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Theorem mpt2difsnif 6368
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
mpt2difsnif  |-  ( i  e.  ( A  \  { X } ) ,  j  e.  B  |->  if ( i  =  X ,  C ,  D
) )  =  ( i  e.  ( A 
\  { X }
) ,  j  e.  B  |->  D )

Proof of Theorem mpt2difsnif
StepHypRef Expression
1 eldifsn 4141 . . . . 5  |-  ( i  e.  ( A  \  { X } )  <->  ( i  e.  A  /\  i  =/=  X ) )
2 df-ne 2651 . . . . . . 7  |-  ( i  =/=  X  <->  -.  i  =  X )
32biimpi 194 . . . . . 6  |-  ( i  =/=  X  ->  -.  i  =  X )
43adantl 464 . . . . 5  |-  ( ( i  e.  A  /\  i  =/=  X )  ->  -.  i  =  X
)
51, 4sylbi 195 . . . 4  |-  ( i  e.  ( A  \  { X } )  ->  -.  i  =  X
)
65adantr 463 . . 3  |-  ( ( i  e.  ( A 
\  { X }
)  /\  j  e.  B )  ->  -.  i  =  X )
76iffalsed 3940 . 2  |-  ( ( i  e.  ( A 
\  { X }
)  /\  j  e.  B )  ->  if ( i  =  X ,  C ,  D
)  =  D )
87mpt2eq3ia 6335 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 setvar class
Syntax hints:   -. wn 3    /\ wa 367    = wceq 1398    e. wcel 1823    =/= wne 2649    \ cdif 3458   ifcif 3929   {csn 4016    |-> cmpt2 6272
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-v 3108  df-dif 3464  df-if 3930  df-sn 4017  df-oprab 6274  df-mpt2 6275
This theorem is referenced by:  smadiadetglem1  19343
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