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Theorem cbvmpt 3879
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 1437 . . . 4  |-  F/ w
( x  e.  A  /\  z  =  B
)
2 nfv 1437 . . . . 5  |-  F/ x  w  e.  A
3 nfs1v 1831 . . . . 5  |-  F/ x [ w  /  x ] z  =  B
42, 3nfan 1473 . . . 4  |-  F/ x
( w  e.  A  /\  [ w  /  x ] z  =  B )
5 eleq1 2116 . . . . 5  |-  ( x  =  w  ->  (
x  e.  A  <->  w  e.  A ) )
6 sbequ12 1670 . . . . 5  |-  ( x  =  w  ->  (
z  =  B  <->  [ w  /  x ] z  =  B ) )
75, 6anbi12d 450 . . . 4  |-  ( x  =  w  ->  (
( x  e.  A  /\  z  =  B
)  <->  ( w  e.  A  /\  [ w  /  x ] z  =  B ) ) )
81, 4, 7cbvopab1 3858 . . 3  |-  { <. x ,  z >.  |  ( x  e.  A  /\  z  =  B ) }  =  { <. w ,  z >.  |  ( w  e.  A  /\  [ w  /  x ]
z  =  B ) }
9 nfv 1437 . . . . 5  |-  F/ y  w  e.  A
10 cbvmpt.1 . . . . . . 7  |-  F/_ y B
1110nfeq2 2205 . . . . . 6  |-  F/ y  z  =  B
1211nfsb 1838 . . . . 5  |-  F/ y [ w  /  x ] z  =  B
139, 12nfan 1473 . . . 4  |-  F/ y ( w  e.  A  /\  [ w  /  x ] z  =  B )
14 nfv 1437 . . . 4  |-  F/ w
( y  e.  A  /\  z  =  C
)
15 eleq1 2116 . . . . 5  |-  ( w  =  y  ->  (
w  e.  A  <->  y  e.  A ) )
16 sbequ 1737 . . . . . 6  |-  ( w  =  y  ->  ( [ w  /  x ] z  =  B  <->  [ y  /  x ] z  =  B ) )
17 cbvmpt.2 . . . . . . . 8  |-  F/_ x C
1817nfeq2 2205 . . . . . . 7  |-  F/ x  z  =  C
19 cbvmpt.3 . . . . . . . 8  |-  ( x  =  y  ->  B  =  C )
2019eqeq2d 2067 . . . . . . 7  |-  ( x  =  y  ->  (
z  =  B  <->  z  =  C ) )
2118, 20sbie 1690 . . . . . 6  |-  ( [ y  /  x ]
z  =  B  <->  z  =  C )
2216, 21syl6bb 189 . . . . 5  |-  ( w  =  y  ->  ( [ w  /  x ] z  =  B  <-> 
z  =  C ) )
2315, 22anbi12d 450 . . . 4  |-  ( w  =  y  ->  (
( w  e.  A  /\  [ w  /  x ] z  =  B )  <->  ( y  e.  A  /\  z  =  C ) ) )
2413, 14, 23cbvopab1 3858 . . 3  |-  { <. w ,  z >.  |  ( w  e.  A  /\  [ w  /  x ]
z  =  B ) }  =  { <. y ,  z >.  |  ( y  e.  A  /\  z  =  C ) }
258, 24eqtri 2076 . 2  |-  { <. x ,  z >.  |  ( x  e.  A  /\  z  =  B ) }  =  { <. y ,  z >.  |  ( y  e.  A  /\  z  =  C ) }
26 df-mpt 3848 . 2  |-  ( x  e.  A  |->  B )  =  { <. x ,  z >.  |  ( x  e.  A  /\  z  =  B ) }
27 df-mpt 3848 . 2  |-  ( y  e.  A  |->  C )  =  { <. y ,  z >.  |  ( y  e.  A  /\  z  =  C ) }
2825, 26, 273eqtr4i 2086 1  |-  ( x  e.  A  |->  B )  =  ( y  e.  A  |->  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    = wceq 1259    e. wcel 1409   [wsb 1661   F/_wnfc 2181   {copab 3845    |-> cmpt 3846
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2950  df-sn 3409  df-pr 3410  df-op 3412  df-opab 3847  df-mpt 3848
This theorem is referenced by:  cbvmptv  3880  dffn5imf  5256  fvmpts  5278  fvmpt2  5282  mptfvex  5284  fmptcof  5359  fmptcos  5360  fliftfuns  5466  offval2  5754  qliftfuns  6221
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