MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cbvmpt2x Structured version   Unicode version

Theorem cbvmpt2x 6356
Description: Rule to change the bound variable in a maps-to function, using implicit substitution. This version of cbvmpt2 6357 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 1728 . . . . 5  |-  F/ z  x  e.  A
2 cbvmpt2x.1 . . . . . 6  |-  F/_ z B
32nfcri 2557 . . . . 5  |-  F/ z  y  e.  B
41, 3nfan 1956 . . . 4  |-  F/ z ( x  e.  A  /\  y  e.  B
)
5 cbvmpt2x.3 . . . . 5  |-  F/_ z C
65nfeq2 2581 . . . 4  |-  F/ z  u  =  C
74, 6nfan 1956 . . 3  |-  F/ z ( ( x  e.  A  /\  y  e.  B )  /\  u  =  C )
8 nfv 1728 . . . . 5  |-  F/ w  x  e.  A
9 nfcv 2564 . . . . . 6  |-  F/_ w B
109nfcri 2557 . . . . 5  |-  F/ w  y  e.  B
118, 10nfan 1956 . . . 4  |-  F/ w
( x  e.  A  /\  y  e.  B
)
12 cbvmpt2x.4 . . . . 5  |-  F/_ w C
1312nfeq2 2581 . . . 4  |-  F/ w  u  =  C
1411, 13nfan 1956 . . 3  |-  F/ w
( ( x  e.  A  /\  y  e.  B )  /\  u  =  C )
15 nfv 1728 . . . . 5  |-  F/ x  z  e.  A
16 cbvmpt2x.2 . . . . . 6  |-  F/_ x D
1716nfcri 2557 . . . . 5  |-  F/ x  w  e.  D
1815, 17nfan 1956 . . . 4  |-  F/ x
( z  e.  A  /\  w  e.  D
)
19 cbvmpt2x.5 . . . . 5  |-  F/_ x E
2019nfeq2 2581 . . . 4  |-  F/ x  u  =  E
2118, 20nfan 1956 . . 3  |-  F/ x
( ( z  e.  A  /\  w  e.  D )  /\  u  =  E )
22 nfv 1728 . . . 4  |-  F/ y ( z  e.  A  /\  w  e.  D
)
23 cbvmpt2x.6 . . . . 5  |-  F/_ y E
2423nfeq2 2581 . . . 4  |-  F/ y  u  =  E
2522, 24nfan 1956 . . 3  |-  F/ y ( ( z  e.  A  /\  w  e.  D )  /\  u  =  E )
26 eleq1 2474 . . . . . 6  |-  ( x  =  z  ->  (
x  e.  A  <->  z  e.  A ) )
2726adantr 463 . . . . 5  |-  ( ( x  =  z  /\  y  =  w )  ->  ( x  e.  A  <->  z  e.  A ) )
28 cbvmpt2x.7 . . . . . . 7  |-  ( x  =  z  ->  B  =  D )
2928eleq2d 2472 . . . . . 6  |-  ( x  =  z  ->  (
y  e.  B  <->  y  e.  D ) )
30 eleq1 2474 . . . . . 6  |-  ( y  =  w  ->  (
y  e.  D  <->  w  e.  D ) )
3129, 30sylan9bb 698 . . . . 5  |-  ( ( x  =  z  /\  y  =  w )  ->  ( y  e.  B  <->  w  e.  D ) )
3227, 31anbi12d 709 . . . 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 2416 . . . 4  |-  ( ( x  =  z  /\  y  =  w )  ->  ( u  =  C  <-> 
u  =  E ) )
3532, 34anbi12d 709 . . 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 6352 . 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 6283 . 2  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  { <. <. x ,  y >. ,  u >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  u  =  C ) }
38 df-mpt2 6283 . 2  |-  ( z  e.  A ,  w  e.  D  |->  E )  =  { <. <. z ,  w >. ,  u >.  |  ( ( z  e.  A  /\  w  e.  D )  /\  u  =  E ) }
3936, 37, 383eqtr4i 2441 1  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  ( z  e.  A ,  w  e.  D  |->  E )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842   F/_wnfc 2550   {coprab 6279    |-> cmpt2 6280
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pr 4630
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-rab 2763  df-v 3061  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-opab 4454  df-oprab 6282  df-mpt2 6283
This theorem is referenced by:  cbvmpt2  6357  mpt2mptsx  6847  dmmpt2ssx  6849  gsumcom2  17324  ptcmpg  20849
  Copyright terms: Public domain W3C validator