |-> <. I , J >. ) = ( z e. D |-> <. I , J >. )
11	10	fnmpt 5338	. . . 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 6165	. . . . . . . 8 |- ( a e. ( A X. B ) <-> ( a e. ( _V X. _V ) /\ ( ( 1st ` a ) e. A /\ ( 2nd ` a ) e. B ) ) )
14	13	anbi1i 458	. . . . . . 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 401	. . . . . . . . 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 3021	. . . . . . . . . . . 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 3021	. . . . . . . . . . 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 3005	. . . . . . . . . . . . . 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 3005	. . . . . . . . . . . . . . . 16 |- ( [. ( 2nd ` a ) / y ]. ( x e. A /\ y e. B ) <-> ( [. ( 2nd ` a ) / y ]. x e. A /\ [. ( 2nd ` a ) / y ]. y e. B ) )
21		vex 2740	. . . . . . . . . . . . . . . . . . 19 |- a e. _V
22		2ndexg 6163	. . . . . . . . . . . . . . . . . . 19 |- ( a e. _V -> ( 2nd ` a ) e. _V )
23	21, 22	ax-mp 5	. . . . . . . . . . . . . . . . . 18 |- ( 2nd ` a ) e. _V
24		sbcg 3032	. . . . . . . . . . . . . . . . . 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 3025	. . . . . . . . . . . . . . . . 17 |- ( [. ( 2nd ` a ) / y ]. y e. B <-> ( 2nd ` a ) e. B )
27	25, 26	anbi12i 460	. . . . . . . . . . . . . . . 16 |- ( ( [. ( 2nd ` a ) / y ]. x e. A /\ [. ( 2nd ` a ) / y ]. y e. B ) <-> ( x e. A /\ ( 2nd ` a ) e. B ) )
28	20, 27	bitri 184	. . . . . . . . . . . . . . 15 |- ( [. ( 2nd ` a ) / y ]. ( x e. A /\ y e. B ) <-> ( x e. A /\ ( 2nd ` a ) e. B ) )
29		sbceq2g 3079	. . . . . . . . . . . . . . . 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 460	. . . . . . . . . . . . . 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 184	. . . . . . . . . . . . 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 3022	. . . . . . . . . . . 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 3005	. . . . . . . . . . . 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 3005	. . . . . . . . . . . . . 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 3025	. . . . . . . . . . . . . . 15 |- ( [. ( 1st ` a ) / x ]. x e. A <-> ( 1st ` a ) e. A )
37		1stexg 6162	. . . . . . . . . . . . . . . . 17 |- ( a e. _V -> ( 1st ` a ) e. _V )
38	21, 37	ax-mp 5	. . . . . . . . . . . . . . . 16 |- ( 1st ` a ) e. _V
39		sbcg 3032	. . . . . . . . . . . . . . . 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 460	. . . . . . . . . . . . . 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 184	. . . . . . . . . . . . 13 |- ( [. ( 1st ` a ) / x ]. ( x e. A /\ ( 2nd ` a ) e. B ) <-> ( ( 1st ` a ) e. A /\ ( 2nd ` a ) e. B ) )
43		sbceq2g 3079	. . . . . . . . . . . . . 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 460	. . . . . . . . . . . 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 206	. . . . . . . . . . 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 3005	. . . . . . . . . . . . . 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 3032	. . . . . . . . . . . . . . . 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 3005	. . . . . . . . . . . . . . . 16 |- ( [. ( 2nd ` a ) / y ]. ( x = I /\ y = J ) <-> ( [. ( 2nd ` a ) / y ]. x = I /\ [. ( 2nd ` a ) / y ]. y = J ) )
51		sbcg 3032	. . . . . . . . . . . . . . . . . 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 3077	. . . . . . . . . . . . . . . . . . 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 3085	. . . . . . . . . . . . . . . . . . . 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 2185	. . . . . . . . . . . . . . . . . 18 |- ( [_ ( 2nd ` a ) / y ]_ y = J <-> ( 2nd ` a ) = J )
58	54, 57	bitri 184	. . . . . . . . . . . . . . . . 17 |- ( [. ( 2nd ` a ) / y ]. y = J <-> ( 2nd ` a ) = J )
59	52, 58	anbi12i 460	. . . . . . . . . . . . . . . 16 |- ( ( [. ( 2nd ` a ) / y ]. x = I /\ [. ( 2nd ` a ) / y ]. y = J ) <-> ( x = I /\ ( 2nd ` a ) = J ) )
60	50, 59	bitri 184	. . . . . . . . . . . . . . 15 |- ( [. ( 2nd ` a ) / y ]. ( x = I /\ y = J ) <-> ( x = I /\ ( 2nd ` a ) = J ) )
61	49, 60	anbi12i 460	. . . . . . . . . . . . . 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 184	. . . . . . . . . . . . 13 |- ( [. ( 2nd ` a ) / y ]. ( z e. D /\ ( x = I /\ y = J ) ) <-> ( z e. D /\ ( x = I /\ ( 2nd ` a ) = J ) ) )
63	62	sbcbii 3022	. . . . . . . . . . . 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 3005	. . . . . . . . . . . 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 3032	. . . . . . . . . . . . . 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 3005	. . . . . . . . . . . . . 14 |- ( [. ( 1st ` a ) / x ]. ( x = I /\ ( 2nd ` a ) = J ) <-> ( [. ( 1st ` a ) / x ]. x = I /\ [. ( 1st ` a ) / x ]. ( 2nd ` a ) = J ) )
68		sbceq1g 3077	. . . . . . . . . . . . . . . . 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 3085	. . . . . . . . . . . . . . . . . 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 2185	. . . . . . . . . . . . . . . 16 |- ( [_ ( 1st ` a ) / x ]_ x = I <-> ( 1st ` a ) = I )
73	69, 72	bitri 184	. . . . . . . . . . . . . . 15 |- ( [. ( 1st ` a ) / x ]. x = I <-> ( 1st ` a ) = I )
74		sbcg 3032	. . . . . . . . . . . . . . . 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 460	. . . . . . . . . . . . . 14 |- ( ( [. ( 1st ` a ) / x ]. x = I /\ [. ( 1st ` a ) / x ]. ( 2nd ` a ) = J ) <-> ( ( 1st ` a ) = I /\ ( 2nd ` a ) = J ) )
77	67, 76	bitri 184	. . . . . . . . . . . . 13 |- ( [. ( 1st ` a ) / x ]. ( x = I /\ ( 2nd ` a ) = J ) <-> ( ( 1st ` a ) = I /\ ( 2nd ` a ) = J ) )
78	66, 77	anbi12i 460	. . . . . . . . . . . 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 206	. . . . . . . . . . 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 222	. . . . . . . . . 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 464	. . . . . . . . 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	bitrid 192	. . . . . . . 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 4731	. . . . . . . . . . . 12 |- ( X X. Y ) C_ ( _V X. _V )
84		simprr 531	. . . . . . . . . . . . 13 |- ( ( ph /\ ( z e. D /\ a = <. I , J >. ) ) -> a = <. I , J >. )
85	8	adantrr 479	. . . . . . . . . . . . 13 |- ( ( ph /\ ( z e. D /\ a = <. I , J >. ) ) -> <. I , J >. e. ( X X. Y ) )
86	84, 85	eqeltrd 2254	. . . . . . . . . . . 12 |- ( ( ph /\ ( z e. D /\ a = <. I , J >. ) ) -> a e. ( X X. Y ) )
87	83, 86	sselid 3153	. . . . . . . . . . 11 |- ( ( ph /\ ( z e. D /\ a = <. I , J >. ) ) -> a e. ( _V X. _V ) )
88	87	ex 115	. . . . . . . . . 10 |- ( ph -> ( ( z e. D /\ a = <. I , J >. ) -> a e. ( _V X. _V ) ) )
89	88	pm4.71rd 394	. . . . . . . . 9 |- ( ph -> ( ( z e. D /\ a = <. I , J >. ) <-> ( a e. ( _V X. _V ) /\ ( z e. D /\ a = <. I , J >. ) ) ) )
90		eqop 6172	. . . . . . . . . . 11 |- ( a e. ( _V X. _V ) -> ( a = <. I , J >. <-> ( ( 1st ` a ) = I /\ ( 2nd ` a ) = J ) ) )
91	90	anbi2d 464	. . . . . . . . . 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 454	. . . . . . . . 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 197	. . . . . . . 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 188	. . . . . . 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	bitrid 192	. . . . . 6 |- ( ph -> ( ( a e. ( A X. B ) /\ z = [_ ( 1st ` a ) / x ]_ [_ ( 2nd ` a ) / y ]_ C ) <-> ( z e. D /\ a = <. I , J >. ) ) )
96	95	opabbidv 4066	. . . . 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 5874	. . . . . . . 8 |- ( x e. A , y e. B |-> C ) = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = C ) }
98	3, 97	eqtri 2198	. . . . . . 7 |- F = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = C ) }
99	98	cnveqi 4798	. . . . . 6 |- `' F = `' { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = C ) }
100		nfv 1528	. . . . . . . 8 |- F/ x a e. ( A X. B )
101		nfcsb1v 3090	. . . . . . . . 9 |- F/_ x [_ ( 1st ` a ) / x ]_ [_ ( 2nd ` a ) / y ]_ C
102	101	nfeq2 2331	. . . . . . . 8 |- F/ x z = [_ ( 1st ` a ) / x ]_ [_ ( 2nd ` a ) / y ]_ C
103	100, 102	nfan 1565	. . . . . . 7 |- F/ x ( a e. ( A X. B ) /\ z = [_ ( 1st ` a ) / x ]_ [_ ( 2nd ` a ) / y ]_ C )
104		nfv 1528	. . . . . . . 8 |- F/ y a e. ( A X. B )
105		nfcv 2319	. . . . . . . . . 10 |- F/_ y ( 1st ` a )
106		nfcsb1v 3090	. . . . . . . . . 10 |- F/_ y [_ ( 2nd ` a ) / y ]_ C
107	105, 106	nfcsb 3094	. . . . . . . . 9 |- F/_ y [_ ( 1st ` a ) / x ]_ [_ ( 2nd ` a ) / y ]_ C
108	107	nfeq2 2331	. . . . . . . 8 |- F/ y z = [_ ( 1st ` a ) / x ]_ [_ ( 2nd ` a ) / y ]_ C
109	104, 108	nfan 1565	. . . . . . 7 |- F/ y ( a e. ( A X. B ) /\ z = [_ ( 1st ` a ) / x ]_ [_ ( 2nd ` a ) / y ]_ C )
110		eleq1 2240	. . . . . . . . 9 |- ( a = <. x , y >. -> ( a e. ( A X. B ) <-> <. x , y >. e. ( A X. B ) ) )
111		opelxp 4653	. . . . . . . . 9 |- ( <. x , y >. e. ( A X. B ) <-> ( x e. A /\ y e. B ) )
112	110, 111	bitrdi 196	. . . . . . . 8 |- ( a = <. x , y >. -> ( a e. ( A X. B ) <-> ( x e. A /\ y e. B ) ) )
113		csbopeq1a 6183	. . . . . . . . 9 |- ( a = <. x , y >. -> [_ ( 1st ` a ) / x ]_ [_ ( 2nd ` a ) / y ]_ C = C )
114	113	eqeq2d 2189	. . . . . . . 8 |- ( a = <. x , y >. -> ( z = [_ ( 1st ` a ) / x ]_ [_ ( 2nd ` a ) / y ]_ C <-> z = C ) )
115	112, 114	anbi12d 473	. . . . . . 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 4731	. . . . . . . . 9 |- ( A X. B ) C_ ( _V X. _V )
117	116	sseli 3151	. . . . . . . 8 |- ( a e. ( A X. B ) -> a e. ( _V X. _V ) )
118	117	adantr 276	. . . . . . 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 6229	. . . . . 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 2198	. . . . 5 |- `' F = { <. z , a >. | ( a e. ( A X. B ) /\ z = [_ ( 1st ` a ) / x ]_ [_ ( 2nd ` a ) / y ]_ C ) }
121		df-mpt 4063	. . . . 5 |- ( z e. D |-> <. I , J >. ) = { <. z , a >. | ( z e. D /\ a = <. I , J >. ) }
122	96, 120, 121	3eqtr4g 2235	. . . 4 |- ( ph -> `' F = ( z e. D |-> <. I , J >. ) )
123	122	fneq1d 5302	. . 3 |- ( ph -> ( `' F Fn D <-> ( z e. D |-> <. I , J >. ) Fn D ) )
124	12, 123	mpbird 167	. 2 |- ( ph -> `' F Fn D )
125		dff1o4 5465	. 2 |- ( F : ( A X. B ) -1-1-onto-> D <-> ( F Fn ( A X. B ) /\ `' F Fn D ) )
126	5, 124, 125	sylanbrc 417	1 |- ( ph -> F : ( A X. B ) -1-1-onto-> D )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   e. wcel 2148   A.wral 2455   _Vcvv 2737   [.wsbc 2962   [_csb 3057   <.cop 3594   {copab 4060   |-> cmpt 4061   X. cxp 4621   `'ccnv 4622   Fn wfn 5207   -1-1-onto->wf1o 5211   ` cfv 5212   {coprab 5870   e. cmpo 5871   1stc1st 6133   2ndc2nd 6134

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136

This theorem is referenced by:  oddpwdc  12157