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Theorem cbvmpt2x 5940
Description: Rule to change the bound variable in a maps-to function, using implicit substitution. This version of cbvmpt2 5941 allows  B to be a function of  x. (Contributed by NM, 29-Dec-2014.)
Hypotheses
Ref Expression
cbvmpt2x.1  |-  F/_ z B
cbvmpt2x.2  |-  F/_ x D
cbvmpt2x.3  |-  F/_ z C
cbvmpt2x.4  |-  F/_ w C
cbvmpt2x.5  |-  F/_ x E
cbvmpt2x.6  |-  F/_ y E
cbvmpt2x.7  |-  ( x  =  z  ->  B  =  D )
cbvmpt2x.8  |-  ( ( x  =  z  /\  y  =  w )  ->  C  =  E )
Assertion
Ref Expression
cbvmpt2x  |-  ( 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 cbvmpt2x
Dummy variable  u is distinct from all other variables.
StepHypRef Expression
1 nfv 1609 . . . . 5  |-  F/ z  x  e.  A
2 cbvmpt2x.1 . . . . . 6  |-  F/_ z B
32nfcri 2426 . . . . 5  |-  F/ z  y  e.  B
41, 3nfan 1783 . . . 4  |-  F/ z ( x  e.  A  /\  y  e.  B
)
5 cbvmpt2x.3 . . . . 5  |-  F/_ z C
65nfeq2 2443 . . . 4  |-  F/ z  u  =  C
74, 6nfan 1783 . . 3  |-  F/ z ( ( x  e.  A  /\  y  e.  B )  /\  u  =  C )
8 nfv 1609 . . . . 5  |-  F/ w  x  e.  A
9 nfcv 2432 . . . . . 6  |-  F/_ w B
109nfcri 2426 . . . . 5  |-  F/ w  y  e.  B
118, 10nfan 1783 . . . 4  |-  F/ w
( x  e.  A  /\  y  e.  B
)
12 cbvmpt2x.4 . . . . 5  |-  F/_ w C
1312nfeq2 2443 . . . 4  |-  F/ w  u  =  C
1411, 13nfan 1783 . . 3  |-  F/ w
( ( x  e.  A  /\  y  e.  B )  /\  u  =  C )
15 nfv 1609 . . . . 5  |-  F/ x  z  e.  A
16 cbvmpt2x.2 . . . . . 6  |-  F/_ x D
1716nfcri 2426 . . . . 5  |-  F/ x  w  e.  D
1815, 17nfan 1783 . . . 4  |-  F/ x
( z  e.  A  /\  w  e.  D
)
19 cbvmpt2x.5 . . . . 5  |-  F/_ x E
2019nfeq2 2443 . . . 4  |-  F/ x  u  =  E
2118, 20nfan 1783 . . 3  |-  F/ x
( ( z  e.  A  /\  w  e.  D )  /\  u  =  E )
22 nfv 1609 . . . 4  |-  F/ y ( z  e.  A  /\  w  e.  D
)
23 cbvmpt2x.6 . . . . 5  |-  F/_ y E
2423nfeq2 2443 . . . 4  |-  F/ y  u  =  E
2522, 24nfan 1783 . . 3  |-  F/ y ( ( z  e.  A  /\  w  e.  D )  /\  u  =  E )
26 eleq1 2356 . . . . . 6  |-  ( x  =  z  ->  (
x  e.  A  <->  z  e.  A ) )
2726adantr 451 . . . . 5  |-  ( ( x  =  z  /\  y  =  w )  ->  ( x  e.  A  <->  z  e.  A ) )
28 cbvmpt2x.7 . . . . . . 7  |-  ( x  =  z  ->  B  =  D )
2928eleq2d 2363 . . . . . 6  |-  ( x  =  z  ->  (
y  e.  B  <->  y  e.  D ) )
30 eleq1 2356 . . . . . 6  |-  ( y  =  w  ->  (
y  e.  D  <->  w  e.  D ) )
3129, 30sylan9bb 680 . . . . 5  |-  ( ( x  =  z  /\  y  =  w )  ->  ( y  e.  B  <->  w  e.  D ) )
3227, 31anbi12d 691 . . . 4  |-  ( ( x  =  z  /\  y  =  w )  ->  ( ( x  e.  A  /\  y  e.  B )  <->  ( z  e.  A  /\  w  e.  D ) ) )
33 cbvmpt2x.8 . . . . 5  |-  ( ( x  =  z  /\  y  =  w )  ->  C  =  E )
3433eqeq2d 2307 . . . 4  |-  ( ( x  =  z  /\  y  =  w )  ->  ( u  =  C  <-> 
u  =  E ) )
3532, 34anbi12d 691 . . 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 5936 . 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-mpt2 5879 . 2  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  { <. <. x ,  y >. ,  u >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  u  =  C ) }
38 df-mpt2 5879 . 2  |-  ( z  e.  A ,  w  e.  D  |->  E )  =  { <. <. z ,  w >. ,  u >.  |  ( ( z  e.  A  /\  w  e.  D )  /\  u  =  E ) }
3936, 37, 383eqtr4i 2326 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    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   F/_wnfc 2419   {coprab 5875    e. cmpt2 5876
This theorem is referenced by:  cbvmpt2  5941  mpt2mptsx  6203  dmmpt2ssx  6205  gsumcom2  15242  ptcmpg  17767
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-rab 2565  df-v 2803  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-opab 4094  df-oprab 5878  df-mpt2 5879
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