MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  opeliunxp Unicode version

Theorem opeliunxp 4740
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 3907 . . 3  |-  U_ x  e.  A  ( {
x }  X.  B
)  =  { y  |  E. x  e.  A  y  e.  ( { x }  X.  B ) }
21eleq2i 2347 . 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 4237 . . 3  |-  <. x ,  C >.  e.  _V
4 df-rex 2549 . . . . 5  |-  ( E. x  e.  A  y  e.  ( { x }  X.  B )  <->  E. x
( x  e.  A  /\  y  e.  ( { x }  X.  B ) ) )
5 nfv 1605 . . . . . 6  |-  F/ z ( x  e.  A  /\  y  e.  ( { x }  X.  B ) )
6 nfs1v 2045 . . . . . . 7  |-  F/ x [ z  /  x ] x  e.  A
7 nfcv 2419 . . . . . . . . 9  |-  F/_ x { z }
8 nfcsb1v 3113 . . . . . . . . 9  |-  F/_ x [_ z  /  x ]_ B
97, 8nfxp 4715 . . . . . . . 8  |-  F/_ x
( { z }  X.  [_ z  /  x ]_ B )
109nfcri 2413 . . . . . . 7  |-  F/ x  y  e.  ( {
z }  X.  [_ z  /  x ]_ B
)
116, 10nfan 1771 . . . . . 6  |-  F/ x
( [ z  /  x ] x  e.  A  /\  y  e.  ( { z }  X.  [_ z  /  x ]_ B ) )
12 sbequ12 1860 . . . . . . 7  |-  ( x  =  z  ->  (
x  e.  A  <->  [ z  /  x ] x  e.  A ) )
13 sneq 3651 . . . . . . . . 9  |-  ( x  =  z  ->  { x }  =  { z } )
14 csbeq1a 3089 . . . . . . . . 9  |-  ( x  =  z  ->  B  =  [_ z  /  x ]_ B )
1513, 14xpeq12d 4714 . . . . . . . 8  |-  ( x  =  z  ->  ( { x }  X.  B )  =  ( { z }  X.  [_ z  /  x ]_ B ) )
1615eleq2d 2350 . . . . . . 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 1925 . . . . 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 2343 . . . . . 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 1612 . . . 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 2914 . 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 4719 . . . . . 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 3655 . . . . . . 7  |-  ( x  e.  { z }  <-> 
x  =  z )
29 eqcom 2285 . . . . . . 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 1569 . . 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 2791 . . . 4  |-  x  e. 
_V
35 sbequ12r 1861 . . . . 5  |-  ( z  =  x  ->  ( [ z  /  x ] x  e.  A  <->  x  e.  A ) )
3614eqcoms 2286 . . . . . . 7  |-  ( z  =  x  ->  B  =  [_ z  /  x ]_ B )
3736eqcomd 2288 . . . . . 6  |-  ( z  =  x  ->  [_ z  /  x ]_ B  =  B )
3837eleq2d 2350 . . . . 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 2823 . . 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 1528    = wceq 1623   [wsb 1629    e. wcel 1684   {cab 2269   E.wrex 2544   [_csb 3081   {csn 3640   <.cop 3643   U_ciun 3905    X. cxp 4687
This theorem is referenced by:  eliunxp  4823  opeliunxp2  4824  gsum2d2lem  15224  gsum2d2  15225  gsumcom2  15226  dprdval  15238  ptbasfi  17276  dvbsss  19252
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-iun 3907  df-opab 4078  df-xp 4695
  Copyright terms: Public domain W3C validator