MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mpt2eq123dva Structured version   Unicode version

Theorem mpt2eq123dva 6137
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 2449 . . . . 5  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  B ) )  -> 
( z  =  C  <-> 
z  =  F ) )
32pm5.32da 624 . . . 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 2505 . . . . . . 7  |-  ( (
ph  /\  x  e.  A )  ->  (
y  e.  B  <->  y  e.  E ) )
65pm5.32da 624 . . . . . 6  |-  ( ph  ->  ( ( x  e.  A  /\  y  e.  B )  <->  ( x  e.  A  /\  y  e.  E ) ) )
7 mpt2eq123dv.1 . . . . . . . 8  |-  ( ph  ->  A  =  D )
87eleq2d 2505 . . . . . . 7  |-  ( ph  ->  ( x  e.  A  <->  x  e.  D ) )
98anbi1d 687 . . . . . 6  |-  ( ph  ->  ( ( x  e.  A  /\  y  e.  E )  <->  ( x  e.  D  /\  y  e.  E ) ) )
106, 9bitrd 246 . . . . 5  |-  ( ph  ->  ( ( x  e.  A  /\  y  e.  B )  <->  ( x  e.  D  /\  y  e.  E ) ) )
1110anbi1d 687 . . . 4  |-  ( ph  ->  ( ( ( x  e.  A  /\  y  e.  B )  /\  z  =  F )  <->  ( (
x  e.  D  /\  y  e.  E )  /\  z  =  F
) ) )
123, 11bitrd 246 . . 3  |-  ( ph  ->  ( ( ( x  e.  A  /\  y  e.  B )  /\  z  =  C )  <->  ( (
x  e.  D  /\  y  e.  E )  /\  z  =  F
) ) )
1312oprabbidv 6130 . 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 6088 . 2  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  A  /\  y  e.  B )  /\  z  =  C
) }
15 df-mpt2 6088 . 2  |-  ( x  e.  D ,  y  e.  E  |->  F )  =  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  D  /\  y  e.  E )  /\  z  =  F
) }
1613, 14, 153eqtr4g 2495 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 360    = wceq 1653    e. wcel 1726   {coprab 6084    e. cmpt2 6085
This theorem is referenced by:  mpt2eq123dv  6138  natpropd  14175  fucpropd  14176  curfpropd  14332  hofpropd  14366
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-oprab 6087  df-mpt2 6088
  Copyright terms: Public domain W3C validator