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Theorem cbvmpt 3993
Description: Rule to change the bound variable in a maps-to function, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable conditions. (Contributed by NM, 11-Sep-2011.)
Hypotheses
Ref Expression
cbvmpt.1  |-  F/_ y B
cbvmpt.2  |-  F/_ x C
cbvmpt.3  |-  ( x  =  y  ->  B  =  C )
Assertion
Ref Expression
cbvmpt  |-  ( x  e.  A  |->  B )  =  ( y  e.  A  |->  C )
Distinct variable groups:    x, A    y, A
Allowed substitution hints:    B( x, y)    C( x, y)

Proof of Theorem cbvmpt
Dummy variables  z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 1493 . . . 4  |-  F/ w
( x  e.  A  /\  z  =  B
)
2 nfv 1493 . . . . 5  |-  F/ x  w  e.  A
3 nfs1v 1892 . . . . 5  |-  F/ x [ w  /  x ] z  =  B
42, 3nfan 1529 . . . 4  |-  F/ x
( w  e.  A  /\  [ w  /  x ] z  =  B )
5 eleq1 2180 . . . . 5  |-  ( x  =  w  ->  (
x  e.  A  <->  w  e.  A ) )
6 sbequ12 1729 . . . . 5  |-  ( x  =  w  ->  (
z  =  B  <->  [ w  /  x ] z  =  B ) )
75, 6anbi12d 464 . . . 4  |-  ( x  =  w  ->  (
( x  e.  A  /\  z  =  B
)  <->  ( w  e.  A  /\  [ w  /  x ] z  =  B ) ) )
81, 4, 7cbvopab1 3971 . . 3  |-  { <. x ,  z >.  |  ( x  e.  A  /\  z  =  B ) }  =  { <. w ,  z >.  |  ( w  e.  A  /\  [ w  /  x ]
z  =  B ) }
9 nfv 1493 . . . . 5  |-  F/ y  w  e.  A
10 cbvmpt.1 . . . . . . 7  |-  F/_ y B
1110nfeq2 2270 . . . . . 6  |-  F/ y  z  =  B
1211nfsb 1899 . . . . 5  |-  F/ y [ w  /  x ] z  =  B
139, 12nfan 1529 . . . 4  |-  F/ y ( w  e.  A  /\  [ w  /  x ] z  =  B )
14 nfv 1493 . . . 4  |-  F/ w
( y  e.  A  /\  z  =  C
)
15 eleq1 2180 . . . . 5  |-  ( w  =  y  ->  (
w  e.  A  <->  y  e.  A ) )
16 sbequ 1796 . . . . . 6  |-  ( w  =  y  ->  ( [ w  /  x ] z  =  B  <->  [ y  /  x ] z  =  B ) )
17 cbvmpt.2 . . . . . . . 8  |-  F/_ x C
1817nfeq2 2270 . . . . . . 7  |-  F/ x  z  =  C
19 cbvmpt.3 . . . . . . . 8  |-  ( x  =  y  ->  B  =  C )
2019eqeq2d 2129 . . . . . . 7  |-  ( x  =  y  ->  (
z  =  B  <->  z  =  C ) )
2118, 20sbie 1749 . . . . . 6  |-  ( [ y  /  x ]
z  =  B  <->  z  =  C )
2216, 21syl6bb 195 . . . . 5  |-  ( w  =  y  ->  ( [ w  /  x ] z  =  B  <-> 
z  =  C ) )
2315, 22anbi12d 464 . . . 4  |-  ( w  =  y  ->  (
( w  e.  A  /\  [ w  /  x ] z  =  B )  <->  ( y  e.  A  /\  z  =  C ) ) )
2413, 14, 23cbvopab1 3971 . . 3  |-  { <. w ,  z >.  |  ( w  e.  A  /\  [ w  /  x ]
z  =  B ) }  =  { <. y ,  z >.  |  ( y  e.  A  /\  z  =  C ) }
258, 24eqtri 2138 . 2  |-  { <. x ,  z >.  |  ( x  e.  A  /\  z  =  B ) }  =  { <. y ,  z >.  |  ( y  e.  A  /\  z  =  C ) }
26 df-mpt 3961 . 2  |-  ( x  e.  A  |->  B )  =  { <. x ,  z >.  |  ( x  e.  A  /\  z  =  B ) }
27 df-mpt 3961 . 2  |-  ( y  e.  A  |->  C )  =  { <. y ,  z >.  |  ( y  e.  A  /\  z  =  C ) }
2825, 26, 273eqtr4i 2148 1  |-  ( x  e.  A  |->  B )  =  ( y  e.  A  |->  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1316    e. wcel 1465   [wsb 1720   F/_wnfc 2245   {copab 3958    |-> cmpt 3959
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-un 3045  df-sn 3503  df-pr 3504  df-op 3506  df-opab 3960  df-mpt 3961
This theorem is referenced by:  cbvmptv  3994  dffn5imf  5444  fvmpts  5467  fvmpt2  5472  mptfvex  5474  fmptcof  5555  fmptcos  5556  fliftfuns  5667  offval2  5965  qliftfuns  6481  summodclem2a  11118  zsumdc  11121  fsum3cvg2  11131  cnmpt1t  12381  fsumcncntop  12652  limcmpted  12728
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