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Theorem cbvmpt 4537
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 1683 . . . 4  |-  F/ w
( x  e.  A  /\  z  =  B
)
2 nfv 1683 . . . . 5  |-  F/ x  w  e.  A
3 nfs1v 2164 . . . . 5  |-  F/ x [ w  /  x ] z  =  B
42, 3nfan 1875 . . . 4  |-  F/ x
( w  e.  A  /\  [ w  /  x ] z  =  B )
5 eleq1 2539 . . . . 5  |-  ( x  =  w  ->  (
x  e.  A  <->  w  e.  A ) )
6 sbequ12 1961 . . . . 5  |-  ( x  =  w  ->  (
z  =  B  <->  [ w  /  x ] z  =  B ) )
75, 6anbi12d 710 . . . 4  |-  ( x  =  w  ->  (
( x  e.  A  /\  z  =  B
)  <->  ( w  e.  A  /\  [ w  /  x ] z  =  B ) ) )
81, 4, 7cbvopab1 4517 . . 3  |-  { <. x ,  z >.  |  ( x  e.  A  /\  z  =  B ) }  =  { <. w ,  z >.  |  ( w  e.  A  /\  [ w  /  x ]
z  =  B ) }
9 nfv 1683 . . . . 5  |-  F/ y  w  e.  A
10 cbvmpt.1 . . . . . . 7  |-  F/_ y B
1110nfeq2 2646 . . . . . 6  |-  F/ y  z  =  B
1211nfsb 2168 . . . . 5  |-  F/ y [ w  /  x ] z  =  B
139, 12nfan 1875 . . . 4  |-  F/ y ( w  e.  A  /\  [ w  /  x ] z  =  B )
14 nfv 1683 . . . 4  |-  F/ w
( y  e.  A  /\  z  =  C
)
15 eleq1 2539 . . . . 5  |-  ( w  =  y  ->  (
w  e.  A  <->  y  e.  A ) )
16 sbequ 2090 . . . . . 6  |-  ( w  =  y  ->  ( [ w  /  x ] z  =  B  <->  [ y  /  x ] z  =  B ) )
17 cbvmpt.2 . . . . . . . 8  |-  F/_ x C
1817nfeq2 2646 . . . . . . 7  |-  F/ x  z  =  C
19 cbvmpt.3 . . . . . . . 8  |-  ( x  =  y  ->  B  =  C )
2019eqeq2d 2481 . . . . . . 7  |-  ( x  =  y  ->  (
z  =  B  <->  z  =  C ) )
2118, 20sbie 2123 . . . . . 6  |-  ( [ y  /  x ]
z  =  B  <->  z  =  C )
2216, 21syl6bb 261 . . . . 5  |-  ( w  =  y  ->  ( [ w  /  x ] z  =  B  <-> 
z  =  C ) )
2315, 22anbi12d 710 . . . 4  |-  ( w  =  y  ->  (
( w  e.  A  /\  [ w  /  x ] z  =  B )  <->  ( y  e.  A  /\  z  =  C ) ) )
2413, 14, 23cbvopab1 4517 . . 3  |-  { <. w ,  z >.  |  ( w  e.  A  /\  [ w  /  x ]
z  =  B ) }  =  { <. y ,  z >.  |  ( y  e.  A  /\  z  =  C ) }
258, 24eqtri 2496 . 2  |-  { <. x ,  z >.  |  ( x  e.  A  /\  z  =  B ) }  =  { <. y ,  z >.  |  ( y  e.  A  /\  z  =  C ) }
26 df-mpt 4507 . 2  |-  ( x  e.  A  |->  B )  =  { <. x ,  z >.  |  ( x  e.  A  /\  z  =  B ) }
27 df-mpt 4507 . 2  |-  ( y  e.  A  |->  C )  =  { <. y ,  z >.  |  ( y  e.  A  /\  z  =  C ) }
2825, 26, 273eqtr4i 2506 1  |-  ( x  e.  A  |->  B )  =  ( y  e.  A  |->  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379   [wsb 1711    e. wcel 1767   F/_wnfc 2615   {copab 4504    |-> cmpt 4505
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-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
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-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-opab 4506  df-mpt 4507
This theorem is referenced by:  cbvmptv  4538  dffn5f  5922  fvmpts  5952  fvmpt2i  5956  fvmptex  5960  fmptcof  6055  fmptcos  6056  fliftfuns  6200  offval2  6540  ofmpteq  6542  mpt2curryvald  6999  qliftfuns  7398  axcc2  8817  ac6num  8859  seqof2  12133  summolem2a  13500  zsum  13503  fsumcvg2  13512  fsumrlim  13588  pcmptdvds  14272  prdsdsval2  14739  gsumconstf  16758  gsummpt1n0  16795  gsum2d2  16805  dprd2d2  16895  gsumdixpOLD  17058  gsumdixp  17059  psrass1lem  17828  coe1fzgsumdlem  18142  gsumply1eq  18146  evl1gsumdlem  18191  madugsum  18940  cnmpt1t  19929  cnmpt2k  19952  elmptrab  20091  flfcnp2  20271  prdsxmet  20635  fsumcn  21137  ovoliunlem3  21678  ovoliun  21679  ovoliun2  21680  voliun  21727  mbfpos  21821  mbfposb  21823  i1fposd  21877  itg2cnlem1  21931  isibl2  21936  cbvitg  21945  itgss3  21984  itgfsum  21996  itgabs  22004  itgcn  22012  limcmpt  22050  dvmptfsum  22139  lhop2  22179  dvfsumle  22185  dvfsumlem2  22191  itgsubstlem  22212  itgsubst  22213  itgulm2  22566  rlimcnp2  23052  xrge0tmd  27592  cbvprod  28652  prodmolem2a  28671  zprod  28674  fprod  28678  mbfposadd  29667  itgabsnc  29689  ftc1cnnclem  29693  ftc2nc  29704  mzpsubst  30313  rabdiophlem2  30367  aomclem8  30639  fsumcnf  31002  cncfmptss  31165  mulc1cncfg  31167  expcnfg  31170  icccncfext  31254  cncficcgt0  31255  cncfiooicclem1  31260  dvsinax  31269  iblsplitf  31316  itgiccshift  31326  itgsbtaddcnst  31328  stoweidlem21  31349  stirlinglem4  31405  stirlinglem13  31414  stirlinglem15  31416  dirkerval  31419  dirkerval2  31422  dirkercncflem2  31432  dirkercncflem3  31433  dirkercncflem4  31434  dirkercncf  31435  fourierdlem81  31516  fourierdlem92  31527  fourierdlem93  31528  fourierdlem101  31536  fourierdlem103  31538  fourierdlem104  31539  fourierdlem111  31546  fourierd  31551  fourierclimd  31552
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