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Theorem cbvmpox 5929
Description: Rule to change the bound variable in a maps-to function, using implicit substitution. This version of cbvmpo 5930 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 1521 . . . . 5  |-  F/ z  x  e.  A
2 cbvmpox.1 . . . . . 6  |-  F/_ z B
32nfcri 2306 . . . . 5  |-  F/ z  y  e.  B
41, 3nfan 1558 . . . 4  |-  F/ z ( x  e.  A  /\  y  e.  B
)
5 cbvmpox.3 . . . . 5  |-  F/_ z C
65nfeq2 2324 . . . 4  |-  F/ z  u  =  C
74, 6nfan 1558 . . 3  |-  F/ z ( ( x  e.  A  /\  y  e.  B )  /\  u  =  C )
8 nfv 1521 . . . . 5  |-  F/ w  x  e.  A
9 nfcv 2312 . . . . . 6  |-  F/_ w B
109nfcri 2306 . . . . 5  |-  F/ w  y  e.  B
118, 10nfan 1558 . . . 4  |-  F/ w
( x  e.  A  /\  y  e.  B
)
12 cbvmpox.4 . . . . 5  |-  F/_ w C
1312nfeq2 2324 . . . 4  |-  F/ w  u  =  C
1411, 13nfan 1558 . . 3  |-  F/ w
( ( x  e.  A  /\  y  e.  B )  /\  u  =  C )
15 nfv 1521 . . . . 5  |-  F/ x  z  e.  A
16 cbvmpox.2 . . . . . 6  |-  F/_ x D
1716nfcri 2306 . . . . 5  |-  F/ x  w  e.  D
1815, 17nfan 1558 . . . 4  |-  F/ x
( z  e.  A  /\  w  e.  D
)
19 cbvmpox.5 . . . . 5  |-  F/_ x E
2019nfeq2 2324 . . . 4  |-  F/ x  u  =  E
2118, 20nfan 1558 . . 3  |-  F/ x
( ( z  e.  A  /\  w  e.  D )  /\  u  =  E )
22 nfv 1521 . . . 4  |-  F/ y ( z  e.  A  /\  w  e.  D
)
23 cbvmpox.6 . . . . 5  |-  F/_ y E
2423nfeq2 2324 . . . 4  |-  F/ y  u  =  E
2522, 24nfan 1558 . . 3  |-  F/ y ( ( z  e.  A  /\  w  e.  D )  /\  u  =  E )
26 eleq1 2233 . . . . . 6  |-  ( x  =  z  ->  (
x  e.  A  <->  z  e.  A ) )
2726adantr 274 . . . . 5  |-  ( ( x  =  z  /\  y  =  w )  ->  ( x  e.  A  <->  z  e.  A ) )
28 cbvmpox.7 . . . . . . 7  |-  ( x  =  z  ->  B  =  D )
2928eleq2d 2240 . . . . . 6  |-  ( x  =  z  ->  (
y  e.  B  <->  y  e.  D ) )
30 eleq1 2233 . . . . . 6  |-  ( y  =  w  ->  (
y  e.  D  <->  w  e.  D ) )
3129, 30sylan9bb 459 . . . . 5  |-  ( ( x  =  z  /\  y  =  w )  ->  ( y  e.  B  <->  w  e.  D ) )
3227, 31anbi12d 470 . . . 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 2182 . . . 4  |-  ( ( x  =  z  /\  y  =  w )  ->  ( u  =  C  <-> 
u  =  E ) )
3532, 34anbi12d 470 . . 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 5925 . 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 5856 . 2  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  { <. <. x ,  y >. ,  u >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  u  =  C ) }
38 df-mpo 5856 . 2  |-  ( z  e.  A ,  w  e.  D  |->  E )  =  { <. <. z ,  w >. ,  u >.  |  ( ( z  e.  A  /\  w  e.  D )  /\  u  =  E ) }
3936, 37, 383eqtr4i 2201 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 1348    e. wcel 2141   F/_wnfc 2299   {coprab 5852    e. cmpo 5853
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-opab 4049  df-oprab 5855  df-mpo 5856
This theorem is referenced by:  cbvmpo  5930  mpomptsx  6174  dmmpossx  6176
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