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Theorem cbvmptv 3879
Description: Rule to change the bound variable in a maps-to function, using implicit substitution. (Contributed by Mario Carneiro, 19-Feb-2013.)
Hypothesis
Ref Expression
cbvmptv.1  |-  ( x  =  y  ->  B  =  C )
Assertion
Ref Expression
cbvmptv  |-  ( x  e.  A  |->  B )  =  ( y  e.  A  |->  C )
Distinct variable groups:    x, A    y, A    y, B    x, C
Allowed substitution hints:    B( x)    C( y)

Proof of Theorem cbvmptv
StepHypRef Expression
1 nfcv 2194 . 2  |-  F/_ y B
2 nfcv 2194 . 2  |-  F/_ x C
3 cbvmptv.1 . 2  |-  ( x  =  y  ->  B  =  C )
41, 2, 3cbvmpt 3878 1  |-  ( x  e.  A  |->  B )  =  ( y  e.  A  |->  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1259    |-> cmpt 3845
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 2949  df-sn 3408  df-pr 3409  df-op 3411  df-opab 3846  df-mpt 3847
This theorem is referenced by:  frecsuc  6021  caucvgsrlembnd  6942  frec2uzzd  9349  frec2uzsucd  9350  climcvg1n  10099
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