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Theorem mpteq12dv 4069
Description: An equality inference for the maps-to notation. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 16-Dec-2013.)
Hypotheses
Ref Expression
mpteq12dv.1  |-  ( ph  ->  A  =  C )
mpteq12dv.2  |-  ( ph  ->  B  =  D )
Assertion
Ref Expression
mpteq12dv  |-  ( ph  ->  ( x  e.  A  |->  B )  =  ( x  e.  C  |->  D ) )
Distinct variable group:    ph, x
Allowed substitution hints:    A( x)    B( x)    C( x)    D( x)

Proof of Theorem mpteq12dv
StepHypRef Expression
1 mpteq12dv.1 . 2  |-  ( ph  ->  A  =  C )
2 mpteq12dv.2 . . 3  |-  ( ph  ->  B  =  D )
32adantr 274 . 2  |-  ( (
ph  /\  x  e.  A )  ->  B  =  D )
41, 3mpteq12dva 4068 1  |-  ( ph  ->  ( x  e.  A  |->  B )  =  ( x  e.  C  |->  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1348    e. wcel 2141    |-> cmpt 4048
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-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-ral 2453  df-opab 4049  df-mpt 4050
This theorem is referenced by:  mpteq12i  4075  offval  6066  offval3  6111  odzval  12188  restval  12578  ntrfval  12859  clsfval  12860  neifval  12899  cnpfval  12954  cnprcl2k  12965  reldvg  13407  dvfvalap  13409  eldvap  13410
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