Theorem f1od2 29499

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 2971	. . 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	fnmpt2 7238	. . 3 |- ( A. x e. A A. y e. B C e. W -> F Fn ( A X. B ) )
5	2, 4	syl 17	. 2 |- ( ph -> F Fn ( A X. B ) )
6		f1od2.3	. . . . . 6 |- ( ( ph /\ z e. D ) -> ( I e. X /\ J e. Y ) )
7		opelxpi 5148	. . . . . 6 |- ( ( I e. X /\ J e. Y ) -> <. I , J >. e. ( X X. Y ) )
8	6, 7	syl 17	. . . . 5 |- ( ( ph /\ z e. D ) -> <. I , J >. e. ( X X. Y ) )
9	8	ralrimiva 2966	. . . 4 |- ( ph -> A. z e. D <. I , J >. e. ( X X. Y ) )
10		eqid 2622	. . . . 5 |- ( z e. D |-> <. I , J >. ) = ( z e. D |-> <. I , J >. )
11	10	fnmpt 6020	. . . 4 |- ( A. z e. D <. I , J >. e. ( X X. Y ) -> ( z e. D |-> <. I , J >. ) Fn D )
12	9, 11	syl 17	. . 3 |- ( ph -> ( z e. D |-> <. I , J >. ) Fn D )
13		elxp7 7201	. . . . . . . 8 |- ( a e. ( A X. B ) <-> ( a e. ( _V X. _V ) /\ ( ( 1st ` a ) e. A /\ ( 2nd ` a ) e. B ) ) )
14	13	anbi1i 731	. . . . . . 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 681	. . . . . . . . 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 3490	. . . . . . . . . . . 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 3490	. . . . . . . . . . 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 3478	. . . . . . . . . . . . . 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 3478	. . . . . . . . . . . . . . . 16 |- ( [. ( 2nd ` a ) / y ]. ( x e. A /\ y e. B ) <-> ( [. ( 2nd ` a ) / y ]. x e. A /\ [. ( 2nd ` a ) / y ]. y e. B ) )
21		fvex 6201	. . . . . . . . . . . . . . . . . 18 |- ( 2nd ` a ) e. _V
22		sbcg 3503	. . . . . . . . . . . . . . . . . 18 |- ( ( 2nd ` a ) e. _V -> ( [. ( 2nd ` a ) / y ]. x e. A <-> x e. A ) )
23	21, 22	ax-mp 5	. . . . . . . . . . . . . . . . 17 |- ( [. ( 2nd ` a ) / y ]. x e. A <-> x e. A )
24		sbcel1v 3495	. . . . . . . . . . . . . . . . 17 |- ( [. ( 2nd ` a ) / y ]. y e. B <-> ( 2nd ` a ) e. B )
25	23, 24	anbi12i 733	. . . . . . . . . . . . . . . 16 |- ( ( [. ( 2nd ` a ) / y ]. x e. A /\ [. ( 2nd ` a ) / y ]. y e. B ) <-> ( x e. A /\ ( 2nd ` a ) e. B ) )
26	20, 25	bitri 264	. . . . . . . . . . . . . . 15 |- ( [. ( 2nd ` a ) / y ]. ( x e. A /\ y e. B ) <-> ( x e. A /\ ( 2nd ` a ) e. B ) )
27		sbceq2g 3990	. . . . . . . . . . . . . . . 16 |- ( ( 2nd ` a ) e. _V -> ( [. ( 2nd ` a ) / y ]. z = C <-> z = [_ ( 2nd ` a ) / y ]_ C ) )
28	21, 27	ax-mp 5	. . . . . . . . . . . . . . 15 |- ( [. ( 2nd ` a ) / y ]. z = C <-> z = [_ ( 2nd ` a ) / y ]_ C )
29	26, 28	anbi12i 733	. . . . . . . . . . . . . 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 ) )
30	19, 29	bitri 264	. . . . . . . . . . . . 13 |- ( [. ( 2nd ` a ) / y ]. ( ( x e. A /\ y e. B ) /\ z = C ) <-> ( ( x e. A /\ ( 2nd ` a ) e. B ) /\ z = [_ ( 2nd ` a ) / y ]_ C ) )
31	30	sbcbii 3491	. . . . . . . . . . . 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 ) )
32		sbcan 3478	. . . . . . . . . . . 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 ) )
33		sbcan 3478	. . . . . . . . . . . . . 14 |- ( [. ( 1st ` a ) / x ]. ( x e. A /\ ( 2nd ` a ) e. B ) <-> ( [. ( 1st ` a ) / x ]. x e. A /\ [. ( 1st ` a ) / x ]. ( 2nd ` a ) e. B ) )
34		sbcel1v 3495	. . . . . . . . . . . . . . 15 |- ( [. ( 1st ` a ) / x ]. x e. A <-> ( 1st ` a ) e. A )
35		fvex 6201	. . . . . . . . . . . . . . . 16 |- ( 1st ` a ) e. _V
36		sbcg 3503	. . . . . . . . . . . . . . . 16 |- ( ( 1st ` a ) e. _V -> ( [. ( 1st ` a ) / x ]. ( 2nd ` a ) e. B <-> ( 2nd ` a ) e. B ) )
37	35, 36	ax-mp 5	. . . . . . . . . . . . . . 15 |- ( [. ( 1st ` a ) / x ]. ( 2nd ` a ) e. B <-> ( 2nd ` a ) e. B )
38	34, 37	anbi12i 733	. . . . . . . . . . . . . 14 |- ( ( [. ( 1st ` a ) / x ]. x e. A /\ [. ( 1st ` a ) / x ]. ( 2nd ` a ) e. B ) <-> ( ( 1st ` a ) e. A /\ ( 2nd ` a ) e. B ) )
39	33, 38	bitri 264	. . . . . . . . . . . . 13 |- ( [. ( 1st ` a ) / x ]. ( x e. A /\ ( 2nd ` a ) e. B ) <-> ( ( 1st ` a ) e. A /\ ( 2nd ` a ) e. B ) )
40		sbceq2g 3990	. . . . . . . . . . . . . 14 |- ( ( 1st ` a ) e. _V -> ( [. ( 1st ` a ) / x ]. z = [_ ( 2nd ` a ) / y ]_ C <-> z = [_ ( 1st ` a ) / x ]_ [_ ( 2nd ` a ) / y ]_ C ) )
41	35, 40	ax-mp 5	. . . . . . . . . . . . 13 |- ( [. ( 1st ` a ) / x ]. z = [_ ( 2nd ` a ) / y ]_ C <-> z = [_ ( 1st ` a ) / x ]_ [_ ( 2nd ` a ) / y ]_ C )
42	39, 41	anbi12i 733	. . . . . . . . . . . 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 ) )
43	31, 32, 42	3bitri 286	. . . . . . . . . . 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 ) )
44		sbcan 3478	. . . . . . . . . . . . . 14 |- ( [. ( 2nd ` a ) / y ]. ( z e. D /\ ( x = I /\ y = J ) ) <-> ( [. ( 2nd ` a ) / y ]. z e. D /\ [. ( 2nd ` a ) / y ]. ( x = I /\ y = J ) ) )
45		sbcg 3503	. . . . . . . . . . . . . . . 16 |- ( ( 2nd ` a ) e. _V -> ( [. ( 2nd ` a ) / y ]. z e. D <-> z e. D ) )
46	21, 45	ax-mp 5	. . . . . . . . . . . . . . 15 |- ( [. ( 2nd ` a ) / y ]. z e. D <-> z e. D )
47		sbcan 3478	. . . . . . . . . . . . . . . 16 |- ( [. ( 2nd ` a ) / y ]. ( x = I /\ y = J ) <-> ( [. ( 2nd ` a ) / y ]. x = I /\ [. ( 2nd ` a ) / y ]. y = J ) )
48		sbcg 3503	. . . . . . . . . . . . . . . . . 18 |- ( ( 2nd ` a ) e. _V -> ( [. ( 2nd ` a ) / y ]. x = I <-> x = I ) )
49	21, 48	ax-mp 5	. . . . . . . . . . . . . . . . 17 |- ( [. ( 2nd ` a ) / y ]. x = I <-> x = I )
50		sbceq1g 3988	. . . . . . . . . . . . . . . . . . 19 |- ( ( 2nd ` a ) e. _V -> ( [. ( 2nd ` a ) / y ]. y = J <-> [_ ( 2nd ` a ) / y ]_ y = J ) )
51	21, 50	ax-mp 5	. . . . . . . . . . . . . . . . . 18 |- ( [. ( 2nd ` a ) / y ]. y = J <-> [_ ( 2nd ` a ) / y ]_ y = J )
52		csbvarg 4003	. . . . . . . . . . . . . . . . . . . 20 |- ( ( 2nd ` a ) e. _V -> [_ ( 2nd ` a ) / y ]_ y = ( 2nd ` a ) )
53	21, 52	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 |- [_ ( 2nd ` a ) / y ]_ y = ( 2nd ` a )
54	53	eqeq1i 2627	. . . . . . . . . . . . . . . . . 18 |- ( [_ ( 2nd ` a ) / y ]_ y = J <-> ( 2nd ` a ) = J )
55	51, 54	bitri 264	. . . . . . . . . . . . . . . . 17 |- ( [. ( 2nd ` a ) / y ]. y = J <-> ( 2nd ` a ) = J )
56	49, 55	anbi12i 733	. . . . . . . . . . . . . . . 16 |- ( ( [. ( 2nd ` a ) / y ]. x = I /\ [. ( 2nd ` a ) / y ]. y = J ) <-> ( x = I /\ ( 2nd ` a ) = J ) )
57	47, 56	bitri 264	. . . . . . . . . . . . . . 15 |- ( [. ( 2nd ` a ) / y ]. ( x = I /\ y = J ) <-> ( x = I /\ ( 2nd ` a ) = J ) )
58	46, 57	anbi12i 733	. . . . . . . . . . . . . 14 |- ( ( [. ( 2nd ` a ) / y ]. z e. D /\ [. ( 2nd ` a ) / y ]. ( x = I /\ y = J ) ) <-> ( z e. D /\ ( x = I /\ ( 2nd ` a ) = J ) ) )
59	44, 58	bitri 264	. . . . . . . . . . . . 13 |- ( [. ( 2nd ` a ) / y ]. ( z e. D /\ ( x = I /\ y = J ) ) <-> ( z e. D /\ ( x = I /\ ( 2nd ` a ) = J ) ) )
60	59	sbcbii 3491	. . . . . . . . . . . 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 ) ) )
61		sbcan 3478	. . . . . . . . . . . 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 ) ) )
62		sbcg 3503	. . . . . . . . . . . . . 14 |- ( ( 1st ` a ) e. _V -> ( [. ( 1st ` a ) / x ]. z e. D <-> z e. D ) )
63	35, 62	ax-mp 5	. . . . . . . . . . . . 13 |- ( [. ( 1st ` a ) / x ]. z e. D <-> z e. D )
64		sbcan 3478	. . . . . . . . . . . . . 14 |- ( [. ( 1st ` a ) / x ]. ( x = I /\ ( 2nd ` a ) = J ) <-> ( [. ( 1st ` a ) / x ]. x = I /\ [. ( 1st ` a ) / x ]. ( 2nd ` a ) = J ) )
65		sbceq1g 3988	. . . . . . . . . . . . . . . . 17 |- ( ( 1st ` a ) e. _V -> ( [. ( 1st ` a ) / x ]. x = I <-> [_ ( 1st ` a ) / x ]_ x = I ) )
66	35, 65	ax-mp 5	. . . . . . . . . . . . . . . 16 |- ( [. ( 1st ` a ) / x ]. x = I <-> [_ ( 1st ` a ) / x ]_ x = I )
67		csbvarg 4003	. . . . . . . . . . . . . . . . . 18 |- ( ( 1st ` a ) e. _V -> [_ ( 1st ` a ) / x ]_ x = ( 1st ` a ) )
68	35, 67	ax-mp 5	. . . . . . . . . . . . . . . . 17 |- [_ ( 1st ` a ) / x ]_ x = ( 1st ` a )
69	68	eqeq1i 2627	. . . . . . . . . . . . . . . 16 |- ( [_ ( 1st ` a ) / x ]_ x = I <-> ( 1st ` a ) = I )
70	66, 69	bitri 264	. . . . . . . . . . . . . . 15 |- ( [. ( 1st ` a ) / x ]. x = I <-> ( 1st ` a ) = I )
71		sbcg 3503	. . . . . . . . . . . . . . . 16 |- ( ( 1st ` a ) e. _V -> ( [. ( 1st ` a ) / x ]. ( 2nd ` a ) = J <-> ( 2nd ` a ) = J ) )
72	35, 71	ax-mp 5	. . . . . . . . . . . . . . 15 |- ( [. ( 1st ` a ) / x ]. ( 2nd ` a ) = J <-> ( 2nd ` a ) = J )
73	70, 72	anbi12i 733	. . . . . . . . . . . . . 14 |- ( ( [. ( 1st ` a ) / x ]. x = I /\ [. ( 1st ` a ) / x ]. ( 2nd ` a ) = J ) <-> ( ( 1st ` a ) = I /\ ( 2nd ` a ) = J ) )
74	64, 73	bitri 264	. . . . . . . . . . . . 13 |- ( [. ( 1st ` a ) / x ]. ( x = I /\ ( 2nd ` a ) = J ) <-> ( ( 1st ` a ) = I /\ ( 2nd ` a ) = J ) )
75	63, 74	anbi12i 733	. . . . . . . . . . . 12 |- ( ( [. ( 1st ` a ) / x ]. z e. D /\ [. ( 1st ` a ) / x ]. ( x = I /\ ( 2nd ` a ) = J ) ) <-> ( z e. D /\ ( ( 1st ` a ) = I /\ ( 2nd ` a ) = J ) ) )
76	60, 61, 75	3bitri 286	. . . . . . . . . . 11 |- ( [. ( 1st ` a ) / x ]. [. ( 2nd ` a ) / y ]. ( z e. D /\ ( x = I /\ y = J ) ) <-> ( z e. D /\ ( ( 1st ` a ) = I /\ ( 2nd ` a ) = J ) ) )
77	18, 43, 76	3bitr3g 302	. . . . . . . . . 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 ) ) ) )
78	77	anbi2d 740	. . . . . . . . 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 ) ) ) ) )
79	15, 78	syl5bb 272	. . . . . . . 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 ) ) ) ) )
80		xpss 5226	. . . . . . . . . . . 12 |- ( X X. Y ) C_ ( _V X. _V )
81		simprr 796	. . . . . . . . . . . . 13 |- ( ( ph /\ ( z e. D /\ a = <. I , J >. ) ) -> a = <. I , J >. )
82	8	adantrr 753	. . . . . . . . . . . . 13 |- ( ( ph /\ ( z e. D /\ a = <. I , J >. ) ) -> <. I , J >. e. ( X X. Y ) )
83	81, 82	eqeltrd 2701	. . . . . . . . . . . 12 |- ( ( ph /\ ( z e. D /\ a = <. I , J >. ) ) -> a e. ( X X. Y ) )
84	80, 83	sseldi 3601	. . . . . . . . . . 11 |- ( ( ph /\ ( z e. D /\ a = <. I , J >. ) ) -> a e. ( _V X. _V ) )
85	84	ex 450	. . . . . . . . . 10 |- ( ph -> ( ( z e. D /\ a = <. I , J >. ) -> a e. ( _V X. _V ) ) )
86	85	pm4.71rd 667	. . . . . . . . 9 |- ( ph -> ( ( z e. D /\ a = <. I , J >. ) <-> ( a e. ( _V X. _V ) /\ ( z e. D /\ a = <. I , J >. ) ) ) )
87		eqop 7208	. . . . . . . . . . 11 |- ( a e. ( _V X. _V ) -> ( a = <. I , J >. <-> ( ( 1st ` a ) = I /\ ( 2nd ` a ) = J ) ) )
88	87	anbi2d 740	. . . . . . . . . 10 |- ( a e. ( _V X. _V ) -> ( ( z e. D /\ a = <. I , J >. ) <-> ( z e. D /\ ( ( 1st ` a ) = I /\ ( 2nd ` a ) = J ) ) ) )
89	88	pm5.32i 669	. . . . . . . . 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 ) ) ) )
90	86, 89	syl6rbb 277	. . . . . . . 8 |- ( ph -> ( ( a e. ( _V X. _V ) /\ ( z e. D /\ ( ( 1st ` a ) = I /\ ( 2nd ` a ) = J ) ) ) <-> ( z e. D /\ a = <. I , J >. ) ) )
91	79, 90	bitrd 268	. . . . . . 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 >. ) ) )
92	14, 91	syl5bb 272	. . . . . 6 |- ( ph -> ( ( a e. ( A X. B ) /\ z = [_ ( 1st ` a ) / x ]_ [_ ( 2nd ` a ) / y ]_ C ) <-> ( z e. D /\ a = <. I , J >. ) ) )
93	92	opabbidv 4716	. . . . 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 >. ) } )
94		df-mpt2 6655	. . . . . . . 8 |- ( x e. A , y e. B |-> C ) = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = C ) }
95	3, 94	eqtri 2644	. . . . . . 7 |- F = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = C ) }
96	95	cnveqi 5297	. . . . . 6 |- `' F = `' { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = C ) }
97		nfv 1843	. . . . . . . 8 |- F/ x a e. ( A X. B )
98		nfcsb1v 3549	. . . . . . . . 9 |- F/_ x [_ ( 1st ` a ) / x ]_ [_ ( 2nd ` a ) / y ]_ C
99	98	nfeq2 2780	. . . . . . . 8 |- F/ x z = [_ ( 1st ` a ) / x ]_ [_ ( 2nd ` a ) / y ]_ C
100	97, 99	nfan 1828	. . . . . . 7 |- F/ x ( a e. ( A X. B ) /\ z = [_ ( 1st ` a ) / x ]_ [_ ( 2nd ` a ) / y ]_ C )
101		nfv 1843	. . . . . . . 8 |- F/ y a e. ( A X. B )
102		nfcv 2764	. . . . . . . . . 10 |- F/_ y ( 1st ` a )
103		nfcsb1v 3549	. . . . . . . . . 10 |- F/_ y [_ ( 2nd ` a ) / y ]_ C
104	102, 103	nfcsb 3551	. . . . . . . . 9 |- F/_ y [_ ( 1st ` a ) / x ]_ [_ ( 2nd ` a ) / y ]_ C
105	104	nfeq2 2780	. . . . . . . 8 |- F/ y z = [_ ( 1st ` a ) / x ]_ [_ ( 2nd ` a ) / y ]_ C
106	101, 105	nfan 1828	. . . . . . 7 |- F/ y ( a e. ( A X. B ) /\ z = [_ ( 1st ` a ) / x ]_ [_ ( 2nd ` a ) / y ]_ C )
107		eleq1 2689	. . . . . . . . 9 |- ( a = <. x , y >. -> ( a e. ( A X. B ) <-> <. x , y >. e. ( A X. B ) ) )
108		opelxp 5146	. . . . . . . . 9 |- ( <. x , y >. e. ( A X. B ) <-> ( x e. A /\ y e. B ) )
109	107, 108	syl6bb 276	. . . . . . . 8 |- ( a = <. x , y >. -> ( a e. ( A X. B ) <-> ( x e. A /\ y e. B ) ) )
110		csbopeq1a 7221	. . . . . . . . 9 |- ( a = <. x , y >. -> [_ ( 1st ` a ) / x ]_ [_ ( 2nd ` a ) / y ]_ C = C )
111	110	eqeq2d 2632	. . . . . . . 8 |- ( a = <. x , y >. -> ( z = [_ ( 1st ` a ) / x ]_ [_ ( 2nd ` a ) / y ]_ C <-> z = C ) )
112	109, 111	anbi12d 747	. . . . . . 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 ) ) )
113		xpss 5226	. . . . . . . . 9 |- ( A X. B ) C_ ( _V X. _V )
114	113	sseli 3599	. . . . . . . 8 |- ( a e. ( A X. B ) -> a e. ( _V X. _V ) )
115	114	adantr 481	. . . . . . 7 |- ( ( a e. ( A X. B ) /\ z = [_ ( 1st ` a ) / x ]_ [_ ( 2nd ` a ) / y ]_ C ) -> a e. ( _V X. _V ) )
116	100, 106, 112, 115	cnvoprab 29498	. . . . . 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 ) }
117	96, 116	eqtri 2644	. . . . 5 |- `' F = { <. z , a >. | ( a e. ( A X. B ) /\ z = [_ ( 1st ` a ) / x ]_ [_ ( 2nd ` a ) / y ]_ C ) }
118		df-mpt 4730	. . . . 5 |- ( z e. D |-> <. I , J >. ) = { <. z , a >. | ( z e. D /\ a = <. I , J >. ) }
119	93, 117, 118	3eqtr4g 2681	. . . 4 |- ( ph -> `' F = ( z e. D |-> <. I , J >. ) )
120	119	fneq1d 5981	. . 3 |- ( ph -> ( `' F Fn D <-> ( z e. D |-> <. I , J >. ) Fn D ) )
121	12, 120	mpbird 247	. 2 |- ( ph -> `' F Fn D )
122		dff1o4 6145	. 2 |- ( F : ( A X. B ) -1-1-onto-> D <-> ( F Fn ( A X. B ) /\ `' F Fn D ) )
123	5, 121, 122	sylanbrc 698	1 |- ( ph -> F : ( A X. B ) -1-1-onto-> D )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   [.wsbc 3435   [_csb 3533   <.cop 4183   {copab 4712   |-> cmpt 4729   X. cxp 5112   `'ccnv 5113   Fn wfn 5883   -1-1-onto->wf1o 5887   ` cfv 5888   {coprab 6651   |-> cmpt2 6652   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-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-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  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-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169

This theorem is referenced by:  oddpwdc  30416