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Theorem opeliunxp 4756
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 df-iun 3923 . . 3  |-  U_ x  e.  A  ( {
x }  X.  B
)  =  { y  |  E. x  e.  A  y  e.  ( { x }  X.  B ) }
21eleq2i 2360 . 2  |-  ( <.
x ,  C >.  e. 
U_ x  e.  A  ( { x }  X.  B )  <->  <. x ,  C >.  e.  { y  |  E. x  e.  A  y  e.  ( { x }  X.  B ) } )
3 opex 4253 . . 3  |-  <. x ,  C >.  e.  _V
4 df-rex 2562 . . . . 5  |-  ( E. x  e.  A  y  e.  ( { x }  X.  B )  <->  E. x
( x  e.  A  /\  y  e.  ( { x }  X.  B ) ) )
5 nfv 1609 . . . . . 6  |-  F/ z ( x  e.  A  /\  y  e.  ( { x }  X.  B ) )
6 nfs1v 2058 . . . . . . 7  |-  F/ x [ z  /  x ] x  e.  A
7 nfcv 2432 . . . . . . . . 9  |-  F/_ x { z }
8 nfcsb1v 3126 . . . . . . . . 9  |-  F/_ x [_ z  /  x ]_ B
97, 8nfxp 4731 . . . . . . . 8  |-  F/_ x
( { z }  X.  [_ z  /  x ]_ B )
109nfcri 2426 . . . . . . 7  |-  F/ x  y  e.  ( {
z }  X.  [_ z  /  x ]_ B
)
116, 10nfan 1783 . . . . . 6  |-  F/ x
( [ z  /  x ] x  e.  A  /\  y  e.  ( { z }  X.  [_ z  /  x ]_ B ) )
12 sbequ12 1872 . . . . . . 7  |-  ( x  =  z  ->  (
x  e.  A  <->  [ z  /  x ] x  e.  A ) )
13 sneq 3664 . . . . . . . . 9  |-  ( x  =  z  ->  { x }  =  { z } )
14 csbeq1a 3102 . . . . . . . . 9  |-  ( x  =  z  ->  B  =  [_ z  /  x ]_ B )
1513, 14xpeq12d 4730 . . . . . . . 8  |-  ( x  =  z  ->  ( { x }  X.  B )  =  ( { z }  X.  [_ z  /  x ]_ B ) )
1615eleq2d 2363 . . . . . . 7  |-  ( x  =  z  ->  (
y  e.  ( { x }  X.  B
)  <->  y  e.  ( { z }  X.  [_ z  /  x ]_ B ) ) )
1712, 16anbi12d 691 . . . . . 6  |-  ( x  =  z  ->  (
( x  e.  A  /\  y  e.  ( { x }  X.  B ) )  <->  ( [
z  /  x ]
x  e.  A  /\  y  e.  ( {
z }  X.  [_ z  /  x ]_ B
) ) ) )
185, 11, 17cbvex 1938 . . . . 5  |-  ( E. x ( x  e.  A  /\  y  e.  ( { x }  X.  B ) )  <->  E. z
( [ z  /  x ] x  e.  A  /\  y  e.  ( { z }  X.  [_ z  /  x ]_ B ) ) )
194, 18bitri 240 . . . 4  |-  ( E. x  e.  A  y  e.  ( { x }  X.  B )  <->  E. z
( [ z  /  x ] x  e.  A  /\  y  e.  ( { z }  X.  [_ z  /  x ]_ B ) ) )
20 eleq1 2356 . . . . . 6  |-  ( y  =  <. x ,  C >.  ->  ( y  e.  ( { z }  X.  [_ z  /  x ]_ B )  <->  <. x ,  C >.  e.  ( { z }  X.  [_ z  /  x ]_ B ) ) )
2120anbi2d 684 . . . . 5  |-  ( 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 ) ) ) )
2221exbidv 1616 . . . 4  |-  ( 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 ) ) ) )
2319, 22syl5bb 248 . . 3  |-  ( 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 ) ) ) )
243, 23elab 2927 . 2  |-  ( <.
x ,  C >.  e. 
{ y  |  E. x  e.  A  y  e.  ( { x }  X.  B ) }  <->  E. z
( [ z  /  x ] x  e.  A  /\  <. x ,  C >.  e.  ( { z }  X.  [_ z  /  x ]_ B ) ) )
25 opelxp 4735 . . . . . 6  |-  ( <.
x ,  C >.  e.  ( { z }  X.  [_ z  /  x ]_ B )  <->  ( x  e.  { z }  /\  C  e.  [_ z  /  x ]_ B ) )
2625anbi2i 675 . . . . 5  |-  ( ( [ 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 772 . . . . 5  |-  ( ( [ 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 elsn 3668 . . . . . . 7  |-  ( x  e.  { z }  <-> 
x  =  z )
29 eqcom 2298 . . . . . . 7  |-  ( x  =  z  <->  z  =  x )
3028, 29bitri 240 . . . . . 6  |-  ( x  e.  { z }  <-> 
z  =  x )
3130anbi1i 676 . . . . 5  |-  ( ( 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 262 . . . 4  |-  ( ( [ 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 1572 . . 3  |-  ( 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 2804 . . . 4  |-  x  e. 
_V
35 sbequ12r 1873 . . . . 5  |-  ( z  =  x  ->  ( [ z  /  x ] x  e.  A  <->  x  e.  A ) )
3614eqcoms 2299 . . . . . . 7  |-  ( z  =  x  ->  B  =  [_ z  /  x ]_ B )
3736eqcomd 2301 . . . . . 6  |-  ( z  =  x  ->  [_ z  /  x ]_ B  =  B )
3837eleq2d 2363 . . . . 5  |-  ( z  =  x  ->  ( C  e.  [_ z  /  x ]_ B  <->  C  e.  B ) )
3935, 38anbi12d 691 . . . 4  |-  ( z  =  x  ->  (
( [ z  /  x ] x  e.  A  /\  C  e.  [_ z  /  x ]_ B )  <-> 
( x  e.  A  /\  C  e.  B
) ) )
4034, 39ceqsexv 2836 . . 3  |-  ( E. z ( z  =  x  /\  ( [ z  /  x ]
x  e.  A  /\  C  e.  [_ z  /  x ]_ B ) )  <-> 
( x  e.  A  /\  C  e.  B
) )
4133, 40bitri 240 . 2  |-  ( E. z ( [ z  /  x ] x  e.  A  /\  <. x ,  C >.  e.  ( { z }  X.  [_ z  /  x ]_ B ) )  <->  ( x  e.  A  /\  C  e.  B ) )
422, 24, 413bitri 262 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:    <-> wb 176    /\ wa 358   E.wex 1531    = wceq 1632   [wsb 1638    e. wcel 1696   {cab 2282   E.wrex 2557   [_csb 3094   {csn 3653   <.cop 3656   U_ciun 3921    X. cxp 4703
This theorem is referenced by:  eliunxp  4839  opeliunxp2  4840  gsum2d2lem  15240  gsum2d2  15241  gsumcom2  15242  dprdval  15254  ptbasfi  17292  dvbsss  19268
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-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
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-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  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-iun 3923  df-opab 4094  df-xp 4711
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