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Theorem mpt2eq123dva 6094
Description: An equality deduction for the maps to notation. (Contributed by Mario Carneiro, 26-Jan-2017.)
Hypotheses
Ref Expression
mpt2eq123dv.1  |-  ( ph  ->  A  =  D )
mpt2eq123dva.2  |-  ( (
ph  /\  x  e.  A )  ->  B  =  E )
mpt2eq123dva.3  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  B ) )  ->  C  =  F )
Assertion
Ref Expression
mpt2eq123dva  |-  ( ph  ->  ( x  e.  A ,  y  e.  B  |->  C )  =  ( x  e.  D , 
y  e.  E  |->  F ) )
Distinct variable groups:    ph, x    ph, y
Allowed substitution hints:    A( x, y)    B( x, y)    C( x, y)    D( x, y)    E( x, y)    F( x, y)

Proof of Theorem mpt2eq123dva
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 mpt2eq123dva.3 . . . . . 6  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  B ) )  ->  C  =  F )
21eqeq2d 2415 . . . . 5  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  B ) )  -> 
( z  =  C  <-> 
z  =  F ) )
32pm5.32da 623 . . . 4  |-  ( ph  ->  ( ( ( x  e.  A  /\  y  e.  B )  /\  z  =  C )  <->  ( (
x  e.  A  /\  y  e.  B )  /\  z  =  F
) ) )
4 mpt2eq123dva.2 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A )  ->  B  =  E )
54eleq2d 2471 . . . . . . 7  |-  ( (
ph  /\  x  e.  A )  ->  (
y  e.  B  <->  y  e.  E ) )
65pm5.32da 623 . . . . . 6  |-  ( ph  ->  ( ( x  e.  A  /\  y  e.  B )  <->  ( x  e.  A  /\  y  e.  E ) ) )
7 mpt2eq123dv.1 . . . . . . . 8  |-  ( ph  ->  A  =  D )
87eleq2d 2471 . . . . . . 7  |-  ( ph  ->  ( x  e.  A  <->  x  e.  D ) )
98anbi1d 686 . . . . . 6  |-  ( ph  ->  ( ( x  e.  A  /\  y  e.  E )  <->  ( x  e.  D  /\  y  e.  E ) ) )
106, 9bitrd 245 . . . . 5  |-  ( ph  ->  ( ( x  e.  A  /\  y  e.  B )  <->  ( x  e.  D  /\  y  e.  E ) ) )
1110anbi1d 686 . . . 4  |-  ( ph  ->  ( ( ( x  e.  A  /\  y  e.  B )  /\  z  =  F )  <->  ( (
x  e.  D  /\  y  e.  E )  /\  z  =  F
) ) )
123, 11bitrd 245 . . 3  |-  ( ph  ->  ( ( ( x  e.  A  /\  y  e.  B )  /\  z  =  C )  <->  ( (
x  e.  D  /\  y  e.  E )  /\  z  =  F
) ) )
1312oprabbidv 6087 . 2  |-  ( ph  ->  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  z  =  C ) }  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  D  /\  y  e.  E )  /\  z  =  F ) } )
14 df-mpt2 6045 . 2  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  A  /\  y  e.  B )  /\  z  =  C
) }
15 df-mpt2 6045 . 2  |-  ( x  e.  D ,  y  e.  E  |->  F )  =  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  D  /\  y  e.  E )  /\  z  =  F
) }
1613, 14, 153eqtr4g 2461 1  |-  ( ph  ->  ( x  e.  A ,  y  e.  B  |->  C )  =  ( x  e.  D , 
y  e.  E  |->  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   {coprab 6041    e. cmpt2 6042
This theorem is referenced by:  mpt2eq123dv  6095  natpropd  14128  fucpropd  14129  curfpropd  14285  hofpropd  14319
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-oprab 6044  df-mpt2 6045
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