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Theorem mpteq12f 4533
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 1898 . . . 4  |-  F/ x A. x  A  =  C
2 nfra1 2838 . . . 4  |-  F/ x A. x  e.  A  B  =  D
31, 2nfan 1929 . . 3  |-  F/ x
( A. x  A  =  C  /\  A. x  e.  A  B  =  D )
4 nfv 1708 . . 3  |-  F/ y ( A. x  A  =  C  /\  A. x  e.  A  B  =  D )
5 rspa 2824 . . . . . 6  |-  ( ( A. x  e.  A  B  =  D  /\  x  e.  A )  ->  B  =  D )
65eqeq2d 2471 . . . . 5  |-  ( ( A. x  e.  A  B  =  D  /\  x  e.  A )  ->  ( y  =  B  <-> 
y  =  D ) )
76pm5.32da 641 . . . 4  |-  ( A. x  e.  A  B  =  D  ->  ( ( x  e.  A  /\  y  =  B )  <->  ( x  e.  A  /\  y  =  D )
) )
8 sp 1860 . . . . . 6  |-  ( A. x  A  =  C  ->  A  =  C )
98eleq2d 2527 . . . . 5  |-  ( A. x  A  =  C  ->  ( x  e.  A  <->  x  e.  C ) )
109anbi1d 704 . . . 4  |-  ( A. x  A  =  C  ->  ( ( x  e.  A  /\  y  =  D )  <->  ( x  e.  C  /\  y  =  D ) ) )
117, 10sylan9bbr 700 . . 3  |-  ( ( A. x  A  =  C  /\  A. x  e.  A  B  =  D )  ->  (
( x  e.  A  /\  y  =  B
)  <->  ( x  e.  C  /\  y  =  D ) ) )
123, 4, 11opabbid 4519 . 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 ) } )
13 df-mpt 4517 . 2  |-  ( x  e.  A  |->  B )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }
14 df-mpt 4517 . 2  |-  ( x  e.  C  |->  D )  =  { <. x ,  y >.  |  ( x  e.  C  /\  y  =  D ) }
1512, 13, 143eqtr4g 2523 1  |-  ( ( A. x  A  =  C  /\  A. x  e.  A  B  =  D )  ->  (
x  e.  A  |->  B )  =  ( x  e.  C  |->  D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369   A.wal 1393    = wceq 1395    e. wcel 1819   A.wral 2807   {copab 4514    |-> cmpt 4515
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-ral 2812  df-opab 4516  df-mpt 4517
This theorem is referenced by:  mpteq12dva  4534  mpteq12  4536  mpteq2ia  4539  mpteq2da  4542  esumeq12dvaf  28197  refsum2cnlem1  31573
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