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Theorem mpteq12dva 4250
Description: An equality inference for the maps to notation. (Contributed by Mario Carneiro, 26-Jan-2017.)
Hypotheses
Ref Expression
mpteq12dv.1  |-  ( ph  ->  A  =  C )
mpteq12dva.2  |-  ( (
ph  /\  x  e.  A )  ->  B  =  D )
Assertion
Ref Expression
mpteq12dva  |-  ( 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 mpteq12dva
StepHypRef Expression
1 mpteq12dv.1 . . 3  |-  ( ph  ->  A  =  C )
21alrimiv 1638 . 2  |-  ( ph  ->  A. x  A  =  C )
3 mpteq12dva.2 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  B  =  D )
43ralrimiva 2753 . 2  |-  ( ph  ->  A. x  e.  A  B  =  D )
5 mpteq12f 4249 . 2  |-  ( ( A. x  A  =  C  /\  A. x  e.  A  B  =  D )  ->  (
x  e.  A  |->  B )  =  ( x  e.  C  |->  D ) )
62, 4, 5syl2anc 643 1  |-  ( ph  ->  ( x  e.  A  |->  B )  =  ( x  e.  C  |->  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   A.wal 1546    = wceq 1649    e. wcel 1721   A.wral 2670    e. cmpt 4230
This theorem is referenced by:  mpteq12dv  4251  cidpropd  13895  monpropd  13922  fucpropd  14133  curfpropd  14289  hofpropd  14323  yonffthlem  14338  ofcfval  24438
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2395  df-cleq 2401  df-clel 2404  df-ral 2675  df-opab 4231  df-mpt 4232
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