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Theorem cbvmpt2x2 32404
Description: Rule to change the bound variable in a maps-to function, using implicit substitution. This version of cbvmpt2 6371 allows  A to be a function of  y, analogous to cbvmpt2x 6370. (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 1683 . . . . 5  |-  F/ w  x  e.  A
2 nfv 1683 . . . . 5  |-  F/ w  y  e.  B
31, 2nfan 1875 . . . 4  |-  F/ w
( x  e.  A  /\  y  e.  B
)
4 cbvmpt2x2.4 . . . . 5  |-  F/_ w C
54nfeq2 2646 . . . 4  |-  F/ w  u  =  C
63, 5nfan 1875 . . 3  |-  F/ w
( ( x  e.  A  /\  y  e.  B )  /\  u  =  C )
7 cbvmpt2x2.1 . . . . . 6  |-  F/_ z A
87nfcri 2622 . . . . 5  |-  F/ z  x  e.  A
9 nfv 1683 . . . . 5  |-  F/ z  y  e.  B
108, 9nfan 1875 . . . 4  |-  F/ z ( x  e.  A  /\  y  e.  B
)
11 cbvmpt2x2.3 . . . . 5  |-  F/_ z C
1211nfeq2 2646 . . . 4  |-  F/ z  u  =  C
1310, 12nfan 1875 . . 3  |-  F/ z ( ( x  e.  A  /\  y  e.  B )  /\  u  =  C )
14 nfv 1683 . . . . 5  |-  F/ x  w  e.  D
15 nfv 1683 . . . . 5  |-  F/ x  z  e.  B
1614, 15nfan 1875 . . . 4  |-  F/ x
( w  e.  D  /\  z  e.  B
)
17 cbvmpt2x2.5 . . . . 5  |-  F/_ x E
1817nfeq2 2646 . . . 4  |-  F/ x  u  =  E
1916, 18nfan 1875 . . 3  |-  F/ x
( ( w  e.  D  /\  z  e.  B )  /\  u  =  E )
20 cbvmpt2x2.2 . . . . . 6  |-  F/_ y D
2120nfcri 2622 . . . . 5  |-  F/ y  w  e.  D
22 nfv 1683 . . . . 5  |-  F/ y  z  e.  B
2321, 22nfan 1875 . . . 4  |-  F/ y ( w  e.  D  /\  z  e.  B
)
24 cbvmpt2x2.6 . . . . 5  |-  F/_ y E
2524nfeq2 2646 . . . 4  |-  F/ y  u  =  E
2623, 25nfan 1875 . . 3  |-  F/ y ( ( w  e.  D  /\  z  e.  B )  /\  u  =  E )
27 eleq1 2539 . . . . . 6  |-  ( x  =  w  ->  (
x  e.  A  <->  w  e.  A ) )
28 cbvmpt2x2.7 . . . . . . 7  |-  ( y  =  z  ->  A  =  D )
2928eleq2d 2537 . . . . . 6  |-  ( y  =  z  ->  (
w  e.  A  <->  w  e.  D ) )
3027, 29sylan9bb 699 . . . . 5  |-  ( ( x  =  w  /\  y  =  z )  ->  ( x  e.  A  <->  w  e.  D ) )
31 simpr 461 . . . . . 6  |-  ( ( x  =  w  /\  y  =  z )  ->  y  =  z )
3231eleq1d 2536 . . . . 5  |-  ( ( x  =  w  /\  y  =  z )  ->  ( y  e.  B  <->  z  e.  B ) )
3330, 32anbi12d 710 . . . 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 453 . . . . 5  |-  ( ( x  =  w  /\  y  =  z )  ->  C  =  E )
3635eqeq2d 2481 . . . 4  |-  ( ( x  =  w  /\  y  =  z )  ->  ( u  =  C  <-> 
u  =  E ) )
3733, 36anbi12d 710 . . 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 6366 . 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 6300 . 2  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  { <. <. x ,  y >. ,  u >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  u  =  C ) }
40 df-mpt2 6300 . 2  |-  ( w  e.  D ,  z  e.  B  |->  E )  =  { <. <. w ,  z >. ,  u >.  |  ( ( w  e.  D  /\  z  e.  B )  /\  u  =  E ) }
4138, 39, 403eqtr4i 2506 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 369    = wceq 1379    e. wcel 1767   F/_wnfc 2615   {coprab 6296    |-> cmpt2 6297
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-opab 4512  df-oprab 6299  df-mpt2 6300
This theorem is referenced by:  dmmpt2ssx2  32405
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