ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  mpoeq3dva Unicode version

Theorem mpoeq3dva 5917
Description: Slightly more general equality inference for the maps-to notation. (Contributed by NM, 17-Oct-2013.)
Hypothesis
Ref Expression
mpoeq3dva.1  |-  ( (
ph  /\  x  e.  A  /\  y  e.  B
)  ->  C  =  D )
Assertion
Ref Expression
mpoeq3dva  |-  ( ph  ->  ( x  e.  A ,  y  e.  B  |->  C )  =  ( x  e.  A , 
y  e.  B  |->  D ) )
Distinct variable groups:    ph, x    ph, y
Allowed substitution hints:    A( x, y)    B( x, y)    C( x, y)    D( x, y)

Proof of Theorem mpoeq3dva
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 mpoeq3dva.1 . . . . . 6  |-  ( (
ph  /\  x  e.  A  /\  y  e.  B
)  ->  C  =  D )
213expb 1199 . . . . 5  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  B ) )  ->  C  =  D )
32eqeq2d 2182 . . . 4  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  B ) )  -> 
( z  =  C  <-> 
z  =  D ) )
43pm5.32da 449 . . 3  |-  ( ph  ->  ( ( ( x  e.  A  /\  y  e.  B )  /\  z  =  C )  <->  ( (
x  e.  A  /\  y  e.  B )  /\  z  =  D
) ) )
54oprabbidv 5907 . 2  |-  ( ph  ->  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  z  =  C ) }  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  z  =  D ) } )
6 df-mpo 5858 . 2  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  A  /\  y  e.  B )  /\  z  =  C
) }
7 df-mpo 5858 . 2  |-  ( x  e.  A ,  y  e.  B  |->  D )  =  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  A  /\  y  e.  B )  /\  z  =  D
) }
85, 6, 73eqtr4g 2228 1  |-  ( ph  ->  ( x  e.  A ,  y  e.  B  |->  C )  =  ( x  e.  A , 
y  e.  B  |->  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 973    = wceq 1348    e. wcel 2141   {coprab 5854    e. cmpo 5855
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-oprab 5857  df-mpo 5858
This theorem is referenced by:  mpoeq3ia  5918  mpoeq3dv  5919  ofeq  6063  fmpoco  6195  mapxpen  6826  seqeq2  10405  seqeq3  10406  cnmpt2t  13087  cnmpt22  13088  cnmptcom  13092
  Copyright terms: Public domain W3C validator