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Theorem opeliunxp 4522
Description: Membership in a union of cross products. (Contributed by Mario Carneiro, 29-Dec-2014.) (Revised by Mario Carneiro, 1-Jan-2017.)
Assertion
Ref Expression
opeliunxp  |-  ( <.
x ,  C >.  e. 
U_ x  e.  A  ( { x }  X.  B )  <->  ( x  e.  A  /\  C  e.  B ) )

Proof of Theorem opeliunxp
Dummy variables  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 2644 . 2  |-  ( <.
x ,  C >.  e. 
U_ x  e.  A  ( { x }  X.  B )  ->  <. x ,  C >.  e.  _V )
2 opexg 4079 . 2  |-  ( ( x  e.  A  /\  C  e.  B )  -> 
<. x ,  C >.  e. 
_V )
3 df-rex 2376 . . . . . 6  |-  ( E. x  e.  A  y  e.  ( { x }  X.  B )  <->  E. x
( x  e.  A  /\  y  e.  ( { x }  X.  B ) ) )
4 nfv 1473 . . . . . . 7  |-  F/ z ( x  e.  A  /\  y  e.  ( { x }  X.  B ) )
5 nfs1v 1870 . . . . . . . 8  |-  F/ x [ z  /  x ] x  e.  A
6 nfcv 2235 . . . . . . . . . 10  |-  F/_ x { z }
7 nfcsb1v 2977 . . . . . . . . . 10  |-  F/_ x [_ z  /  x ]_ B
86, 7nfxp 4494 . . . . . . . . 9  |-  F/_ x
( { z }  X.  [_ z  /  x ]_ B )
98nfcri 2229 . . . . . . . 8  |-  F/ x  y  e.  ( {
z }  X.  [_ z  /  x ]_ B
)
105, 9nfan 1509 . . . . . . 7  |-  F/ x
( [ z  /  x ] x  e.  A  /\  y  e.  ( { z }  X.  [_ z  /  x ]_ B ) )
11 sbequ12 1708 . . . . . . . 8  |-  ( x  =  z  ->  (
x  e.  A  <->  [ z  /  x ] x  e.  A ) )
12 sneq 3477 . . . . . . . . . 10  |-  ( x  =  z  ->  { x }  =  { z } )
13 csbeq1a 2955 . . . . . . . . . 10  |-  ( x  =  z  ->  B  =  [_ z  /  x ]_ B )
1412, 13xpeq12d 4492 . . . . . . . . 9  |-  ( x  =  z  ->  ( { x }  X.  B )  =  ( { z }  X.  [_ z  /  x ]_ B ) )
1514eleq2d 2164 . . . . . . . 8  |-  ( x  =  z  ->  (
y  e.  ( { x }  X.  B
)  <->  y  e.  ( { z }  X.  [_ z  /  x ]_ B ) ) )
1611, 15anbi12d 458 . . . . . . 7  |-  ( x  =  z  ->  (
( x  e.  A  /\  y  e.  ( { x }  X.  B ) )  <->  ( [
z  /  x ]
x  e.  A  /\  y  e.  ( {
z }  X.  [_ z  /  x ]_ B
) ) ) )
174, 10, 16cbvex 1693 . . . . . 6  |-  ( E. x ( x  e.  A  /\  y  e.  ( { x }  X.  B ) )  <->  E. z
( [ z  /  x ] x  e.  A  /\  y  e.  ( { z }  X.  [_ z  /  x ]_ B ) ) )
183, 17bitri 183 . . . . 5  |-  ( E. x  e.  A  y  e.  ( { x }  X.  B )  <->  E. z
( [ z  /  x ] x  e.  A  /\  y  e.  ( { z }  X.  [_ z  /  x ]_ B ) ) )
19 eleq1 2157 . . . . . . 7  |-  ( y  =  <. x ,  C >.  ->  ( y  e.  ( { z }  X.  [_ z  /  x ]_ B )  <->  <. x ,  C >.  e.  ( { z }  X.  [_ z  /  x ]_ B ) ) )
2019anbi2d 453 . . . . . 6  |-  ( y  =  <. x ,  C >.  ->  ( ( [ z  /  x ]
x  e.  A  /\  y  e.  ( {
z }  X.  [_ z  /  x ]_ B
) )  <->  ( [
z  /  x ]
x  e.  A  /\  <.
x ,  C >.  e.  ( { z }  X.  [_ z  /  x ]_ B ) ) ) )
2120exbidv 1760 . . . . 5  |-  ( y  =  <. x ,  C >.  ->  ( E. z
( [ z  /  x ] x  e.  A  /\  y  e.  ( { z }  X.  [_ z  /  x ]_ B ) )  <->  E. z
( [ z  /  x ] x  e.  A  /\  <. x ,  C >.  e.  ( { z }  X.  [_ z  /  x ]_ B ) ) ) )
2218, 21syl5bb 191 . . . 4  |-  ( y  =  <. x ,  C >.  ->  ( E. x  e.  A  y  e.  ( { x }  X.  B )  <->  E. z
( [ z  /  x ] x  e.  A  /\  <. x ,  C >.  e.  ( { z }  X.  [_ z  /  x ]_ B ) ) ) )
23 df-iun 3754 . . . 4  |-  U_ x  e.  A  ( {
x }  X.  B
)  =  { y  |  E. x  e.  A  y  e.  ( { x }  X.  B ) }
2422, 23elab2g 2776 . . 3  |-  ( <.
x ,  C >.  e. 
_V  ->  ( <. x ,  C >.  e.  U_ x  e.  A  ( {
x }  X.  B
)  <->  E. z ( [ z  /  x ]
x  e.  A  /\  <.
x ,  C >.  e.  ( { z }  X.  [_ z  /  x ]_ B ) ) ) )
25 opelxp 4497 . . . . . . 7  |-  ( <.
x ,  C >.  e.  ( { z }  X.  [_ z  /  x ]_ B )  <->  ( x  e.  { z }  /\  C  e.  [_ z  /  x ]_ B ) )
2625anbi2i 446 . . . . . 6  |-  ( ( [ z  /  x ] x  e.  A  /\  <. x ,  C >.  e.  ( { z }  X.  [_ z  /  x ]_ B ) )  <->  ( [ z  /  x ] x  e.  A  /\  (
x  e.  { z }  /\  C  e. 
[_ z  /  x ]_ B ) ) )
27 an12 529 . . . . . 6  |-  ( ( [ z  /  x ] x  e.  A  /\  ( x  e.  {
z }  /\  C  e.  [_ z  /  x ]_ B ) )  <->  ( x  e.  { z }  /\  ( [ z  /  x ] x  e.  A  /\  C  e.  [_ z  /  x ]_ B ) ) )
28 velsn 3483 . . . . . . . 8  |-  ( x  e.  { z }  <-> 
x  =  z )
29 equcom 1646 . . . . . . . 8  |-  ( x  =  z  <->  z  =  x )
3028, 29bitri 183 . . . . . . 7  |-  ( x  e.  { z }  <-> 
z  =  x )
3130anbi1i 447 . . . . . 6  |-  ( ( x  e.  { z }  /\  ( [ z  /  x ]
x  e.  A  /\  C  e.  [_ z  /  x ]_ B ) )  <-> 
( z  =  x  /\  ( [ z  /  x ] x  e.  A  /\  C  e. 
[_ z  /  x ]_ B ) ) )
3226, 27, 313bitri 205 . . . . 5  |-  ( ( [ z  /  x ] x  e.  A  /\  <. x ,  C >.  e.  ( { z }  X.  [_ z  /  x ]_ B ) )  <->  ( z  =  x  /\  ( [ z  /  x ]
x  e.  A  /\  C  e.  [_ z  /  x ]_ B ) ) )
3332exbii 1548 . . . 4  |-  ( E. z ( [ z  /  x ] x  e.  A  /\  <. x ,  C >.  e.  ( { z }  X.  [_ z  /  x ]_ B ) )  <->  E. z
( z  =  x  /\  ( [ z  /  x ] x  e.  A  /\  C  e. 
[_ z  /  x ]_ B ) ) )
34 vex 2636 . . . . 5  |-  x  e. 
_V
35 sbequ12r 1709 . . . . . 6  |-  ( z  =  x  ->  ( [ z  /  x ] x  e.  A  <->  x  e.  A ) )
3613equcoms 1648 . . . . . . . 8  |-  ( z  =  x  ->  B  =  [_ z  /  x ]_ B )
3736eqcomd 2100 . . . . . . 7  |-  ( z  =  x  ->  [_ z  /  x ]_ B  =  B )
3837eleq2d 2164 . . . . . 6  |-  ( z  =  x  ->  ( C  e.  [_ z  /  x ]_ B  <->  C  e.  B ) )
3935, 38anbi12d 458 . . . . 5  |-  ( z  =  x  ->  (
( [ z  /  x ] x  e.  A  /\  C  e.  [_ z  /  x ]_ B )  <-> 
( x  e.  A  /\  C  e.  B
) ) )
4034, 39ceqsexv 2672 . . . 4  |-  ( E. z ( z  =  x  /\  ( [ z  /  x ]
x  e.  A  /\  C  e.  [_ z  /  x ]_ B ) )  <-> 
( x  e.  A  /\  C  e.  B
) )
4133, 40bitri 183 . . 3  |-  ( E. z ( [ z  /  x ] x  e.  A  /\  <. x ,  C >.  e.  ( { z }  X.  [_ z  /  x ]_ B ) )  <->  ( x  e.  A  /\  C  e.  B ) )
4224, 41syl6bb 195 . 2  |-  ( <.
x ,  C >.  e. 
_V  ->  ( <. x ,  C >.  e.  U_ x  e.  A  ( {
x }  X.  B
)  <->  ( x  e.  A  /\  C  e.  B ) ) )
431, 2, 42pm5.21nii 658 1  |-  ( <.
x ,  C >.  e. 
U_ x  e.  A  ( { x }  X.  B )  <->  ( x  e.  A  /\  C  e.  B ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    = wceq 1296   E.wex 1433    e. wcel 1445   [wsb 1699   E.wrex 2371   _Vcvv 2633   [_csb 2947   {csn 3466   <.cop 3469   U_ciun 3752    X. cxp 4465
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-sbc 2855  df-csb 2948  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-iun 3754  df-opab 3922  df-xp 4473
This theorem is referenced by:  eliunxp  4606  opeliunxp2  4607  opeliunxp2f  6041
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