Theorem xpf1o 6738

Description: Construct a bijection on a Cartesian product given bijections on the factors. (Contributed by Mario Carneiro, 30-May-2015.)

Hypotheses

Ref	Expression
xpf1o.1	|- ( ph -> ( x e. A |-> X ) : A -1-1-onto-> B )
xpf1o.2	|- ( ph -> ( y e. C |-> Y ) : C -1-1-onto-> D )

Assertion

Ref	Expression
xpf1o	|- ( ph -> ( x e. A , y e. C |-> <. X , Y >. ) : ( A X. C ) -1-1-onto-> ( B X. D ) )

Distinct variable groups:   x, y, A   x, C, y   y, X   x, B   y, D   x, Y

Allowed substitution hints:   ph( x, y)   B( y)   D( x)   X( x)   Y( y)

Proof of Theorem xpf1o

Dummy variables t s u v w z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xp1st 6063	. . . . . 6 |- ( u e. ( A X. C ) -> ( 1st ` u ) e. A )
2	1	adantl 275	. . . . 5 |- ( ( ph /\ u e. ( A X. C ) ) -> ( 1st ` u ) e. A )
3		xpf1o.1	. . . . . . . 8 |- ( ph -> ( x e. A |-> X ) : A -1-1-onto-> B )
4		eqid 2139	. . . . . . . . 9 |- ( x e. A |-> X ) = ( x e. A |-> X )
5	4	f1ompt 5571	. . . . . . . 8 |- ( ( x e. A |-> X ) : A -1-1-onto-> B <-> ( A. x e. A X e. B /\ A. z e. B E! x e. A z = X ) )
6	3, 5	sylib 121	. . . . . . 7 |- ( ph -> ( A. x e. A X e. B /\ A. z e. B E! x e. A z = X ) )
7	6	simpld 111	. . . . . 6 |- ( ph -> A. x e. A X e. B )
8	7	adantr 274	. . . . 5 |- ( ( ph /\ u e. ( A X. C ) ) -> A. x e. A X e. B )
9		nfcsb1v 3035	. . . . . . 7 |- F/_ x [_ ( 1st ` u ) / x ]_ X
10	9	nfel1 2292	. . . . . 6 |- F/ x [_ ( 1st ` u ) / x ]_ X e. B
11		csbeq1a 3012	. . . . . . 7 |- ( x = ( 1st ` u ) -> X = [_ ( 1st ` u ) / x ]_ X )
12	11	eleq1d 2208	. . . . . 6 |- ( x = ( 1st ` u ) -> ( X e. B <-> [_ ( 1st ` u ) / x ]_ X e. B ) )
13	10, 12	rspc 2783	. . . . 5 |- ( ( 1st ` u ) e. A -> ( A. x e. A X e. B -> [_ ( 1st ` u ) / x ]_ X e. B ) )
14	2, 8, 13	sylc 62	. . . 4 |- ( ( ph /\ u e. ( A X. C ) ) -> [_ ( 1st ` u ) / x ]_ X e. B )
15		xp2nd 6064	. . . . . 6 |- ( u e. ( A X. C ) -> ( 2nd ` u ) e. C )
16	15	adantl 275	. . . . 5 |- ( ( ph /\ u e. ( A X. C ) ) -> ( 2nd ` u ) e. C )
17		xpf1o.2	. . . . . . . 8 |- ( ph -> ( y e. C |-> Y ) : C -1-1-onto-> D )
18		eqid 2139	. . . . . . . . 9 |- ( y e. C |-> Y ) = ( y e. C |-> Y )
19	18	f1ompt 5571	. . . . . . . 8 |- ( ( y e. C |-> Y ) : C -1-1-onto-> D <-> ( A. y e. C Y e. D /\ A. w e. D E! y e. C w = Y ) )
20	17, 19	sylib 121	. . . . . . 7 |- ( ph -> ( A. y e. C Y e. D /\ A. w e. D E! y e. C w = Y ) )
21	20	simpld 111	. . . . . 6 |- ( ph -> A. y e. C Y e. D )
22	21	adantr 274	. . . . 5 |- ( ( ph /\ u e. ( A X. C ) ) -> A. y e. C Y e. D )
23		nfcsb1v 3035	. . . . . . 7 |- F/_ y [_ ( 2nd ` u ) / y ]_ Y
24	23	nfel1 2292	. . . . . 6 |- F/ y [_ ( 2nd ` u ) / y ]_ Y e. D
25		csbeq1a 3012	. . . . . . 7 |- ( y = ( 2nd ` u ) -> Y = [_ ( 2nd ` u ) / y ]_ Y )
26	25	eleq1d 2208	. . . . . 6 |- ( y = ( 2nd ` u ) -> ( Y e. D <-> [_ ( 2nd ` u ) / y ]_ Y e. D ) )
27	24, 26	rspc 2783	. . . . 5 |- ( ( 2nd ` u ) e. C -> ( A. y e. C Y e. D -> [_ ( 2nd ` u ) / y ]_ Y e. D ) )
28	16, 22, 27	sylc 62	. . . 4 |- ( ( ph /\ u e. ( A X. C ) ) -> [_ ( 2nd ` u ) / y ]_ Y e. D )
29		opelxpi 4571	. . . 4 |- ( ( [_ ( 1st ` u ) / x ]_ X e. B /\ [_ ( 2nd ` u ) / y ]_ Y e. D ) -> <. [_ ( 1st ` u ) / x ]_ X , [_ ( 2nd ` u ) / y ]_ Y >. e. ( B X. D ) )
30	14, 28, 29	syl2anc 408	. . 3 |- ( ( ph /\ u e. ( A X. C ) ) -> <. [_ ( 1st ` u ) / x ]_ X , [_ ( 2nd ` u ) / y ]_ Y >. e. ( B X. D ) )
31	30	ralrimiva 2505	. 2 |- ( ph -> A. u e. ( A X. C ) <. [_ ( 1st ` u ) / x ]_ X , [_ ( 2nd ` u ) / y ]_ Y >. e. ( B X. D ) )
32	6	simprd 113	. . . . . . . . . 10 |- ( ph -> A. z e. B E! x e. A z = X )
33	32	r19.21bi 2520	. . . . . . . . 9 |- ( ( ph /\ z e. B ) -> E! x e. A z = X )
34		reu6 2873	. . . . . . . . 9 |- ( E! x e. A z = X <-> E. s e. A A. x e. A ( z = X <-> x = s ) )
35	33, 34	sylib 121	. . . . . . . 8 |- ( ( ph /\ z e. B ) -> E. s e. A A. x e. A ( z = X <-> x = s ) )
36	20	simprd 113	. . . . . . . . . 10 |- ( ph -> A. w e. D E! y e. C w = Y )
37	36	r19.21bi 2520	. . . . . . . . 9 |- ( ( ph /\ w e. D ) -> E! y e. C w = Y )
38		reu6 2873	. . . . . . . . 9 |- ( E! y e. C w = Y <-> E. t e. C A. y e. C ( w = Y <-> y = t ) )
39	37, 38	sylib 121	. . . . . . . 8 |- ( ( ph /\ w e. D ) -> E. t e. C A. y e. C ( w = Y <-> y = t ) )
40	35, 39	anim12dan 589	. . . . . . 7 |- ( ( ph /\ ( z e. B /\ w e. D ) ) -> ( E. s e. A A. x e. A ( z = X <-> x = s ) /\ E. t e. C A. y e. C ( w = Y <-> y = t ) ) )
41		reeanv 2600	. . . . . . . 8 |- ( E. s e. A E. t e. C ( A. x e. A ( z = X <-> x = s ) /\ A. y e. C ( w = Y <-> y = t ) ) <-> ( E. s e. A A. x e. A ( z = X <-> x = s ) /\ E. t e. C A. y e. C ( w = Y <-> y = t ) ) )
42		pm4.38 594	. . . . . . . . . . . . . . 15 |- ( ( ( z = X <-> x = s ) /\ ( w = Y <-> y = t ) ) -> ( ( z = X /\ w = Y ) <-> ( x = s /\ y = t ) ) )
43	42	ex 114	. . . . . . . . . . . . . 14 |- ( ( z = X <-> x = s ) -> ( ( w = Y <-> y = t ) -> ( ( z = X /\ w = Y ) <-> ( x = s /\ y = t ) ) ) )
44	43	ralimdv 2500	. . . . . . . . . . . . 13 |- ( ( z = X <-> x = s ) -> ( A. y e. C ( w = Y <-> y = t ) -> A. y e. C ( ( z = X /\ w = Y ) <-> ( x = s /\ y = t ) ) ) )
45	44	com12 30	. . . . . . . . . . . 12 |- ( A. y e. C ( w = Y <-> y = t ) -> ( ( z = X <-> x = s ) -> A. y e. C ( ( z = X /\ w = Y ) <-> ( x = s /\ y = t ) ) ) )
46	45	ralimdv 2500	. . . . . . . . . . 11 |- ( A. y e. C ( w = Y <-> y = t ) -> ( A. x e. A ( z = X <-> x = s ) -> A. x e. A A. y e. C ( ( z = X /\ w = Y ) <-> ( x = s /\ y = t ) ) ) )
47	46	impcom 124	. . . . . . . . . 10 |- ( ( A. x e. A ( z = X <-> x = s ) /\ A. y e. C ( w = Y <-> y = t ) ) -> A. x e. A A. y e. C ( ( z = X /\ w = Y ) <-> ( x = s /\ y = t ) ) )
48	47	reximi 2529	. . . . . . . . 9 |- ( E. t e. C ( A. x e. A ( z = X <-> x = s ) /\ A. y e. C ( w = Y <-> y = t ) ) -> E. t e. C A. x e. A A. y e. C ( ( z = X /\ w = Y ) <-> ( x = s /\ y = t ) ) )
49	48	reximi 2529	. . . . . . . 8 |- ( E. s e. A E. t e. C ( A. x e. A ( z = X <-> x = s ) /\ A. y e. C ( w = Y <-> y = t ) ) -> E. s e. A E. t e. C A. x e. A A. y e. C ( ( z = X /\ w = Y ) <-> ( x = s /\ y = t ) ) )
50	41, 49	sylbir 134	. . . . . . 7 |- ( ( E. s e. A A. x e. A ( z = X <-> x = s ) /\ E. t e. C A. y e. C ( w = Y <-> y = t ) ) -> E. s e. A E. t e. C A. x e. A A. y e. C ( ( z = X /\ w = Y ) <-> ( x = s /\ y = t ) ) )
51	40, 50	syl 14	. . . . . 6 |- ( ( ph /\ ( z e. B /\ w e. D ) ) -> E. s e. A E. t e. C A. x e. A A. y e. C ( ( z = X /\ w = Y ) <-> ( x = s /\ y = t ) ) )
52		vex 2689	. . . . . . . . . . . . . . 15 |- s e. _V
53		vex 2689	. . . . . . . . . . . . . . 15 |- t e. _V
54	52, 53	op1std 6046	. . . . . . . . . . . . . 14 |- ( u = <. s , t >. -> ( 1st ` u ) = s )
55	54	csbeq1d 3010	. . . . . . . . . . . . 13 |- ( u = <. s , t >. -> [_ ( 1st ` u ) / x ]_ X = [_ s / x ]_ X )
56	55	eqeq2d 2151	. . . . . . . . . . . 12 |- ( u = <. s , t >. -> ( z = [_ ( 1st ` u ) / x ]_ X <-> z = [_ s / x ]_ X ) )
57	52, 53	op2ndd 6047	. . . . . . . . . . . . . 14 |- ( u = <. s , t >. -> ( 2nd ` u ) = t )
58	57	csbeq1d 3010	. . . . . . . . . . . . 13 |- ( u = <. s , t >. -> [_ ( 2nd ` u ) / y ]_ Y = [_ t / y ]_ Y )
59	58	eqeq2d 2151	. . . . . . . . . . . 12 |- ( u = <. s , t >. -> ( w = [_ ( 2nd ` u ) / y ]_ Y <-> w = [_ t / y ]_ Y ) )
60	56, 59	anbi12d 464	. . . . . . . . . . 11 |- ( u = <. s , t >. -> ( ( z = [_ ( 1st ` u ) / x ]_ X /\ w = [_ ( 2nd ` u ) / y ]_ Y ) <-> ( z = [_ s / x ]_ X /\ w = [_ t / y ]_ Y ) ) )
61		eqeq1 2146	. . . . . . . . . . 11 |- ( u = <. s , t >. -> ( u = v <-> <. s , t >. = v ) )
62	60, 61	bibi12d 234	. . . . . . . . . 10 |- ( u = <. s , t >. -> ( ( ( z = [_ ( 1st ` u ) / x ]_ X /\ w = [_ ( 2nd ` u ) / y ]_ Y ) <-> u = v ) <-> ( ( z = [_ s / x ]_ X /\ w = [_ t / y ]_ Y ) <-> <. s , t >. = v ) ) )
63	62	ralxp 4682	. . . . . . . . 9 |- ( A. u e. ( A X. C ) ( ( z = [_ ( 1st ` u ) / x ]_ X /\ w = [_ ( 2nd ` u ) / y ]_ Y ) <-> u = v ) <-> A. s e. A A. t e. C ( ( z = [_ s / x ]_ X /\ w = [_ t / y ]_ Y ) <-> <. s , t >. = v ) )
64		nfv 1508	. . . . . . . . . 10 |- F/ s A. y e. C ( ( z = X /\ w = Y ) <-> <. x , y >. = v )
65		nfcv 2281	. . . . . . . . . . 11 |- F/_ x C
66		nfcsb1v 3035	. . . . . . . . . . . . . 14 |- F/_ x [_ s / x ]_ X
67	66	nfeq2 2293	. . . . . . . . . . . . 13 |- F/ x z = [_ s / x ]_ X
68		nfv 1508	. . . . . . . . . . . . 13 |- F/ x w = [_ t / y ]_ Y
69	67, 68	nfan 1544	. . . . . . . . . . . 12 |- F/ x ( z = [_ s / x ]_ X /\ w = [_ t / y ]_ Y )
70		nfv 1508	. . . . . . . . . . . 12 |- F/ x <. s , t >. = v
71	69, 70	nfbi 1568	. . . . . . . . . . 11 |- F/ x ( ( z = [_ s / x ]_ X /\ w = [_ t / y ]_ Y ) <-> <. s , t >. = v )
72	65, 71	nfralxy 2471	. . . . . . . . . 10 |- F/ x A. t e. C ( ( z = [_ s / x ]_ X /\ w = [_ t / y ]_ Y ) <-> <. s , t >. = v )
73		nfv 1508	. . . . . . . . . . . 12 |- F/ t ( ( z = X /\ w = Y ) <-> <. x , y >. = v )
74		nfv 1508	. . . . . . . . . . . . . 14 |- F/ y z = X
75		nfcsb1v 3035	. . . . . . . . . . . . . . 15 |- F/_ y [_ t / y ]_ Y
76	75	nfeq2 2293	. . . . . . . . . . . . . 14 |- F/ y w = [_ t / y ]_ Y
77	74, 76	nfan 1544	. . . . . . . . . . . . 13 |- F/ y ( z = X /\ w = [_ t / y ]_ Y )
78		nfv 1508	. . . . . . . . . . . . 13 |- F/ y <. x , t >. = v
79	77, 78	nfbi 1568	. . . . . . . . . . . 12 |- F/ y ( ( z = X /\ w = [_ t / y ]_ Y ) <-> <. x , t >. = v )
80		csbeq1a 3012	. . . . . . . . . . . . . . 15 |- ( y = t -> Y = [_ t / y ]_ Y )
81	80	eqeq2d 2151	. . . . . . . . . . . . . 14 |- ( y = t -> ( w = Y <-> w = [_ t / y ]_ Y ) )
82	81	anbi2d 459	. . . . . . . . . . . . 13 |- ( y = t -> ( ( z = X /\ w = Y ) <-> ( z = X /\ w = [_ t / y ]_ Y ) ) )
83		opeq2 3706	. . . . . . . . . . . . . 14 |- ( y = t -> <. x , y >. = <. x , t >. )
84	83	eqeq1d 2148	. . . . . . . . . . . . 13 |- ( y = t -> ( <. x , y >. = v <-> <. x , t >. = v ) )
85	82, 84	bibi12d 234	. . . . . . . . . . . 12 |- ( y = t -> ( ( ( z = X /\ w = Y ) <-> <. x , y >. = v ) <-> ( ( z = X /\ w = [_ t / y ]_ Y ) <-> <. x , t >. = v ) ) )
86	73, 79, 85	cbvral 2650	. . . . . . . . . . 11 |- ( A. y e. C ( ( z = X /\ w = Y ) <-> <. x , y >. = v ) <-> A. t e. C ( ( z = X /\ w = [_ t / y ]_ Y ) <-> <. x , t >. = v ) )
87		csbeq1a 3012	. . . . . . . . . . . . . . 15 |- ( x = s -> X = [_ s / x ]_ X )
88	87	eqeq2d 2151	. . . . . . . . . . . . . 14 |- ( x = s -> ( z = X <-> z = [_ s / x ]_ X ) )
89	88	anbi1d 460	. . . . . . . . . . . . 13 |- ( x = s -> ( ( z = X /\ w = [_ t / y ]_ Y ) <-> ( z = [_ s / x ]_ X /\ w = [_ t / y ]_ Y ) ) )
90		opeq1 3705	. . . . . . . . . . . . . 14 |- ( x = s -> <. x , t >. = <. s , t >. )
91	90	eqeq1d 2148	. . . . . . . . . . . . 13 |- ( x = s -> ( <. x , t >. = v <-> <. s , t >. = v ) )
92	89, 91	bibi12d 234	. . . . . . . . . . . 12 |- ( x = s -> ( ( ( z = X /\ w = [_ t / y ]_ Y ) <-> <. x , t >. = v ) <-> ( ( z = [_ s / x ]_ X /\ w = [_ t / y ]_ Y ) <-> <. s , t >. = v ) ) )
93	92	ralbidv 2437	. . . . . . . . . . 11 |- ( x = s -> ( A. t e. C ( ( z = X /\ w = [_ t / y ]_ Y ) <-> <. x , t >. = v ) <-> A. t e. C ( ( z = [_ s / x ]_ X /\ w = [_ t / y ]_ Y ) <-> <. s , t >. = v ) ) )
94	86, 93	syl5bb 191	. . . . . . . . . 10 |- ( x = s -> ( A. y e. C ( ( z = X /\ w = Y ) <-> <. x , y >. = v ) <-> A. t e. C ( ( z = [_ s / x ]_ X /\ w = [_ t / y ]_ Y ) <-> <. s , t >. = v ) ) )
95	64, 72, 94	cbvral 2650	. . . . . . . . 9 |- ( A. x e. A A. y e. C ( ( z = X /\ w = Y ) <-> <. x , y >. = v ) <-> A. s e. A A. t e. C ( ( z = [_ s / x ]_ X /\ w = [_ t / y ]_ Y ) <-> <. s , t >. = v ) )
96	63, 95	bitr4i 186	. . . . . . . 8 |- ( A. u e. ( A X. C ) ( ( z = [_ ( 1st ` u ) / x ]_ X /\ w = [_ ( 2nd ` u ) / y ]_ Y ) <-> u = v ) <-> A. x e. A A. y e. C ( ( z = X /\ w = Y ) <-> <. x , y >. = v ) )
97		eqeq2 2149	. . . . . . . . . . 11 |- ( v = <. s , t >. -> ( <. x , y >. = v <-> <. x , y >. = <. s , t >. ) )
98		vex 2689	. . . . . . . . . . . 12 |- x e. _V
99		vex 2689	. . . . . . . . . . . 12 |- y e. _V
100	98, 99	opth 4159	. . . . . . . . . . 11 |- ( <. x , y >. = <. s , t >. <-> ( x = s /\ y = t ) )
101	97, 100	syl6bb 195	. . . . . . . . . 10 |- ( v = <. s , t >. -> ( <. x , y >. = v <-> ( x = s /\ y = t ) ) )
102	101	bibi2d 231	. . . . . . . . 9 |- ( v = <. s , t >. -> ( ( ( z = X /\ w = Y ) <-> <. x , y >. = v ) <-> ( ( z = X /\ w = Y ) <-> ( x = s /\ y = t ) ) ) )
103	102	2ralbidv 2459	. . . . . . . 8 |- ( v = <. s , t >. -> ( A. x e. A A. y e. C ( ( z = X /\ w = Y ) <-> <. x , y >. = v ) <-> A. x e. A A. y e. C ( ( z = X /\ w = Y ) <-> ( x = s /\ y = t ) ) ) )
104	96, 103	syl5bb 191	. . . . . . 7 |- ( v = <. s , t >. -> ( A. u e. ( A X. C ) ( ( z = [_ ( 1st ` u ) / x ]_ X /\ w = [_ ( 2nd ` u ) / y ]_ Y ) <-> u = v ) <-> A. x e. A A. y e. C ( ( z = X /\ w = Y ) <-> ( x = s /\ y = t ) ) ) )
105	104	rexxp 4683	. . . . . 6 |- ( E. v e. ( A X. C ) A. u e. ( A X. C ) ( ( z = [_ ( 1st ` u ) / x ]_ X /\ w = [_ ( 2nd ` u ) / y ]_ Y ) <-> u = v ) <-> E. s e. A E. t e. C A. x e. A A. y e. C ( ( z = X /\ w = Y ) <-> ( x = s /\ y = t ) ) )
106	51, 105	sylibr 133	. . . . 5 |- ( ( ph /\ ( z e. B /\ w e. D ) ) -> E. v e. ( A X. C ) A. u e. ( A X. C ) ( ( z = [_ ( 1st ` u ) / x ]_ X /\ w = [_ ( 2nd ` u ) / y ]_ Y ) <-> u = v ) )
107		reu6 2873	. . . . 5 |- ( E! u e. ( A X. C ) ( z = [_ ( 1st ` u ) / x ]_ X /\ w = [_ ( 2nd ` u ) / y ]_ Y ) <-> E. v e. ( A X. C ) A. u e. ( A X. C ) ( ( z = [_ ( 1st ` u ) / x ]_ X /\ w = [_ ( 2nd ` u ) / y ]_ Y ) <-> u = v ) )
108	106, 107	sylibr 133	. . . 4 |- ( ( ph /\ ( z e. B /\ w e. D ) ) -> E! u e. ( A X. C ) ( z = [_ ( 1st ` u ) / x ]_ X /\ w = [_ ( 2nd ` u ) / y ]_ Y ) )
109	108	ralrimivva 2514	. . 3 |- ( ph -> A. z e. B A. w e. D E! u e. ( A X. C ) ( z = [_ ( 1st ` u ) / x ]_ X /\ w = [_ ( 2nd ` u ) / y ]_ Y ) )
110		eqeq1 2146	. . . . . 6 |- ( v = <. z , w >. -> ( v = <. [_ ( 1st ` u ) / x ]_ X , [_ ( 2nd ` u ) / y ]_ Y >. <-> <. z , w >. = <. [_ ( 1st ` u ) / x ]_ X , [_ ( 2nd ` u ) / y ]_ Y >. ) )
111		vex 2689	. . . . . . 7 |- z e. _V
112		vex 2689	. . . . . . 7 |- w e. _V
113	111, 112	opth 4159	. . . . . 6 |- ( <. z , w >. = <. [_ ( 1st ` u ) / x ]_ X , [_ ( 2nd ` u ) / y ]_ Y >. <-> ( z = [_ ( 1st ` u ) / x ]_ X /\ w = [_ ( 2nd ` u ) / y ]_ Y ) )
114	110, 113	syl6bb 195	. . . . 5 |- ( v = <. z , w >. -> ( v = <. [_ ( 1st ` u ) / x ]_ X , [_ ( 2nd ` u ) / y ]_ Y >. <-> ( z = [_ ( 1st ` u ) / x ]_ X /\ w = [_ ( 2nd ` u ) / y ]_ Y ) ) )
115	114	reubidv 2614	. . . 4 |- ( v = <. z , w >. -> ( E! u e. ( A X. C ) v = <. [_ ( 1st ` u ) / x ]_ X , [_ ( 2nd ` u ) / y ]_ Y >. <-> E! u e. ( A X. C ) ( z = [_ ( 1st ` u ) / x ]_ X /\ w = [_ ( 2nd ` u ) / y ]_ Y ) ) )
116	115	ralxp 4682	. . 3 |- ( A. v e. ( B X. D ) E! u e. ( A X. C ) v = <. [_ ( 1st ` u ) / x ]_ X , [_ ( 2nd ` u ) / y ]_ Y >. <-> A. z e. B A. w e. D E! u e. ( A X. C ) ( z = [_ ( 1st ` u ) / x ]_ X /\ w = [_ ( 2nd ` u ) / y ]_ Y ) )
117	109, 116	sylibr 133	. 2 |- ( ph -> A. v e. ( B X. D ) E! u e. ( A X. C ) v = <. [_ ( 1st ` u ) / x ]_ X , [_ ( 2nd ` u ) / y ]_ Y >. )
118		nfcv 2281	. . . . 5 |- F/_ z <. X , Y >.
119		nfcv 2281	. . . . 5 |- F/_ w <. X , Y >.
120		nfcsb1v 3035	. . . . . 6 |- F/_ x [_ z / x ]_ X
121		nfcv 2281	. . . . . 6 |- F/_ x [_ w / y ]_ Y
122	120, 121	nfop 3721	. . . . 5 |- F/_ x <. [_ z / x ]_ X , [_ w / y ]_ Y >.
123		nfcv 2281	. . . . . 6 |- F/_ y [_ z / x ]_ X
124		nfcsb1v 3035	. . . . . 6 |- F/_ y [_ w / y ]_ Y
125	123, 124	nfop 3721	. . . . 5 |- F/_ y <. [_ z / x ]_ X , [_ w / y ]_ Y >.
126		csbeq1a 3012	. . . . . 6 |- ( x = z -> X = [_ z / x ]_ X )
127		csbeq1a 3012	. . . . . 6 |- ( y = w -> Y = [_ w / y ]_ Y )
128		opeq12 3707	. . . . . 6 |- ( ( X = [_ z / x ]_ X /\ Y = [_ w / y ]_ Y ) -> <. X , Y >. = <. [_ z / x ]_ X , [_ w / y ]_ Y >. )
129	126, 127, 128	syl2an 287	. . . . 5 |- ( ( x = z /\ y = w ) -> <. X , Y >. = <. [_ z / x ]_ X , [_ w / y ]_ Y >. )
130	118, 119, 122, 125, 129	cbvmpo 5850	. . . 4 |- ( x e. A , y e. C |-> <. X , Y >. ) = ( z e. A , w e. C |-> <. [_ z / x ]_ X , [_ w / y ]_ Y >. )
131	111, 112	op1std 6046	. . . . . . 7 |- ( u = <. z , w >. -> ( 1st ` u ) = z )
132	131	csbeq1d 3010	. . . . . 6 |- ( u = <. z , w >. -> [_ ( 1st ` u ) / x ]_ X = [_ z / x ]_ X )
133	111, 112	op2ndd 6047	. . . . . . 7 |- ( u = <. z , w >. -> ( 2nd ` u ) = w )
134	133	csbeq1d 3010	. . . . . 6 |- ( u = <. z , w >. -> [_ ( 2nd ` u ) / y ]_ Y = [_ w / y ]_ Y )
135	132, 134	opeq12d 3713	. . . . 5 |- ( u = <. z , w >. -> <. [_ ( 1st ` u ) / x ]_ X , [_ ( 2nd ` u ) / y ]_ Y >. = <. [_ z / x ]_ X , [_ w / y ]_ Y >. )
136	135	mpompt 5863	. . . 4 |- ( u e. ( A X. C ) |-> <. [_ ( 1st ` u ) / x ]_ X , [_ ( 2nd ` u ) / y ]_ Y >. ) = ( z e. A , w e. C |-> <. [_ z / x ]_ X , [_ w / y ]_ Y >. )
137	130, 136	eqtr4i 2163	. . 3 |- ( x e. A , y e. C |-> <. X , Y >. ) = ( u e. ( A X. C ) |-> <. [_ ( 1st ` u ) / x ]_ X , [_ ( 2nd ` u ) / y ]_ Y >. )
138	137	f1ompt 5571	. 2 |- ( ( x e. A , y e. C |-> <. X , Y >. ) : ( A X. C ) -1-1-onto-> ( B X. D ) <-> ( A. u e. ( A X. C ) <. [_ ( 1st ` u ) / x ]_ X , [_ ( 2nd ` u ) / y ]_ Y >. e. ( B X. D ) /\ A. v e. ( B X. D ) E! u e. ( A X. C ) v = <. [_ ( 1st ` u ) / x ]_ X , [_ ( 2nd ` u ) / y ]_ Y >. ) )
139	31, 117, 138	sylanbrc 413	1 |- ( ph -> ( x e. A , y e. C |-> <. X , Y >. ) : ( A X. C ) -1-1-onto-> ( B X. D ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1331   e. wcel 1480   A.wral 2416   E.wrex 2417   E!wreu 2418   [_csb 3003   <.cop 3530   |-> cmpt 3989   X. cxp 4537   -1-1-onto->wf1o 5122   ` cfv 5123   e. cmpo 5776   1stc1st 6036   2ndc2nd 6037

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039

This theorem is referenced by:  xpen  6739