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Theorem mpteq12f 4277
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 1806 . . . 4  |-  F/ x A. x  A  =  C
2 nfra1 2748 . . . 4  |-  F/ x A. x  e.  A  B  =  D
31, 2nfan 1846 . . 3  |-  F/ x
( A. x  A  =  C  /\  A. x  e.  A  B  =  D )
4 nfv 1629 . . 3  |-  F/ y ( A. x  A  =  C  /\  A. x  e.  A  B  =  D )
5 rsp 2758 . . . . . . 7  |-  ( A. x  e.  A  B  =  D  ->  ( x  e.  A  ->  B  =  D ) )
65imp 419 . . . . . 6  |-  ( ( A. x  e.  A  B  =  D  /\  x  e.  A )  ->  B  =  D )
76eqeq2d 2446 . . . . 5  |-  ( ( A. x  e.  A  B  =  D  /\  x  e.  A )  ->  ( y  =  B  <-> 
y  =  D ) )
87pm5.32da 623 . . . 4  |-  ( A. x  e.  A  B  =  D  ->  ( ( x  e.  A  /\  y  =  B )  <->  ( x  e.  A  /\  y  =  D )
) )
9 sp 1763 . . . . . 6  |-  ( A. x  A  =  C  ->  A  =  C )
109eleq2d 2502 . . . . 5  |-  ( A. x  A  =  C  ->  ( x  e.  A  <->  x  e.  C ) )
1110anbi1d 686 . . . 4  |-  ( A. x  A  =  C  ->  ( ( x  e.  A  /\  y  =  D )  <->  ( x  e.  C  /\  y  =  D ) ) )
128, 11sylan9bbr 682 . . 3  |-  ( ( A. x  A  =  C  /\  A. x  e.  A  B  =  D )  ->  (
( x  e.  A  /\  y  =  B
)  <->  ( x  e.  C  /\  y  =  D ) ) )
133, 4, 12opabbid 4262 . 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 4260 . 2  |-  ( x  e.  A  |->  B )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }
15 df-mpt 4260 . 2  |-  ( x  e.  C  |->  D )  =  { <. x ,  y >.  |  ( x  e.  C  /\  y  =  D ) }
1613, 14, 153eqtr4g 2492 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 359   A.wal 1549    = wceq 1652    e. wcel 1725   A.wral 2697   {copab 4257    e. cmpt 4258
This theorem is referenced by:  mpteq12dva  4278  mpteq12  4280  mpteq2ia  4283  mpteq2da  4286  esumeq12dvaf  24420  refsum2cnlem1  27675
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-ral 2702  df-opab 4259  df-mpt 4260
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