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Theorem mpteq12f 3978
Description: An equality theorem for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
Assertion
Ref Expression
mpteq12f  |-  ( ( A. x  A  =  C  /\  A. x  e.  A  B  =  D )  ->  (
x  e.  A  |->  B )  =  ( x  e.  C  |->  D ) )

Proof of Theorem mpteq12f
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 nfa1 1506 . . . 4  |-  F/ x A. x  A  =  C
2 nfra1 2443 . . . 4  |-  F/ x A. x  e.  A  B  =  D
31, 2nfan 1529 . . 3  |-  F/ x
( A. x  A  =  C  /\  A. x  e.  A  B  =  D )
4 nfv 1493 . . 3  |-  F/ y ( A. x  A  =  C  /\  A. x  e.  A  B  =  D )
5 rsp 2457 . . . . . . 7  |-  ( A. x  e.  A  B  =  D  ->  ( x  e.  A  ->  B  =  D ) )
65imp 123 . . . . . 6  |-  ( ( A. x  e.  A  B  =  D  /\  x  e.  A )  ->  B  =  D )
76eqeq2d 2129 . . . . 5  |-  ( ( A. x  e.  A  B  =  D  /\  x  e.  A )  ->  ( y  =  B  <-> 
y  =  D ) )
87pm5.32da 447 . . . 4  |-  ( A. x  e.  A  B  =  D  ->  ( ( x  e.  A  /\  y  =  B )  <->  ( x  e.  A  /\  y  =  D )
) )
9 sp 1473 . . . . . 6  |-  ( A. x  A  =  C  ->  A  =  C )
109eleq2d 2187 . . . . 5  |-  ( A. x  A  =  C  ->  ( x  e.  A  <->  x  e.  C ) )
1110anbi1d 460 . . . 4  |-  ( A. x  A  =  C  ->  ( ( x  e.  A  /\  y  =  D )  <->  ( x  e.  C  /\  y  =  D ) ) )
128, 11sylan9bbr 458 . . 3  |-  ( ( A. x  A  =  C  /\  A. x  e.  A  B  =  D )  ->  (
( x  e.  A  /\  y  =  B
)  <->  ( x  e.  C  /\  y  =  D ) ) )
133, 4, 12opabbid 3963 . 2  |-  ( ( A. x  A  =  C  /\  A. x  e.  A  B  =  D )  ->  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }  =  { <. x ,  y >.  |  ( x  e.  C  /\  y  =  D ) } )
14 df-mpt 3961 . 2  |-  ( x  e.  A  |->  B )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }
15 df-mpt 3961 . 2  |-  ( x  e.  C  |->  D )  =  { <. x ,  y >.  |  ( x  e.  C  /\  y  =  D ) }
1613, 14, 153eqtr4g 2175 1  |-  ( ( A. x  A  =  C  /\  A. x  e.  A  B  =  D )  ->  (
x  e.  A  |->  B )  =  ( x  e.  C  |->  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103   A.wal 1314    = wceq 1316    e. wcel 1465   A.wral 2393   {copab 3958    |-> cmpt 3959
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 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-11 1469  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-ral 2398  df-opab 3960  df-mpt 3961
This theorem is referenced by:  mpteq12dva  3979  mpteq12  3981  mpteq2ia  3984  mpteq2da  3987
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