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Theorem cbvmpt2x2 40625
Description: Rule to change the bound variable in a maps-to function, using implicit substitution. This version of cbvmpt2 6389 allows  A to be a function of  y, analogous to cbvmpt2x 6388. (Contributed by AV, 30-Mar-2019.)
Hypotheses
Ref Expression
cbvmpt2x2.1  |-  F/_ z A
cbvmpt2x2.2  |-  F/_ y D
cbvmpt2x2.3  |-  F/_ z C
cbvmpt2x2.4  |-  F/_ w C
cbvmpt2x2.5  |-  F/_ x E
cbvmpt2x2.6  |-  F/_ y E
cbvmpt2x2.7  |-  ( y  =  z  ->  A  =  D )
cbvmpt2x2.8  |-  ( ( y  =  z  /\  x  =  w )  ->  C  =  E )
Assertion
Ref Expression
cbvmpt2x2  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  ( w  e.  D ,  z  e.  B  |->  E )
Distinct variable groups:    x, w, y, z    w, A    w, B, x, y, z    x, D
Allowed substitution hints:    A( x, y, z)    C( x, y, z, w)    D( y, z, w)    E( x, y, z, w)

Proof of Theorem cbvmpt2x2
Dummy variable  u is distinct from all other variables.
StepHypRef Expression
1 nfv 1769 . . . . 5  |-  F/ w  x  e.  A
2 nfv 1769 . . . . 5  |-  F/ w  y  e.  B
31, 2nfan 2031 . . . 4  |-  F/ w
( x  e.  A  /\  y  e.  B
)
4 cbvmpt2x2.4 . . . . 5  |-  F/_ w C
54nfeq2 2627 . . . 4  |-  F/ w  u  =  C
63, 5nfan 2031 . . 3  |-  F/ w
( ( x  e.  A  /\  y  e.  B )  /\  u  =  C )
7 cbvmpt2x2.1 . . . . . 6  |-  F/_ z A
87nfcri 2606 . . . . 5  |-  F/ z  x  e.  A
9 nfv 1769 . . . . 5  |-  F/ z  y  e.  B
108, 9nfan 2031 . . . 4  |-  F/ z ( x  e.  A  /\  y  e.  B
)
11 cbvmpt2x2.3 . . . . 5  |-  F/_ z C
1211nfeq2 2627 . . . 4  |-  F/ z  u  =  C
1310, 12nfan 2031 . . 3  |-  F/ z ( ( x  e.  A  /\  y  e.  B )  /\  u  =  C )
14 nfv 1769 . . . . 5  |-  F/ x  w  e.  D
15 nfv 1769 . . . . 5  |-  F/ x  z  e.  B
1614, 15nfan 2031 . . . 4  |-  F/ x
( w  e.  D  /\  z  e.  B
)
17 cbvmpt2x2.5 . . . . 5  |-  F/_ x E
1817nfeq2 2627 . . . 4  |-  F/ x  u  =  E
1916, 18nfan 2031 . . 3  |-  F/ x
( ( w  e.  D  /\  z  e.  B )  /\  u  =  E )
20 cbvmpt2x2.2 . . . . . 6  |-  F/_ y D
2120nfcri 2606 . . . . 5  |-  F/ y  w  e.  D
22 nfv 1769 . . . . 5  |-  F/ y  z  e.  B
2321, 22nfan 2031 . . . 4  |-  F/ y ( w  e.  D  /\  z  e.  B
)
24 cbvmpt2x2.6 . . . . 5  |-  F/_ y E
2524nfeq2 2627 . . . 4  |-  F/ y  u  =  E
2623, 25nfan 2031 . . 3  |-  F/ y ( ( w  e.  D  /\  z  e.  B )  /\  u  =  E )
27 eleq1 2537 . . . . . 6  |-  ( x  =  w  ->  (
x  e.  A  <->  w  e.  A ) )
28 cbvmpt2x2.7 . . . . . . 7  |-  ( y  =  z  ->  A  =  D )
2928eleq2d 2534 . . . . . 6  |-  ( y  =  z  ->  (
w  e.  A  <->  w  e.  D ) )
3027, 29sylan9bb 714 . . . . 5  |-  ( ( x  =  w  /\  y  =  z )  ->  ( x  e.  A  <->  w  e.  D ) )
31 simpr 468 . . . . . 6  |-  ( ( x  =  w  /\  y  =  z )  ->  y  =  z )
3231eleq1d 2533 . . . . 5  |-  ( ( x  =  w  /\  y  =  z )  ->  ( y  e.  B  <->  z  e.  B ) )
3330, 32anbi12d 725 . . . 4  |-  ( ( x  =  w  /\  y  =  z )  ->  ( ( x  e.  A  /\  y  e.  B )  <->  ( w  e.  D  /\  z  e.  B ) ) )
34 cbvmpt2x2.8 . . . . . 6  |-  ( ( y  =  z  /\  x  =  w )  ->  C  =  E )
3534ancoms 460 . . . . 5  |-  ( ( x  =  w  /\  y  =  z )  ->  C  =  E )
3635eqeq2d 2481 . . . 4  |-  ( ( x  =  w  /\  y  =  z )  ->  ( u  =  C  <-> 
u  =  E ) )
3733, 36anbi12d 725 . . 3  |-  ( ( x  =  w  /\  y  =  z )  ->  ( ( ( x  e.  A  /\  y  e.  B )  /\  u  =  C )  <->  ( (
w  e.  D  /\  z  e.  B )  /\  u  =  E
) ) )
386, 13, 19, 26, 37cbvoprab12 6384 . 2  |-  { <. <.
x ,  y >. ,  u >.  |  (
( x  e.  A  /\  y  e.  B
)  /\  u  =  C ) }  =  { <. <. w ,  z
>. ,  u >.  |  ( ( w  e.  D  /\  z  e.  B )  /\  u  =  E ) }
39 df-mpt2 6313 . 2  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  { <. <. x ,  y >. ,  u >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  u  =  C ) }
40 df-mpt2 6313 . 2  |-  ( w  e.  D ,  z  e.  B  |->  E )  =  { <. <. w ,  z >. ,  u >.  |  ( ( w  e.  D  /\  z  e.  B )  /\  u  =  E ) }
4138, 39, 403eqtr4i 2503 1  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  ( w  e.  D ,  z  e.  B  |->  E )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    = wceq 1452    e. wcel 1904   F/_wnfc 2599   {coprab 6309    |-> cmpt2 6310
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-opab 4455  df-oprab 6312  df-mpt2 6313
This theorem is referenced by:  dmmpt2ssx2  40626
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