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

Theorem mpodifsnif 5943
Description: A mapping with two arguments with the first argument from a difference set with a singleton and a conditional as result. (Contributed by AV, 13-Feb-2019.)
Assertion
Ref Expression
mpodifsnif  |-  ( i  e.  ( A  \  { X } ) ,  j  e.  B  |->  if ( i  =  X ,  C ,  D
) )  =  ( i  e.  ( A 
\  { X }
) ,  j  e.  B  |->  D )

Proof of Theorem mpodifsnif
StepHypRef Expression
1 eldifsn 3708 . . . . 5  |-  ( i  e.  ( A  \  { X } )  <->  ( i  e.  A  /\  i  =/=  X ) )
2 neneq 2362 . . . . 5  |-  ( i  =/=  X  ->  -.  i  =  X )
31, 2simplbiim 385 . . . 4  |-  ( i  e.  ( A  \  { X } )  ->  -.  i  =  X
)
43adantr 274 . . 3  |-  ( ( i  e.  ( A 
\  { X }
)  /\  j  e.  B )  ->  -.  i  =  X )
54iffalsed 3535 . 2  |-  ( ( i  e.  ( A 
\  { X }
)  /\  j  e.  B )  ->  if ( i  =  X ,  C ,  D
)  =  D )
65mpoeq3ia 5915 1  |-  ( i  e.  ( A  \  { X } ) ,  j  e.  B  |->  if ( i  =  X ,  C ,  D
) )  =  ( i  e.  ( A 
\  { X }
) ,  j  e.  B  |->  D )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 103    = wceq 1348    e. wcel 2141    =/= wne 2340    \ cdif 3118   ifcif 3525   {csn 3581    e. cmpo 5852
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-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  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-clel 2166  df-nfc 2301  df-ne 2341  df-v 2732  df-dif 3123  df-if 3526  df-sn 3587  df-oprab 5854  df-mpo 5855
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator