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Theorem mpteq12f 4112
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 1768 . . . 4  |-  F/ x A. x  A  =  C
2 nfra1 2606 . . . 4  |-  F/ x A. x  e.  A  B  =  D
31, 2nfan 1783 . . 3  |-  F/ x
( A. x  A  =  C  /\  A. x  e.  A  B  =  D )
4 nfv 1609 . . 3  |-  F/ y ( A. x  A  =  C  /\  A. x  e.  A  B  =  D )
5 rsp 2616 . . . . . . 7  |-  ( A. x  e.  A  B  =  D  ->  ( x  e.  A  ->  B  =  D ) )
65imp 418 . . . . . 6  |-  ( ( A. x  e.  A  B  =  D  /\  x  e.  A )  ->  B  =  D )
76eqeq2d 2307 . . . . 5  |-  ( ( A. x  e.  A  B  =  D  /\  x  e.  A )  ->  ( y  =  B  <-> 
y  =  D ) )
87pm5.32da 622 . . . 4  |-  ( A. x  e.  A  B  =  D  ->  ( ( x  e.  A  /\  y  =  B )  <->  ( x  e.  A  /\  y  =  D )
) )
9 sp 1728 . . . . . 6  |-  ( A. x  A  =  C  ->  A  =  C )
109eleq2d 2363 . . . . 5  |-  ( A. x  A  =  C  ->  ( x  e.  A  <->  x  e.  C ) )
1110anbi1d 685 . . . 4  |-  ( A. x  A  =  C  ->  ( ( x  e.  A  /\  y  =  D )  <->  ( x  e.  C  /\  y  =  D ) ) )
128, 11sylan9bbr 681 . . 3  |-  ( ( A. x  A  =  C  /\  A. x  e.  A  B  =  D )  ->  (
( x  e.  A  /\  y  =  B
)  <->  ( x  e.  C  /\  y  =  D ) ) )
133, 4, 12opabbid 4097 . 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 4095 . 2  |-  ( x  e.  A  |->  B )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }
15 df-mpt 4095 . 2  |-  ( x  e.  C  |->  D )  =  { <. x ,  y >.  |  ( x  e.  C  /\  y  =  D ) }
1613, 14, 153eqtr4g 2353 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 358   A.wal 1530    = wceq 1632    e. wcel 1696   A.wral 2556   {copab 4092    e. cmpt 4093
This theorem is referenced by:  mpteq12dva  4113  mpteq12  4115  mpteq2ia  4118  mpteq2da  4121  esumeq12dvaf  23429  refsum2cnlem1  27811  stoweidlem19  27871
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-ral 2561  df-opab 4094  df-mpt 4095
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