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Theorem opeliunxp 4594
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 2697 . 2  |-  ( <.
x ,  C >.  e. 
U_ x  e.  A  ( { x }  X.  B )  ->  <. x ,  C >.  e.  _V )
2 opexg 4150 . 2  |-  ( ( x  e.  A  /\  C  e.  B )  -> 
<. x ,  C >.  e. 
_V )
3 df-rex 2422 . . . . . 6  |-  ( E. x  e.  A  y  e.  ( { x }  X.  B )  <->  E. x
( x  e.  A  /\  y  e.  ( { x }  X.  B ) ) )
4 nfv 1508 . . . . . . 7  |-  F/ z ( x  e.  A  /\  y  e.  ( { x }  X.  B ) )
5 nfs1v 1912 . . . . . . . 8  |-  F/ x [ z  /  x ] x  e.  A
6 nfcv 2281 . . . . . . . . . 10  |-  F/_ x { z }
7 nfcsb1v 3035 . . . . . . . . . 10  |-  F/_ x [_ z  /  x ]_ B
86, 7nfxp 4566 . . . . . . . . 9  |-  F/_ x
( { z }  X.  [_ z  /  x ]_ B )
98nfcri 2275 . . . . . . . 8  |-  F/ x  y  e.  ( {
z }  X.  [_ z  /  x ]_ B
)
105, 9nfan 1544 . . . . . . 7  |-  F/ x
( [ z  /  x ] x  e.  A  /\  y  e.  ( { z }  X.  [_ z  /  x ]_ B ) )
11 sbequ12 1744 . . . . . . . 8  |-  ( x  =  z  ->  (
x  e.  A  <->  [ z  /  x ] x  e.  A ) )
12 sneq 3538 . . . . . . . . . 10  |-  ( x  =  z  ->  { x }  =  { z } )
13 csbeq1a 3012 . . . . . . . . . 10  |-  ( x  =  z  ->  B  =  [_ z  /  x ]_ B )
1412, 13xpeq12d 4564 . . . . . . . . 9  |-  ( x  =  z  ->  ( { x }  X.  B )  =  ( { z }  X.  [_ z  /  x ]_ B ) )
1514eleq2d 2209 . . . . . . . 8  |-  ( x  =  z  ->  (
y  e.  ( { x }  X.  B
)  <->  y  e.  ( { z }  X.  [_ z  /  x ]_ B ) ) )
1611, 15anbi12d 464 . . . . . . 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 1729 . . . . . 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 2202 . . . . . . 7  |-  ( y  =  <. x ,  C >.  ->  ( y  e.  ( { z }  X.  [_ z  /  x ]_ B )  <->  <. x ,  C >.  e.  ( { z }  X.  [_ z  /  x ]_ B ) ) )
2019anbi2d 459 . . . . . 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 1797 . . . . 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 3815 . . . 4  |-  U_ x  e.  A  ( {
x }  X.  B
)  =  { y  |  E. x  e.  A  y  e.  ( { x }  X.  B ) }
2422, 23elab2g 2831 . . 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 4569 . . . . . . 7  |-  ( <.
x ,  C >.  e.  ( { z }  X.  [_ z  /  x ]_ B )  <->  ( x  e.  { z }  /\  C  e.  [_ z  /  x ]_ B ) )
2625anbi2i 452 . . . . . 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 550 . . . . . 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 3544 . . . . . . . 8  |-  ( x  e.  { z }  <-> 
x  =  z )
29 equcom 1682 . . . . . . . 8  |-  ( x  =  z  <->  z  =  x )
3028, 29bitri 183 . . . . . . 7  |-  ( x  e.  { z }  <-> 
z  =  x )
3130anbi1i 453 . . . . . 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 1584 . . . 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 2689 . . . . 5  |-  x  e. 
_V
35 sbequ12r 1745 . . . . . 6  |-  ( z  =  x  ->  ( [ z  /  x ] x  e.  A  <->  x  e.  A ) )
3613equcoms 1684 . . . . . . . 8  |-  ( z  =  x  ->  B  =  [_ z  /  x ]_ B )
3736eqcomd 2145 . . . . . . 7  |-  ( z  =  x  ->  [_ z  /  x ]_ B  =  B )
3837eleq2d 2209 . . . . . 6  |-  ( z  =  x  ->  ( C  e.  [_ z  /  x ]_ B  <->  C  e.  B ) )
3935, 38anbi12d 464 . . . . 5  |-  ( z  =  x  ->  (
( [ z  /  x ] x  e.  A  /\  C  e.  [_ z  /  x ]_ B )  <-> 
( x  e.  A  /\  C  e.  B
) ) )
4034, 39ceqsexv 2725 . . . 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 693 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 1331   E.wex 1468    e. wcel 1480   [wsb 1735   E.wrex 2417   _Vcvv 2686   [_csb 3003   {csn 3527   <.cop 3530   U_ciun 3813    X. cxp 4537
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-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
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  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-iun 3815  df-opab 3990  df-xp 4545
This theorem is referenced by:  eliunxp  4678  opeliunxp2  4679  opeliunxp2f  6135
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