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Theorem mpt2difsnif 6288
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 4103 . . . . 5  |-  ( i  e.  ( A  \  { X } )  <->  ( i  e.  A  /\  i  =/=  X ) )
2 df-ne 2647 . . . . . . 7  |-  ( i  =/=  X  <->  -.  i  =  X )
32biimpi 194 . . . . . 6  |-  ( i  =/=  X  ->  -.  i  =  X )
43adantl 466 . . . . 5  |-  ( ( i  e.  A  /\  i  =/=  X )  ->  -.  i  =  X
)
51, 4sylbi 195 . . . 4  |-  ( i  e.  ( A  \  { X } )  ->  -.  i  =  X
)
65adantr 465 . . 3  |-  ( ( i  e.  ( A 
\  { X }
)  /\  j  e.  B )  ->  -.  i  =  X )
7 iffalse 3902 . . 3  |-  ( -.  i  =  X  ->  if ( i  =  X ,  C ,  D
)  =  D )
86, 7syl 16 . 2  |-  ( ( i  e.  ( A 
\  { X }
)  /\  j  e.  B )  ->  if ( i  =  X ,  C ,  D
)  =  D )
98mpt2eq3ia 6255 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 369    = wceq 1370    e. wcel 1758    =/= wne 2645    \ cdif 3428   ifcif 3894   {csn 3980    |-> cmpt2 6197
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-v 3074  df-dif 3434  df-if 3895  df-sn 3981  df-oprab 6199  df-mpt2 6200
This theorem is referenced by:  smadiadetglem1  18604
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