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Theorem xpf1o 7228
Description: Construct a bijection on a cross 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.
StepHypRef Expression
1 xp1st 6335 . . . . . 6  |-  ( u  e.  ( A  X.  C )  ->  ( 1st `  u )  e.  A )
21adantl 453 . . . . 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 2404 . . . . . . . . 9  |-  ( x  e.  A  |->  X )  =  ( x  e.  A  |->  X )
54f1ompt 5850 . . . . . . . 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 ) )
63, 5sylib 189 . . . . . . 7  |-  ( ph  ->  ( A. x  e.  A  X  e.  B  /\  A. z  e.  B  E! x  e.  A  z  =  X )
)
76simpld 446 . . . . . 6  |-  ( ph  ->  A. x  e.  A  X  e.  B )
87adantr 452 . . . . 5  |-  ( (
ph  /\  u  e.  ( A  X.  C
) )  ->  A. x  e.  A  X  e.  B )
9 nfcsb1v 3243 . . . . . . 7  |-  F/_ x [_ ( 1st `  u
)  /  x ]_ X
109nfel1 2550 . . . . . 6  |-  F/ x [_ ( 1st `  u
)  /  x ]_ X  e.  B
11 csbeq1a 3219 . . . . . . 7  |-  ( x  =  ( 1st `  u
)  ->  X  =  [_ ( 1st `  u
)  /  x ]_ X )
1211eleq1d 2470 . . . . . 6  |-  ( x  =  ( 1st `  u
)  ->  ( X  e.  B  <->  [_ ( 1st `  u
)  /  x ]_ X  e.  B )
)
1310, 12rspc 3006 . . . . 5  |-  ( ( 1st `  u )  e.  A  ->  ( A. x  e.  A  X  e.  B  ->  [_ ( 1st `  u
)  /  x ]_ X  e.  B )
)
142, 8, 13sylc 58 . . . 4  |-  ( (
ph  /\  u  e.  ( A  X.  C
) )  ->  [_ ( 1st `  u )  /  x ]_ X  e.  B
)
15 xp2nd 6336 . . . . . 6  |-  ( u  e.  ( A  X.  C )  ->  ( 2nd `  u )  e.  C )
1615adantl 453 . . . . 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 2404 . . . . . . . . 9  |-  ( y  e.  C  |->  Y )  =  ( y  e.  C  |->  Y )
1918f1ompt 5850 . . . . . . . 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 ) )
2017, 19sylib 189 . . . . . . 7  |-  ( ph  ->  ( A. y  e.  C  Y  e.  D  /\  A. w  e.  D  E! y  e.  C  w  =  Y )
)
2120simpld 446 . . . . . 6  |-  ( ph  ->  A. y  e.  C  Y  e.  D )
2221adantr 452 . . . . 5  |-  ( (
ph  /\  u  e.  ( A  X.  C
) )  ->  A. y  e.  C  Y  e.  D )
23 nfcsb1v 3243 . . . . . . 7  |-  F/_ y [_ ( 2nd `  u
)  /  y ]_ Y
2423nfel1 2550 . . . . . 6  |-  F/ y
[_ ( 2nd `  u
)  /  y ]_ Y  e.  D
25 csbeq1a 3219 . . . . . . 7  |-  ( y  =  ( 2nd `  u
)  ->  Y  =  [_ ( 2nd `  u
)  /  y ]_ Y )
2625eleq1d 2470 . . . . . 6  |-  ( y  =  ( 2nd `  u
)  ->  ( Y  e.  D  <->  [_ ( 2nd `  u
)  /  y ]_ Y  e.  D )
)
2724, 26rspc 3006 . . . . 5  |-  ( ( 2nd `  u )  e.  C  ->  ( A. y  e.  C  Y  e.  D  ->  [_ ( 2nd `  u
)  /  y ]_ Y  e.  D )
)
2816, 22, 27sylc 58 . . . 4  |-  ( (
ph  /\  u  e.  ( A  X.  C
) )  ->  [_ ( 2nd `  u )  / 
y ]_ Y  e.  D
)
29 opelxpi 4869 . . . 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 ) )
3014, 28, 29syl2anc 643 . . 3  |-  ( (
ph  /\  u  e.  ( A  X.  C
) )  ->  <. [_ ( 1st `  u )  /  x ]_ X ,  [_ ( 2nd `  u )  /  y ]_ Y >.  e.  ( B  X.  D ) )
3130ralrimiva 2749 . 2  |-  ( ph  ->  A. u  e.  ( A  X.  C )
<. [_ ( 1st `  u
)  /  x ]_ X ,  [_ ( 2nd `  u )  /  y ]_ Y >.  e.  ( B  X.  D ) )
326simprd 450 . . . . . . . . . 10  |-  ( ph  ->  A. z  e.  B  E! x  e.  A  z  =  X )
3332r19.21bi 2764 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  B )  ->  E! x  e.  A  z  =  X )
34 reu6 3083 . . . . . . . . 9  |-  ( E! x  e.  A  z  =  X  <->  E. s  e.  A  A. x  e.  A  ( z  =  X  <->  x  =  s
) )
3533, 34sylib 189 . . . . . . . 8  |-  ( (
ph  /\  z  e.  B )  ->  E. s  e.  A  A. x  e.  A  ( z  =  X  <->  x  =  s
) )
3620simprd 450 . . . . . . . . . 10  |-  ( ph  ->  A. w  e.  D  E! y  e.  C  w  =  Y )
3736r19.21bi 2764 . . . . . . . . 9  |-  ( (
ph  /\  w  e.  D )  ->  E! y  e.  C  w  =  Y )
38 reu6 3083 . . . . . . . . 9  |-  ( E! y  e.  C  w  =  Y  <->  E. t  e.  C  A. y  e.  C  ( w  =  Y  <->  y  =  t ) )
3937, 38sylib 189 . . . . . . . 8  |-  ( (
ph  /\  w  e.  D )  ->  E. t  e.  C  A. y  e.  C  ( w  =  Y  <->  y  =  t ) )
4035, 39anim12dan 811 . . . . . . 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 2835 . . . . . . . 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 pm4.38 843 . . . . . . . . . . . . . . 15  |-  ( ( ( z  =  X  <-> 
x  =  s )  /\  ( w  =  Y  <->  y  =  t ) )  ->  (
( z  =  X  /\  w  =  Y )  <->  ( x  =  s  /\  y  =  t ) ) )
4342ex 424 . . . . . . . . . . . . . 14  |-  ( ( z  =  X  <->  x  =  s )  ->  (
( w  =  Y  <-> 
y  =  t )  ->  ( ( z  =  X  /\  w  =  Y )  <->  ( x  =  s  /\  y  =  t ) ) ) )
4443ralimdv 2745 . . . . . . . . . . . . 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 ) ) ) )
4544com12 29 . . . . . . . . . . . 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 ) ) ) )
4645ralimdv 2745 . . . . . . . . . . 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 ) ) ) )
4746impcom 420 . . . . . . . . . 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 )
) )
4847reximi 2773 . . . . . . . . 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 )
) )
4948reximi 2773 . . . . . . . 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 )
) )
5041, 49sylbir 205 . . . . . . 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 )
) )
5140, 50syl 16 . . . . . 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 2919 . . . . . . . . . . . . . . 15  |-  s  e. 
_V
53 vex 2919 . . . . . . . . . . . . . . 15  |-  t  e. 
_V
5452, 53op1std 6316 . . . . . . . . . . . . . 14  |-  ( u  =  <. s ,  t
>.  ->  ( 1st `  u
)  =  s )
5554csbeq1d 3217 . . . . . . . . . . . . 13  |-  ( u  =  <. s ,  t
>.  ->  [_ ( 1st `  u
)  /  x ]_ X  =  [_ s  /  x ]_ X )
5655eqeq2d 2415 . . . . . . . . . . . 12  |-  ( u  =  <. s ,  t
>.  ->  ( z  = 
[_ ( 1st `  u
)  /  x ]_ X 
<->  z  =  [_ s  /  x ]_ X ) )
5752, 53op2ndd 6317 . . . . . . . . . . . . . 14  |-  ( u  =  <. s ,  t
>.  ->  ( 2nd `  u
)  =  t )
5857csbeq1d 3217 . . . . . . . . . . . . 13  |-  ( u  =  <. s ,  t
>.  ->  [_ ( 2nd `  u
)  /  y ]_ Y  =  [_ t  / 
y ]_ Y )
5958eqeq2d 2415 . . . . . . . . . . . 12  |-  ( u  =  <. s ,  t
>.  ->  ( w  = 
[_ ( 2nd `  u
)  /  y ]_ Y 
<->  w  =  [_ t  /  y ]_ Y
) )
6056, 59anbi12d 692 . . . . . . . . . . 11  |-  ( u  =  <. s ,  t
>.  ->  ( ( z  =  [_ ( 1st `  u )  /  x ]_ X  /\  w  =  [_ ( 2nd `  u
)  /  y ]_ Y )  <->  ( z  =  [_ s  /  x ]_ X  /\  w  =  [_ t  /  y ]_ Y ) ) )
61 eqeq1 2410 . . . . . . . . . . 11  |-  ( u  =  <. s ,  t
>.  ->  ( u  =  v  <->  <. s ,  t
>.  =  v )
)
6260, 61bibi12d 313 . . . . . . . . . 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
) ) )
6362ralxp 4975 . . . . . . . . 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 1626 . . . . . . . . . 10  |-  F/ s A. y  e.  C  ( ( z  =  X  /\  w  =  Y )  <->  <. x ,  y >.  =  v
)
65 nfcv 2540 . . . . . . . . . . 11  |-  F/_ x C
66 nfcsb1v 3243 . . . . . . . . . . . . . 14  |-  F/_ x [_ s  /  x ]_ X
6766nfeq2 2551 . . . . . . . . . . . . 13  |-  F/ x  z  =  [_ s  /  x ]_ X
68 nfv 1626 . . . . . . . . . . . . 13  |-  F/ x  w  =  [_ t  / 
y ]_ Y
6967, 68nfan 1842 . . . . . . . . . . . 12  |-  F/ x
( z  =  [_ s  /  x ]_ X  /\  w  =  [_ t  /  y ]_ Y
)
70 nfv 1626 . . . . . . . . . . . 12  |-  F/ x <. s ,  t >.  =  v
7169, 70nfbi 1852 . . . . . . . . . . 11  |-  F/ x
( ( z  = 
[_ s  /  x ]_ X  /\  w  =  [_ t  /  y ]_ Y )  <->  <. s ,  t >.  =  v
)
7265, 71nfral 2719 . . . . . . . . . 10  |-  F/ x A. t  e.  C  ( ( z  = 
[_ s  /  x ]_ X  /\  w  =  [_ t  /  y ]_ Y )  <->  <. s ,  t >.  =  v
)
73 nfv 1626 . . . . . . . . . . . 12  |-  F/ t ( ( z  =  X  /\  w  =  Y )  <->  <. x ,  y >.  =  v
)
74 nfv 1626 . . . . . . . . . . . . . 14  |-  F/ y  z  =  X
75 nfcsb1v 3243 . . . . . . . . . . . . . . 15  |-  F/_ y [_ t  /  y ]_ Y
7675nfeq2 2551 . . . . . . . . . . . . . 14  |-  F/ y  w  =  [_ t  /  y ]_ Y
7774, 76nfan 1842 . . . . . . . . . . . . 13  |-  F/ y ( z  =  X  /\  w  =  [_ t  /  y ]_ Y
)
78 nfv 1626 . . . . . . . . . . . . 13  |-  F/ y
<. x ,  t >.  =  v
7977, 78nfbi 1852 . . . . . . . . . . . 12  |-  F/ y ( ( z  =  X  /\  w  = 
[_ t  /  y ]_ Y )  <->  <. x ,  t >.  =  v
)
80 csbeq1a 3219 . . . . . . . . . . . . . . 15  |-  ( y  =  t  ->  Y  =  [_ t  /  y ]_ Y )
8180eqeq2d 2415 . . . . . . . . . . . . . 14  |-  ( y  =  t  ->  (
w  =  Y  <->  w  =  [_ t  /  y ]_ Y ) )
8281anbi2d 685 . . . . . . . . . . . . 13  |-  ( y  =  t  ->  (
( z  =  X  /\  w  =  Y )  <->  ( z  =  X  /\  w  = 
[_ t  /  y ]_ Y ) ) )
83 opeq2 3945 . . . . . . . . . . . . . 14  |-  ( y  =  t  ->  <. x ,  y >.  =  <. x ,  t >. )
8483eqeq1d 2412 . . . . . . . . . . . . 13  |-  ( y  =  t  ->  ( <. x ,  y >.  =  v  <->  <. x ,  t
>.  =  v )
)
8582, 84bibi12d 313 . . . . . . . . . . . 12  |-  ( y  =  t  ->  (
( ( z  =  X  /\  w  =  Y )  <->  <. x ,  y >.  =  v
)  <->  ( ( z  =  X  /\  w  =  [_ t  /  y ]_ Y )  <->  <. x ,  t >.  =  v
) ) )
8673, 79, 85cbvral 2888 . . . . . . . . . . 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 3219 . . . . . . . . . . . . . . 15  |-  ( x  =  s  ->  X  =  [_ s  /  x ]_ X )
8887eqeq2d 2415 . . . . . . . . . . . . . 14  |-  ( x  =  s  ->  (
z  =  X  <->  z  =  [_ s  /  x ]_ X ) )
8988anbi1d 686 . . . . . . . . . . . . 13  |-  ( x  =  s  ->  (
( z  =  X  /\  w  =  [_ t  /  y ]_ Y
)  <->  ( z  = 
[_ s  /  x ]_ X  /\  w  =  [_ t  /  y ]_ Y ) ) )
90 opeq1 3944 . . . . . . . . . . . . . 14  |-  ( x  =  s  ->  <. x ,  t >.  =  <. s ,  t >. )
9190eqeq1d 2412 . . . . . . . . . . . . 13  |-  ( x  =  s  ->  ( <. x ,  t >.  =  v  <->  <. s ,  t
>.  =  v )
)
9289, 91bibi12d 313 . . . . . . . . . . . 12  |-  ( x  =  s  ->  (
( ( z  =  X  /\  w  = 
[_ t  /  y ]_ Y )  <->  <. x ,  t >.  =  v
)  <->  ( ( z  =  [_ s  /  x ]_ X  /\  w  =  [_ t  /  y ]_ Y )  <->  <. s ,  t >.  =  v
) ) )
9392ralbidv 2686 . . . . . . . . . . 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
) ) )
9486, 93syl5bb 249 . . . . . . . . . 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
) ) )
9564, 72, 94cbvral 2888 . . . . . . . . 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 )
)
9663, 95bitr4i 244 . . . . . . . 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 2413 . . . . . . . . . . 11  |-  ( v  =  <. s ,  t
>.  ->  ( <. x ,  y >.  =  v  <->  <. x ,  y >.  =  <. s ,  t
>. ) )
98 vex 2919 . . . . . . . . . . . 12  |-  x  e. 
_V
99 vex 2919 . . . . . . . . . . . 12  |-  y  e. 
_V
10098, 99opth 4395 . . . . . . . . . . 11  |-  ( <.
x ,  y >.  =  <. s ,  t
>. 
<->  ( x  =  s  /\  y  =  t ) )
10197, 100syl6bb 253 . . . . . . . . . 10  |-  ( v  =  <. s ,  t
>.  ->  ( <. x ,  y >.  =  v  <-> 
( x  =  s  /\  y  =  t ) ) )
102101bibi2d 310 . . . . . . . . 9  |-  ( v  =  <. s ,  t
>.  ->  ( ( ( z  =  X  /\  w  =  Y )  <->  <.
x ,  y >.  =  v )  <->  ( (
z  =  X  /\  w  =  Y )  <->  ( x  =  s  /\  y  =  t )
) ) )
1031022ralbidv 2708 . . . . . . . 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 )
) ) )
10496, 103syl5bb 249 . . . . . . 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 )
) ) )
105104rexxp 4976 . . . . . 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 )
) )
10651, 105sylibr 204 . . . . 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 3083 . . . . 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 ) )
108106, 107sylibr 204 . . . 4  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  D ) )  ->  E! u  e.  ( A  X.  C ) ( z  =  [_ ( 1st `  u )  /  x ]_ X  /\  w  =  [_ ( 2nd `  u
)  /  y ]_ Y ) )
109108ralrimivva 2758 . . 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 2410 . . . . . 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 2919 . . . . . . 7  |-  z  e. 
_V
112 vex 2919 . . . . . . 7  |-  w  e. 
_V
113111, 112opth 4395 . . . . . 6  |-  ( <.
z ,  w >.  = 
<. [_ ( 1st `  u
)  /  x ]_ X ,  [_ ( 2nd `  u )  /  y ]_ Y >.  <->  ( z  = 
[_ ( 1st `  u
)  /  x ]_ X  /\  w  =  [_ ( 2nd `  u )  /  y ]_ Y
) )
114110, 113syl6bb 253 . . . . 5  |-  ( v  =  <. z ,  w >.  ->  ( v  = 
<. [_ ( 1st `  u
)  /  x ]_ X ,  [_ ( 2nd `  u )  /  y ]_ Y >.  <->  ( z  = 
[_ ( 1st `  u
)  /  x ]_ X  /\  w  =  [_ ( 2nd `  u )  /  y ]_ Y
) ) )
115114reubidv 2852 . . . 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 ) ) )
116115ralxp 4975 . . 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 ) )
117109, 116sylibr 204 . 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 2540 . . . . 5  |-  F/_ z <. X ,  Y >.
119 nfcv 2540 . . . . 5  |-  F/_ w <. X ,  Y >.
120 nfcsb1v 3243 . . . . . 6  |-  F/_ x [_ z  /  x ]_ X
121 nfcv 2540 . . . . . 6  |-  F/_ x [_ w  /  y ]_ Y
122120, 121nfop 3960 . . . . 5  |-  F/_ x <. [_ z  /  x ]_ X ,  [_ w  /  y ]_ Y >.
123 nfcv 2540 . . . . . 6  |-  F/_ y [_ z  /  x ]_ X
124 nfcsb1v 3243 . . . . . 6  |-  F/_ y [_ w  /  y ]_ Y
125123, 124nfop 3960 . . . . 5  |-  F/_ y <. [_ z  /  x ]_ X ,  [_ w  /  y ]_ Y >.
126 csbeq1a 3219 . . . . . 6  |-  ( x  =  z  ->  X  =  [_ z  /  x ]_ X )
127 csbeq1a 3219 . . . . . 6  |-  ( y  =  w  ->  Y  =  [_ w  /  y ]_ Y )
128 opeq12 3946 . . . . . 6  |-  ( ( X  =  [_ z  /  x ]_ X  /\  Y  =  [_ w  / 
y ]_ Y )  ->  <. X ,  Y >.  = 
<. [_ z  /  x ]_ X ,  [_ w  /  y ]_ Y >. )
129126, 127, 128syl2an 464 . . . . 5  |-  ( ( x  =  z  /\  y  =  w )  -> 
<. X ,  Y >.  = 
<. [_ z  /  x ]_ X ,  [_ w  /  y ]_ Y >. )
130118, 119, 122, 125, 129cbvmpt2 6110 . . . 4  |-  ( x  e.  A ,  y  e.  C  |->  <. X ,  Y >. )  =  ( z  e.  A ,  w  e.  C  |->  <. [_ z  /  x ]_ X ,  [_ w  /  y ]_ Y >. )
131111, 112op1std 6316 . . . . . . 7  |-  ( u  =  <. z ,  w >.  ->  ( 1st `  u
)  =  z )
132131csbeq1d 3217 . . . . . 6  |-  ( u  =  <. z ,  w >.  ->  [_ ( 1st `  u
)  /  x ]_ X  =  [_ z  /  x ]_ X )
133111, 112op2ndd 6317 . . . . . . 7  |-  ( u  =  <. z ,  w >.  ->  ( 2nd `  u
)  =  w )
134133csbeq1d 3217 . . . . . 6  |-  ( u  =  <. z ,  w >.  ->  [_ ( 2nd `  u
)  /  y ]_ Y  =  [_ w  / 
y ]_ Y )
135132, 134opeq12d 3952 . . . . 5  |-  ( u  =  <. z ,  w >.  ->  <. [_ ( 1st `  u
)  /  x ]_ X ,  [_ ( 2nd `  u )  /  y ]_ Y >.  =  <. [_ z  /  x ]_ X ,  [_ w  / 
y ]_ Y >. )
136135mpt2mpt 6124 . . . 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 >. )
137130, 136eqtr4i 2427 . . 3  |-  ( x  e.  A ,  y  e.  C  |->  <. X ,  Y >. )  =  ( u  e.  ( A  X.  C )  |->  <. [_ ( 1st `  u
)  /  x ]_ X ,  [_ ( 2nd `  u )  /  y ]_ Y >. )
138137f1ompt 5850 . 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 >. ) )
13931, 117, 138sylanbrc 646 1  |-  ( ph  ->  ( x  e.  A ,  y  e.  C  |-> 
<. X ,  Y >. ) : ( A  X.  C ) -1-1-onto-> ( B  X.  D
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2666   E.wrex 2667   E!wreu 2668   [_csb 3211   <.cop 3777    e. cmpt 4226    X. cxp 4835   -1-1-onto->wf1o 5412   ` cfv 5413    e. cmpt2 6042   1stc1st 6306   2ndc2nd 6307
This theorem is referenced by:  infxpenc  7855  pwfseqlem5  8494
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309
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