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Theorem xpf1o 7039
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 6165 . . . . . 6  |-  ( u  e.  ( A  X.  C )  ->  ( 1st `  u )  e.  A )
21adantl 452 . . . . 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 2296 . . . . . . . . 9  |-  ( x  e.  A  |->  X )  =  ( x  e.  A  |->  X )
54f1ompt 5698 . . . . . . . 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 188 . . . . . . 7  |-  ( ph  ->  ( A. x  e.  A  X  e.  B  /\  A. z  e.  B  E! x  e.  A  z  =  X )
)
76simpld 445 . . . . . 6  |-  ( ph  ->  A. x  e.  A  X  e.  B )
87adantr 451 . . . . 5  |-  ( (
ph  /\  u  e.  ( A  X.  C
) )  ->  A. x  e.  A  X  e.  B )
9 nfcsb1v 3126 . . . . . . 7  |-  F/_ x [_ ( 1st `  u
)  /  x ]_ X
109nfel1 2442 . . . . . 6  |-  F/ x [_ ( 1st `  u
)  /  x ]_ X  e.  B
11 csbeq1a 3102 . . . . . . 7  |-  ( x  =  ( 1st `  u
)  ->  X  =  [_ ( 1st `  u
)  /  x ]_ X )
1211eleq1d 2362 . . . . . 6  |-  ( x  =  ( 1st `  u
)  ->  ( X  e.  B  <->  [_ ( 1st `  u
)  /  x ]_ X  e.  B )
)
1310, 12rspc 2891 . . . . 5  |-  ( ( 1st `  u )  e.  A  ->  ( A. x  e.  A  X  e.  B  ->  [_ ( 1st `  u
)  /  x ]_ X  e.  B )
)
142, 8, 13sylc 56 . . . 4  |-  ( (
ph  /\  u  e.  ( A  X.  C
) )  ->  [_ ( 1st `  u )  /  x ]_ X  e.  B
)
15 xp2nd 6166 . . . . . 6  |-  ( u  e.  ( A  X.  C )  ->  ( 2nd `  u )  e.  C )
1615adantl 452 . . . . 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 2296 . . . . . . . . 9  |-  ( y  e.  C  |->  Y )  =  ( y  e.  C  |->  Y )
1918f1ompt 5698 . . . . . . . 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 188 . . . . . . 7  |-  ( ph  ->  ( A. y  e.  C  Y  e.  D  /\  A. w  e.  D  E! y  e.  C  w  =  Y )
)
2120simpld 445 . . . . . 6  |-  ( ph  ->  A. y  e.  C  Y  e.  D )
2221adantr 451 . . . . 5  |-  ( (
ph  /\  u  e.  ( A  X.  C
) )  ->  A. y  e.  C  Y  e.  D )
23 nfcsb1v 3126 . . . . . . 7  |-  F/_ y [_ ( 2nd `  u
)  /  y ]_ Y
2423nfel1 2442 . . . . . 6  |-  F/ y
[_ ( 2nd `  u
)  /  y ]_ Y  e.  D
25 csbeq1a 3102 . . . . . . 7  |-  ( y  =  ( 2nd `  u
)  ->  Y  =  [_ ( 2nd `  u
)  /  y ]_ Y )
2625eleq1d 2362 . . . . . 6  |-  ( y  =  ( 2nd `  u
)  ->  ( Y  e.  D  <->  [_ ( 2nd `  u
)  /  y ]_ Y  e.  D )
)
2724, 26rspc 2891 . . . . 5  |-  ( ( 2nd `  u )  e.  C  ->  ( A. y  e.  C  Y  e.  D  ->  [_ ( 2nd `  u
)  /  y ]_ Y  e.  D )
)
2816, 22, 27sylc 56 . . . 4  |-  ( (
ph  /\  u  e.  ( A  X.  C
) )  ->  [_ ( 2nd `  u )  / 
y ]_ Y  e.  D
)
29 opelxpi 4737 . . . 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 642 . . 3  |-  ( (
ph  /\  u  e.  ( A  X.  C
) )  ->  <. [_ ( 1st `  u )  /  x ]_ X ,  [_ ( 2nd `  u )  /  y ]_ Y >.  e.  ( B  X.  D ) )
3130ralrimiva 2639 . 2  |-  ( ph  ->  A. u  e.  ( A  X.  C )
<. [_ ( 1st `  u
)  /  x ]_ X ,  [_ ( 2nd `  u )  /  y ]_ Y >.  e.  ( B  X.  D ) )
326simprd 449 . . . . . . . . . 10  |-  ( ph  ->  A. z  e.  B  E! x  e.  A  z  =  X )
3332r19.21bi 2654 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  B )  ->  E! x  e.  A  z  =  X )
34 reu6 2967 . . . . . . . . 9  |-  ( E! x  e.  A  z  =  X  <->  E. s  e.  A  A. x  e.  A  ( z  =  X  <->  x  =  s
) )
3533, 34sylib 188 . . . . . . . 8  |-  ( (
ph  /\  z  e.  B )  ->  E. s  e.  A  A. x  e.  A  ( z  =  X  <->  x  =  s
) )
3620simprd 449 . . . . . . . . . 10  |-  ( ph  ->  A. w  e.  D  E! y  e.  C  w  =  Y )
3736r19.21bi 2654 . . . . . . . . 9  |-  ( (
ph  /\  w  e.  D )  ->  E! y  e.  C  w  =  Y )
38 reu6 2967 . . . . . . . . 9  |-  ( E! y  e.  C  w  =  Y  <->  E. t  e.  C  A. y  e.  C  ( w  =  Y  <->  y  =  t ) )
3937, 38sylib 188 . . . . . . . 8  |-  ( (
ph  /\  w  e.  D )  ->  E. t  e.  C  A. y  e.  C  ( w  =  Y  <->  y  =  t ) )
4035, 39anim12dan 810 . . . . . . 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 2720 . . . . . . . 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 pm4.38 842 . . . . . . . . . . . . . . 15  |-  ( ( ( z  =  X  <-> 
x  =  s )  /\  ( w  =  Y  <->  y  =  t ) )  ->  (
( z  =  X  /\  w  =  Y )  <->  ( x  =  s  /\  y  =  t ) ) )
4342ex 423 . . . . . . . . . . . . . 14  |-  ( ( z  =  X  <->  x  =  s )  ->  (
( w  =  Y  <-> 
y  =  t )  ->  ( ( z  =  X  /\  w  =  Y )  <->  ( x  =  s  /\  y  =  t ) ) ) )
4443ralimdv 2635 . . . . . . . . . . . . 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 27 . . . . . . . . . . . 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 2635 . . . . . . . . . . 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 419 . . . . . . . . . 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 2663 . . . . . . . . 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 2663 . . . . . . . 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 204 . . . . . . 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 15 . . . . . 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 2804 . . . . . . . . . . . . . . 15  |-  s  e. 
_V
53 vex 2804 . . . . . . . . . . . . . . 15  |-  t  e. 
_V
5452, 53op1std 6146 . . . . . . . . . . . . . 14  |-  ( u  =  <. s ,  t
>.  ->  ( 1st `  u
)  =  s )
5554csbeq1d 3100 . . . . . . . . . . . . 13  |-  ( u  =  <. s ,  t
>.  ->  [_ ( 1st `  u
)  /  x ]_ X  =  [_ s  /  x ]_ X )
5655eqeq2d 2307 . . . . . . . . . . . 12  |-  ( u  =  <. s ,  t
>.  ->  ( z  = 
[_ ( 1st `  u
)  /  x ]_ X 
<->  z  =  [_ s  /  x ]_ X ) )
5752, 53op2ndd 6147 . . . . . . . . . . . . . 14  |-  ( u  =  <. s ,  t
>.  ->  ( 2nd `  u
)  =  t )
5857csbeq1d 3100 . . . . . . . . . . . . 13  |-  ( u  =  <. s ,  t
>.  ->  [_ ( 2nd `  u
)  /  y ]_ Y  =  [_ t  / 
y ]_ Y )
5958eqeq2d 2307 . . . . . . . . . . . 12  |-  ( u  =  <. s ,  t
>.  ->  ( w  = 
[_ ( 2nd `  u
)  /  y ]_ Y 
<->  w  =  [_ t  /  y ]_ Y
) )
6056, 59anbi12d 691 . . . . . . . . . . 11  |-  ( u  =  <. s ,  t
>.  ->  ( ( z  =  [_ ( 1st `  u )  /  x ]_ X  /\  w  =  [_ ( 2nd `  u
)  /  y ]_ Y )  <->  ( z  =  [_ s  /  x ]_ X  /\  w  =  [_ t  /  y ]_ Y ) ) )
61 eqeq1 2302 . . . . . . . . . . 11  |-  ( u  =  <. s ,  t
>.  ->  ( u  =  v  <->  <. s ,  t
>.  =  v )
)
6260, 61bibi12d 312 . . . . . . . . . 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 4843 . . . . . . . . 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 1609 . . . . . . . . . 10  |-  F/ s A. y  e.  C  ( ( z  =  X  /\  w  =  Y )  <->  <. x ,  y >.  =  v
)
65 nfcv 2432 . . . . . . . . . . 11  |-  F/_ x C
66 nfcsb1v 3126 . . . . . . . . . . . . . 14  |-  F/_ x [_ s  /  x ]_ X
6766nfeq2 2443 . . . . . . . . . . . . 13  |-  F/ x  z  =  [_ s  /  x ]_ X
68 nfv 1609 . . . . . . . . . . . . 13  |-  F/ x  w  =  [_ t  / 
y ]_ Y
6967, 68nfan 1783 . . . . . . . . . . . 12  |-  F/ x
( z  =  [_ s  /  x ]_ X  /\  w  =  [_ t  /  y ]_ Y
)
70 nfv 1609 . . . . . . . . . . . 12  |-  F/ x <. s ,  t >.  =  v
7169, 70nfbi 1784 . . . . . . . . . . 11  |-  F/ x
( ( z  = 
[_ s  /  x ]_ X  /\  w  =  [_ t  /  y ]_ Y )  <->  <. s ,  t >.  =  v
)
7265, 71nfral 2609 . . . . . . . . . 10  |-  F/ x A. t  e.  C  ( ( z  = 
[_ s  /  x ]_ X  /\  w  =  [_ t  /  y ]_ Y )  <->  <. s ,  t >.  =  v
)
73 nfv 1609 . . . . . . . . . . . 12  |-  F/ t ( ( z  =  X  /\  w  =  Y )  <->  <. x ,  y >.  =  v
)
74 nfv 1609 . . . . . . . . . . . . . 14  |-  F/ y  z  =  X
75 nfcsb1v 3126 . . . . . . . . . . . . . . 15  |-  F/_ y [_ t  /  y ]_ Y
7675nfeq2 2443 . . . . . . . . . . . . . 14  |-  F/ y  w  =  [_ t  /  y ]_ Y
7774, 76nfan 1783 . . . . . . . . . . . . 13  |-  F/ y ( z  =  X  /\  w  =  [_ t  /  y ]_ Y
)
78 nfv 1609 . . . . . . . . . . . . 13  |-  F/ y
<. x ,  t >.  =  v
7977, 78nfbi 1784 . . . . . . . . . . . 12  |-  F/ y ( ( z  =  X  /\  w  = 
[_ t  /  y ]_ Y )  <->  <. x ,  t >.  =  v
)
80 csbeq1a 3102 . . . . . . . . . . . . . . 15  |-  ( y  =  t  ->  Y  =  [_ t  /  y ]_ Y )
8180eqeq2d 2307 . . . . . . . . . . . . . 14  |-  ( y  =  t  ->  (
w  =  Y  <->  w  =  [_ t  /  y ]_ Y ) )
8281anbi2d 684 . . . . . . . . . . . . 13  |-  ( y  =  t  ->  (
( z  =  X  /\  w  =  Y )  <->  ( z  =  X  /\  w  = 
[_ t  /  y ]_ Y ) ) )
83 opeq2 3813 . . . . . . . . . . . . . 14  |-  ( y  =  t  ->  <. x ,  y >.  =  <. x ,  t >. )
8483eqeq1d 2304 . . . . . . . . . . . . 13  |-  ( y  =  t  ->  ( <. x ,  y >.  =  v  <->  <. x ,  t
>.  =  v )
)
8582, 84bibi12d 312 . . . . . . . . . . . 12  |-  ( y  =  t  ->  (
( ( z  =  X  /\  w  =  Y )  <->  <. x ,  y >.  =  v
)  <->  ( ( z  =  X  /\  w  =  [_ t  /  y ]_ Y )  <->  <. x ,  t >.  =  v
) ) )
8673, 79, 85cbvral 2773 . . . . . . . . . . 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 3102 . . . . . . . . . . . . . . 15  |-  ( x  =  s  ->  X  =  [_ s  /  x ]_ X )
8887eqeq2d 2307 . . . . . . . . . . . . . 14  |-  ( x  =  s  ->  (
z  =  X  <->  z  =  [_ s  /  x ]_ X ) )
8988anbi1d 685 . . . . . . . . . . . . 13  |-  ( x  =  s  ->  (
( z  =  X  /\  w  =  [_ t  /  y ]_ Y
)  <->  ( z  = 
[_ s  /  x ]_ X  /\  w  =  [_ t  /  y ]_ Y ) ) )
90 opeq1 3812 . . . . . . . . . . . . . 14  |-  ( x  =  s  ->  <. x ,  t >.  =  <. s ,  t >. )
9190eqeq1d 2304 . . . . . . . . . . . . 13  |-  ( x  =  s  ->  ( <. x ,  t >.  =  v  <->  <. s ,  t
>.  =  v )
)
9289, 91bibi12d 312 . . . . . . . . . . . 12  |-  ( x  =  s  ->  (
( ( z  =  X  /\  w  = 
[_ t  /  y ]_ Y )  <->  <. x ,  t >.  =  v
)  <->  ( ( z  =  [_ s  /  x ]_ X  /\  w  =  [_ t  /  y ]_ Y )  <->  <. s ,  t >.  =  v
) ) )
9392ralbidv 2576 . . . . . . . . . . 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 248 . . . . . . . . . 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 2773 . . . . . . . . 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 243 . . . . . . . 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 2305 . . . . . . . . . . 11  |-  ( v  =  <. s ,  t
>.  ->  ( <. x ,  y >.  =  v  <->  <. x ,  y >.  =  <. s ,  t
>. ) )
98 vex 2804 . . . . . . . . . . . 12  |-  x  e. 
_V
99 vex 2804 . . . . . . . . . . . 12  |-  y  e. 
_V
10098, 99opth 4261 . . . . . . . . . . 11  |-  ( <.
x ,  y >.  =  <. s ,  t
>. 
<->  ( x  =  s  /\  y  =  t ) )
10197, 100syl6bb 252 . . . . . . . . . 10  |-  ( v  =  <. s ,  t
>.  ->  ( <. x ,  y >.  =  v  <-> 
( x  =  s  /\  y  =  t ) ) )
102101bibi2d 309 . . . . . . . . 9  |-  ( v  =  <. s ,  t
>.  ->  ( ( ( z  =  X  /\  w  =  Y )  <->  <.
x ,  y >.  =  v )  <->  ( (
z  =  X  /\  w  =  Y )  <->  ( x  =  s  /\  y  =  t )
) ) )
1031022ralbidv 2598 . . . . . . . 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 248 . . . . . . 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 4844 . . . . . 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 203 . . . . 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 2967 . . . . 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 203 . . . 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 2648 . . 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 2302 . . . . . 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 2804 . . . . . . 7  |-  z  e. 
_V
112 vex 2804 . . . . . . 7  |-  w  e. 
_V
113111, 112opth 4261 . . . . . 6  |-  ( <.
z ,  w >.  = 
<. [_ ( 1st `  u
)  /  x ]_ X ,  [_ ( 2nd `  u )  /  y ]_ Y >.  <->  ( z  = 
[_ ( 1st `  u
)  /  x ]_ X  /\  w  =  [_ ( 2nd `  u )  /  y ]_ Y
) )
114110, 113syl6bb 252 . . . . 5  |-  ( v  =  <. z ,  w >.  ->  ( v  = 
<. [_ ( 1st `  u
)  /  x ]_ X ,  [_ ( 2nd `  u )  /  y ]_ Y >.  <->  ( z  = 
[_ ( 1st `  u
)  /  x ]_ X  /\  w  =  [_ ( 2nd `  u )  /  y ]_ Y
) ) )
115114reubidv 2737 . . . 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 4843 . . 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 203 . 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 2432 . . . . 5  |-  F/_ z <. X ,  Y >.
119 nfcv 2432 . . . . 5  |-  F/_ w <. X ,  Y >.
120 nfcsb1v 3126 . . . . . 6  |-  F/_ x [_ z  /  x ]_ X
121 nfcv 2432 . . . . . 6  |-  F/_ x [_ w  /  y ]_ Y
122120, 121nfop 3828 . . . . 5  |-  F/_ x <. [_ z  /  x ]_ X ,  [_ w  /  y ]_ Y >.
123 nfcv 2432 . . . . . 6  |-  F/_ y [_ z  /  x ]_ X
124 nfcsb1v 3126 . . . . . 6  |-  F/_ y [_ w  /  y ]_ Y
125123, 124nfop 3828 . . . . 5  |-  F/_ y <. [_ z  /  x ]_ X ,  [_ w  /  y ]_ Y >.
126 csbeq1a 3102 . . . . . 6  |-  ( x  =  z  ->  X  =  [_ z  /  x ]_ X )
127 csbeq1a 3102 . . . . . 6  |-  ( y  =  w  ->  Y  =  [_ w  /  y ]_ Y )
128 opeq12 3814 . . . . . 6  |-  ( ( X  =  [_ z  /  x ]_ X  /\  Y  =  [_ w  / 
y ]_ Y )  ->  <. X ,  Y >.  = 
<. [_ z  /  x ]_ X ,  [_ w  /  y ]_ Y >. )
129126, 127, 128syl2an 463 . . . . 5  |-  ( ( x  =  z  /\  y  =  w )  -> 
<. X ,  Y >.  = 
<. [_ z  /  x ]_ X ,  [_ w  /  y ]_ Y >. )
130118, 119, 122, 125, 129cbvmpt2 5941 . . . 4  |-  ( x  e.  A ,  y  e.  C  |->  <. X ,  Y >. )  =  ( z  e.  A ,  w  e.  C  |->  <. [_ z  /  x ]_ X ,  [_ w  /  y ]_ Y >. )
131111, 112op1std 6146 . . . . . . 7  |-  ( u  =  <. z ,  w >.  ->  ( 1st `  u
)  =  z )
132131csbeq1d 3100 . . . . . 6  |-  ( u  =  <. z ,  w >.  ->  [_ ( 1st `  u
)  /  x ]_ X  =  [_ z  /  x ]_ X )
133111, 112op2ndd 6147 . . . . . . 7  |-  ( u  =  <. z ,  w >.  ->  ( 2nd `  u
)  =  w )
134133csbeq1d 3100 . . . . . 6  |-  ( u  =  <. z ,  w >.  ->  [_ ( 2nd `  u
)  /  y ]_ Y  =  [_ w  / 
y ]_ Y )
135132, 134opeq12d 3820 . . . . 5  |-  ( u  =  <. z ,  w >.  ->  <. [_ ( 1st `  u
)  /  x ]_ X ,  [_ ( 2nd `  u )  /  y ]_ Y >.  =  <. [_ z  /  x ]_ X ,  [_ w  / 
y ]_ Y >. )
136135mpt2mpt 5955 . . . 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 2319 . . 3  |-  ( x  e.  A ,  y  e.  C  |->  <. X ,  Y >. )  =  ( u  e.  ( A  X.  C )  |->  <. [_ ( 1st `  u
)  /  x ]_ X ,  [_ ( 2nd `  u )  /  y ]_ Y >. )
138137f1ompt 5698 . 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 645 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 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   E!wreu 2558   [_csb 3094   <.cop 3656    e. cmpt 4093    X. cxp 4703   -1-1-onto->wf1o 5270   ` cfv 5271    e. cmpt2 5876   1stc1st 6136   2ndc2nd 6137
This theorem is referenced by:  infxpenc  7661  pwfseqlem5  8301
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139
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