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Theorem mpt2eq123dva 6343
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 2457 . . . . 5  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  B ) )  -> 
( z  =  C  <-> 
z  =  F ) )
32pm5.32da 641 . . . 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 2513 . . . . . . 7  |-  ( (
ph  /\  x  e.  A )  ->  (
y  e.  B  <->  y  e.  E ) )
65pm5.32da 641 . . . . . 6  |-  ( ph  ->  ( ( x  e.  A  /\  y  e.  B )  <->  ( x  e.  A  /\  y  e.  E ) ) )
7 mpt2eq123dv.1 . . . . . . . 8  |-  ( ph  ->  A  =  D )
87eleq2d 2513 . . . . . . 7  |-  ( ph  ->  ( x  e.  A  <->  x  e.  D ) )
98anbi1d 704 . . . . . 6  |-  ( ph  ->  ( ( x  e.  A  /\  y  e.  E )  <->  ( x  e.  D  /\  y  e.  E ) ) )
106, 9bitrd 253 . . . . 5  |-  ( ph  ->  ( ( x  e.  A  /\  y  e.  B )  <->  ( x  e.  D  /\  y  e.  E ) ) )
1110anbi1d 704 . . . 4  |-  ( ph  ->  ( ( ( x  e.  A  /\  y  e.  B )  /\  z  =  F )  <->  ( (
x  e.  D  /\  y  e.  E )  /\  z  =  F
) ) )
123, 11bitrd 253 . . 3  |-  ( ph  ->  ( ( ( x  e.  A  /\  y  e.  B )  /\  z  =  C )  <->  ( (
x  e.  D  /\  y  e.  E )  /\  z  =  F
) ) )
1312oprabbidv 6336 . 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 6286 . 2  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  A  /\  y  e.  B )  /\  z  =  C
) }
15 df-mpt2 6286 . 2  |-  ( x  e.  D ,  y  e.  E  |->  F )  =  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  D  /\  y  e.  E )  /\  z  =  F
) }
1613, 14, 153eqtr4g 2509 1  |-  ( ph  ->  ( x  e.  A ,  y  e.  B  |->  C )  =  ( x  e.  D , 
y  e.  E  |->  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1383    e. wcel 1804   {coprab 6282    |-> cmpt2 6283
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-oprab 6285  df-mpt2 6286
This theorem is referenced by:  mpt2eq123dv  6344  natpropd  15219  fucpropd  15220  curfpropd  15376  hofpropd  15410  istrkgl  23727  eengv  24154  elntg  24159  rngcifuestrc  32545  funcrngcsetc  32546  funcrngcsetcALT  32547  funcringcsetc  32580
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