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Theorem mpteq12f 3865
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 1450 . . . 4  |-  F/ x A. x  A  =  C
2 nfra1 2372 . . . 4  |-  F/ x A. x  e.  A  B  =  D
31, 2nfan 1473 . . 3  |-  F/ x
( A. x  A  =  C  /\  A. x  e.  A  B  =  D )
4 nfv 1437 . . 3  |-  F/ y ( A. x  A  =  C  /\  A. x  e.  A  B  =  D )
5 rsp 2386 . . . . . . 7  |-  ( A. x  e.  A  B  =  D  ->  ( x  e.  A  ->  B  =  D ) )
65imp 119 . . . . . 6  |-  ( ( A. x  e.  A  B  =  D  /\  x  e.  A )  ->  B  =  D )
76eqeq2d 2067 . . . . 5  |-  ( ( A. x  e.  A  B  =  D  /\  x  e.  A )  ->  ( y  =  B  <-> 
y  =  D ) )
87pm5.32da 433 . . . 4  |-  ( A. x  e.  A  B  =  D  ->  ( ( x  e.  A  /\  y  =  B )  <->  ( x  e.  A  /\  y  =  D )
) )
9 sp 1417 . . . . . 6  |-  ( A. x  A  =  C  ->  A  =  C )
109eleq2d 2123 . . . . 5  |-  ( A. x  A  =  C  ->  ( x  e.  A  <->  x  e.  C ) )
1110anbi1d 446 . . . 4  |-  ( A. x  A  =  C  ->  ( ( x  e.  A  /\  y  =  D )  <->  ( x  e.  C  /\  y  =  D ) ) )
128, 11sylan9bbr 444 . . 3  |-  ( ( A. x  A  =  C  /\  A. x  e.  A  B  =  D )  ->  (
( x  e.  A  /\  y  =  B
)  <->  ( x  e.  C  /\  y  =  D ) ) )
133, 4, 12opabbid 3850 . 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 3848 . 2  |-  ( x  e.  A  |->  B )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }
15 df-mpt 3848 . 2  |-  ( x  e.  C  |->  D )  =  { <. x ,  y >.  |  ( x  e.  C  /\  y  =  D ) }
1613, 14, 153eqtr4g 2113 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 101   A.wal 1257    = wceq 1259    e. wcel 1409   A.wral 2323   {copab 3845    |-> cmpt 3846
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-ral 2328  df-opab 3847  df-mpt 3848
This theorem is referenced by:  mpteq12dva  3866  mpteq12  3868  mpteq2ia  3871  mpteq2da  3874
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