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Theorem opeliunxp 4423
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 2583 . 2  |-  ( <.
x ,  C >.  e. 
U_ x  e.  A  ( { x }  X.  B )  ->  <. x ,  C >.  e.  _V )
2 vex 2577 . . 3  |-  x  e. 
_V
3 elex 2583 . . . 4  |-  ( C  e.  B  ->  C  e.  _V )
43adantl 266 . . 3  |-  ( ( x  e.  A  /\  C  e.  B )  ->  C  e.  _V )
5 opexgOLD 3993 . . 3  |-  ( ( x  e.  _V  /\  C  e.  _V )  -> 
<. x ,  C >.  e. 
_V )
62, 4, 5sylancr 399 . 2  |-  ( ( x  e.  A  /\  C  e.  B )  -> 
<. x ,  C >.  e. 
_V )
7 df-rex 2329 . . . . . 6  |-  ( E. x  e.  A  y  e.  ( { x }  X.  B )  <->  E. x
( x  e.  A  /\  y  e.  ( { x }  X.  B ) ) )
8 nfv 1437 . . . . . . 7  |-  F/ z ( x  e.  A  /\  y  e.  ( { x }  X.  B ) )
9 nfs1v 1831 . . . . . . . 8  |-  F/ x [ z  /  x ] x  e.  A
10 nfcv 2194 . . . . . . . . . 10  |-  F/_ x { z }
11 nfcsb1v 2910 . . . . . . . . . 10  |-  F/_ x [_ z  /  x ]_ B
1210, 11nfxp 4399 . . . . . . . . 9  |-  F/_ x
( { z }  X.  [_ z  /  x ]_ B )
1312nfcri 2188 . . . . . . . 8  |-  F/ x  y  e.  ( {
z }  X.  [_ z  /  x ]_ B
)
149, 13nfan 1473 . . . . . . 7  |-  F/ x
( [ z  /  x ] x  e.  A  /\  y  e.  ( { z }  X.  [_ z  /  x ]_ B ) )
15 sbequ12 1670 . . . . . . . 8  |-  ( x  =  z  ->  (
x  e.  A  <->  [ z  /  x ] x  e.  A ) )
16 sneq 3414 . . . . . . . . . 10  |-  ( x  =  z  ->  { x }  =  { z } )
17 csbeq1a 2888 . . . . . . . . . 10  |-  ( x  =  z  ->  B  =  [_ z  /  x ]_ B )
1816, 17xpeq12d 4398 . . . . . . . . 9  |-  ( x  =  z  ->  ( { x }  X.  B )  =  ( { z }  X.  [_ z  /  x ]_ B ) )
1918eleq2d 2123 . . . . . . . 8  |-  ( x  =  z  ->  (
y  e.  ( { x }  X.  B
)  <->  y  e.  ( { z }  X.  [_ z  /  x ]_ B ) ) )
2015, 19anbi12d 450 . . . . . . 7  |-  ( x  =  z  ->  (
( x  e.  A  /\  y  e.  ( { x }  X.  B ) )  <->  ( [
z  /  x ]
x  e.  A  /\  y  e.  ( {
z }  X.  [_ z  /  x ]_ B
) ) ) )
218, 14, 20cbvex 1655 . . . . . 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 ) ) )
227, 21bitri 177 . . . . 5  |-  ( E. x  e.  A  y  e.  ( { x }  X.  B )  <->  E. z
( [ z  /  x ] x  e.  A  /\  y  e.  ( { z }  X.  [_ z  /  x ]_ B ) ) )
23 eleq1 2116 . . . . . . 7  |-  ( y  =  <. x ,  C >.  ->  ( y  e.  ( { z }  X.  [_ z  /  x ]_ B )  <->  <. x ,  C >.  e.  ( { z }  X.  [_ z  /  x ]_ B ) ) )
2423anbi2d 445 . . . . . 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 ) ) ) )
2524exbidv 1722 . . . . 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 ) ) ) )
2622, 25syl5bb 185 . . . 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 ) ) ) )
27 df-iun 3687 . . . 4  |-  U_ x  e.  A  ( {
x }  X.  B
)  =  { y  |  E. x  e.  A  y  e.  ( { x }  X.  B ) }
2826, 27elab2g 2712 . . 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 ) ) ) )
29 opelxp 4402 . . . . . . 7  |-  ( <.
x ,  C >.  e.  ( { z }  X.  [_ z  /  x ]_ B )  <->  ( x  e.  { z }  /\  C  e.  [_ z  /  x ]_ B ) )
3029anbi2i 438 . . . . . 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 ) ) )
31 an12 503 . . . . . 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 ) ) )
32 velsn 3420 . . . . . . . 8  |-  ( x  e.  { z }  <-> 
x  =  z )
33 equcom 1609 . . . . . . . 8  |-  ( x  =  z  <->  z  =  x )
3432, 33bitri 177 . . . . . . 7  |-  ( x  e.  { z }  <-> 
z  =  x )
3534anbi1i 439 . . . . . 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 ) ) )
3630, 31, 353bitri 199 . . . . 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 ) ) )
3736exbii 1512 . . . 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 ) ) )
38 sbequ12r 1671 . . . . . 6  |-  ( z  =  x  ->  ( [ z  /  x ] x  e.  A  <->  x  e.  A ) )
3917equcoms 1610 . . . . . . . 8  |-  ( z  =  x  ->  B  =  [_ z  /  x ]_ B )
4039eqcomd 2061 . . . . . . 7  |-  ( z  =  x  ->  [_ z  /  x ]_ B  =  B )
4140eleq2d 2123 . . . . . 6  |-  ( z  =  x  ->  ( C  e.  [_ z  /  x ]_ B  <->  C  e.  B ) )
4238, 41anbi12d 450 . . . . 5  |-  ( z  =  x  ->  (
( [ z  /  x ] x  e.  A  /\  C  e.  [_ z  /  x ]_ B )  <-> 
( x  e.  A  /\  C  e.  B
) ) )
432, 42ceqsexv 2610 . . . 4  |-  ( E. z ( z  =  x  /\  ( [ z  /  x ]
x  e.  A  /\  C  e.  [_ z  /  x ]_ B ) )  <-> 
( x  e.  A  /\  C  e.  B
) )
4437, 43bitri 177 . . 3  |-  ( E. z ( [ z  /  x ] x  e.  A  /\  <. x ,  C >.  e.  ( { z }  X.  [_ z  /  x ]_ B ) )  <->  ( x  e.  A  /\  C  e.  B ) )
4528, 44syl6bb 189 . 2  |-  ( <.
x ,  C >.  e. 
_V  ->  ( <. x ,  C >.  e.  U_ x  e.  A  ( {
x }  X.  B
)  <->  ( x  e.  A  /\  C  e.  B ) ) )
461, 6, 45pm5.21nii 630 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 101    <-> wb 102    = wceq 1259   E.wex 1397    e. wcel 1409   [wsb 1661   E.wrex 2324   _Vcvv 2574   [_csb 2880   {csn 3403   <.cop 3406   U_ciun 3685    X. cxp 4371
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2788  df-csb 2881  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-iun 3687  df-opab 3847  df-xp 4379
This theorem is referenced by:  eliunxp  4503  opeliunxp2  4504
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