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Theorem cbvmpox 5815
Description: Rule to change the bound variable in a maps-to function, using implicit substitution. This version of cbvmpo 5816 allows  B to be a function of  x. (Contributed by NM, 29-Dec-2014.)
Hypotheses
Ref Expression
cbvmpox.1  |-  F/_ z B
cbvmpox.2  |-  F/_ x D
cbvmpox.3  |-  F/_ z C
cbvmpox.4  |-  F/_ w C
cbvmpox.5  |-  F/_ x E
cbvmpox.6  |-  F/_ y E
cbvmpox.7  |-  ( x  =  z  ->  B  =  D )
cbvmpox.8  |-  ( ( x  =  z  /\  y  =  w )  ->  C  =  E )
Assertion
Ref Expression
cbvmpox  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  ( z  e.  A ,  w  e.  D  |->  E )
Distinct variable groups:    x, w, y, z, A    w, B    y, D
Allowed substitution hints:    B( x, y, z)    C( x, y, z, w)    D( x, z, w)    E( x, y, z, w)

Proof of Theorem cbvmpox
Dummy variable  u is distinct from all other variables.
StepHypRef Expression
1 nfv 1491 . . . . 5  |-  F/ z  x  e.  A
2 cbvmpox.1 . . . . . 6  |-  F/_ z B
32nfcri 2250 . . . . 5  |-  F/ z  y  e.  B
41, 3nfan 1527 . . . 4  |-  F/ z ( x  e.  A  /\  y  e.  B
)
5 cbvmpox.3 . . . . 5  |-  F/_ z C
65nfeq2 2268 . . . 4  |-  F/ z  u  =  C
74, 6nfan 1527 . . 3  |-  F/ z ( ( x  e.  A  /\  y  e.  B )  /\  u  =  C )
8 nfv 1491 . . . . 5  |-  F/ w  x  e.  A
9 nfcv 2256 . . . . . 6  |-  F/_ w B
109nfcri 2250 . . . . 5  |-  F/ w  y  e.  B
118, 10nfan 1527 . . . 4  |-  F/ w
( x  e.  A  /\  y  e.  B
)
12 cbvmpox.4 . . . . 5  |-  F/_ w C
1312nfeq2 2268 . . . 4  |-  F/ w  u  =  C
1411, 13nfan 1527 . . 3  |-  F/ w
( ( x  e.  A  /\  y  e.  B )  /\  u  =  C )
15 nfv 1491 . . . . 5  |-  F/ x  z  e.  A
16 cbvmpox.2 . . . . . 6  |-  F/_ x D
1716nfcri 2250 . . . . 5  |-  F/ x  w  e.  D
1815, 17nfan 1527 . . . 4  |-  F/ x
( z  e.  A  /\  w  e.  D
)
19 cbvmpox.5 . . . . 5  |-  F/_ x E
2019nfeq2 2268 . . . 4  |-  F/ x  u  =  E
2118, 20nfan 1527 . . 3  |-  F/ x
( ( z  e.  A  /\  w  e.  D )  /\  u  =  E )
22 nfv 1491 . . . 4  |-  F/ y ( z  e.  A  /\  w  e.  D
)
23 cbvmpox.6 . . . . 5  |-  F/_ y E
2423nfeq2 2268 . . . 4  |-  F/ y  u  =  E
2522, 24nfan 1527 . . 3  |-  F/ y ( ( z  e.  A  /\  w  e.  D )  /\  u  =  E )
26 eleq1 2178 . . . . . 6  |-  ( x  =  z  ->  (
x  e.  A  <->  z  e.  A ) )
2726adantr 272 . . . . 5  |-  ( ( x  =  z  /\  y  =  w )  ->  ( x  e.  A  <->  z  e.  A ) )
28 cbvmpox.7 . . . . . . 7  |-  ( x  =  z  ->  B  =  D )
2928eleq2d 2185 . . . . . 6  |-  ( x  =  z  ->  (
y  e.  B  <->  y  e.  D ) )
30 eleq1 2178 . . . . . 6  |-  ( y  =  w  ->  (
y  e.  D  <->  w  e.  D ) )
3129, 30sylan9bb 455 . . . . 5  |-  ( ( x  =  z  /\  y  =  w )  ->  ( y  e.  B  <->  w  e.  D ) )
3227, 31anbi12d 462 . . . 4  |-  ( ( x  =  z  /\  y  =  w )  ->  ( ( x  e.  A  /\  y  e.  B )  <->  ( z  e.  A  /\  w  e.  D ) ) )
33 cbvmpox.8 . . . . 5  |-  ( ( x  =  z  /\  y  =  w )  ->  C  =  E )
3433eqeq2d 2127 . . . 4  |-  ( ( x  =  z  /\  y  =  w )  ->  ( u  =  C  <-> 
u  =  E ) )
3532, 34anbi12d 462 . . 3  |-  ( ( x  =  z  /\  y  =  w )  ->  ( ( ( x  e.  A  /\  y  e.  B )  /\  u  =  C )  <->  ( (
z  e.  A  /\  w  e.  D )  /\  u  =  E
) ) )
367, 14, 21, 25, 35cbvoprab12 5811 . 2  |-  { <. <.
x ,  y >. ,  u >.  |  (
( x  e.  A  /\  y  e.  B
)  /\  u  =  C ) }  =  { <. <. z ,  w >. ,  u >.  |  ( ( z  e.  A  /\  w  e.  D
)  /\  u  =  E ) }
37 df-mpo 5745 . 2  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  { <. <. x ,  y >. ,  u >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  u  =  C ) }
38 df-mpo 5745 . 2  |-  ( z  e.  A ,  w  e.  D  |->  E )  =  { <. <. z ,  w >. ,  u >.  |  ( ( z  e.  A  /\  w  e.  D )  /\  u  =  E ) }
3936, 37, 383eqtr4i 2146 1  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  ( z  e.  A ,  w  e.  D  |->  E )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1314    e. wcel 1463   F/_wnfc 2243   {coprab 5741    e. cmpo 5742
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-opab 3958  df-oprab 5744  df-mpo 5745
This theorem is referenced by:  cbvmpo  5816  mpomptsx  6061  dmmpossx  6063
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