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Theorem cbvmptf 4493
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 Thierry Arnoux, 9-Mar-2017.)
Hypotheses
Ref Expression
cbvmptf.1  |-  F/_ x A
cbvmptf.2  |-  F/_ y A
cbvmptf.3  |-  F/_ y B
cbvmptf.4  |-  F/_ x C
cbvmptf.5  |-  ( x  =  y  ->  B  =  C )
Assertion
Ref Expression
cbvmptf  |-  ( x  e.  A  |->  B )  =  ( y  e.  A  |->  C )
Distinct variable group:    x, y
Allowed substitution hints:    A( x, y)    B( x, y)    C( x, y)

Proof of Theorem cbvmptf
Dummy variables  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 1761 . . . 4  |-  F/ w
( x  e.  A  /\  z  =  B
)
2 cbvmptf.1 . . . . . 6  |-  F/_ x A
32nfcri 2586 . . . . 5  |-  F/ x  w  e.  A
4 nfs1v 2266 . . . . 5  |-  F/ x [ w  /  x ] z  =  B
53, 4nfan 2011 . . . 4  |-  F/ x
( w  e.  A  /\  [ w  /  x ] z  =  B )
6 eleq1 2517 . . . . 5  |-  ( x  =  w  ->  (
x  e.  A  <->  w  e.  A ) )
7 sbequ12 2083 . . . . 5  |-  ( x  =  w  ->  (
z  =  B  <->  [ w  /  x ] z  =  B ) )
86, 7anbi12d 717 . . . 4  |-  ( x  =  w  ->  (
( x  e.  A  /\  z  =  B
)  <->  ( w  e.  A  /\  [ w  /  x ] z  =  B ) ) )
91, 5, 8cbvopab1 4473 . . 3  |-  { <. x ,  z >.  |  ( x  e.  A  /\  z  =  B ) }  =  { <. w ,  z >.  |  ( w  e.  A  /\  [ w  /  x ]
z  =  B ) }
10 cbvmptf.2 . . . . . 6  |-  F/_ y A
1110nfcri 2586 . . . . 5  |-  F/ y  w  e.  A
12 cbvmptf.3 . . . . . . 7  |-  F/_ y B
1312nfeq2 2607 . . . . . 6  |-  F/ y  z  =  B
1413nfsb 2269 . . . . 5  |-  F/ y [ w  /  x ] z  =  B
1511, 14nfan 2011 . . . 4  |-  F/ y ( w  e.  A  /\  [ w  /  x ] z  =  B )
16 nfv 1761 . . . 4  |-  F/ w
( y  e.  A  /\  z  =  C
)
17 eleq1 2517 . . . . 5  |-  ( w  =  y  ->  (
w  e.  A  <->  y  e.  A ) )
18 cbvmptf.4 . . . . . . 7  |-  F/_ x C
1918nfeq2 2607 . . . . . 6  |-  F/ x  z  =  C
20 cbvmptf.5 . . . . . . 7  |-  ( x  =  y  ->  B  =  C )
2120eqeq2d 2461 . . . . . 6  |-  ( x  =  y  ->  (
z  =  B  <->  z  =  C ) )
2219, 21sbhypf 3095 . . . . 5  |-  ( w  =  y  ->  ( [ w  /  x ] z  =  B  <-> 
z  =  C ) )
2317, 22anbi12d 717 . . . 4  |-  ( w  =  y  ->  (
( w  e.  A  /\  [ w  /  x ] z  =  B )  <->  ( y  e.  A  /\  z  =  C ) ) )
2415, 16, 23cbvopab1 4473 . . 3  |-  { <. w ,  z >.  |  ( w  e.  A  /\  [ w  /  x ]
z  =  B ) }  =  { <. y ,  z >.  |  ( y  e.  A  /\  z  =  C ) }
259, 24eqtri 2473 . 2  |-  { <. x ,  z >.  |  ( x  e.  A  /\  z  =  B ) }  =  { <. y ,  z >.  |  ( y  e.  A  /\  z  =  C ) }
26 df-mpt 4463 . 2  |-  ( x  e.  A  |->  B )  =  { <. x ,  z >.  |  ( x  e.  A  /\  z  =  B ) }
27 df-mpt 4463 . 2  |-  ( y  e.  A  |->  C )  =  { <. y ,  z >.  |  ( y  e.  A  /\  z  =  C ) }
2825, 26, 273eqtr4i 2483 1  |-  ( x  e.  A  |->  B )  =  ( y  e.  A  |->  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    = wceq 1444   [wsb 1797    e. wcel 1887   F/_wnfc 2579   {copab 4460    |-> cmpt 4461
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-rab 2746  df-v 3047  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-opab 4462  df-mpt 4463
This theorem is referenced by:  fvmpt2f  5949  offval2f  6543  suppss2f  28238  fmptdF  28255  resmptf  28257  acunirnmpt2f  28263  funcnv4mpt  28273  cbvesum  28863  esumpfinvalf  28897  binomcxplemdvbinom  36702  binomcxplemdvsum  36704  binomcxplemnotnn0  36705  sge0iunmptlemre  38257
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