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Theorem mpt2snif 6189
Description: A mapping with two arguments with the first argument from a singleton and a conditional as result. (Contributed by AV, 14-Feb-2019.)
Assertion
Ref Expression
mpt2snif  |-  ( i  e.  { X } ,  j  e.  B  |->  if ( i  =  X ,  C ,  D ) )  =  ( i  e.  { X } ,  j  e.  B  |->  C )

Proof of Theorem mpt2snif
StepHypRef Expression
1 elsni 3907 . . . 4  |-  ( i  e.  { X }  ->  i  =  X )
21adantr 465 . . 3  |-  ( ( i  e.  { X }  /\  j  e.  B
)  ->  i  =  X )
3 iftrue 3802 . . 3  |-  ( i  =  X  ->  if ( i  =  X ,  C ,  D
)  =  C )
42, 3syl 16 . 2  |-  ( ( i  e.  { X }  /\  j  e.  B
)  ->  if (
i  =  X ,  C ,  D )  =  C )
54mpt2eq3ia 6156 1  |-  ( i  e.  { X } ,  j  e.  B  |->  if ( i  =  X ,  C ,  D ) )  =  ( i  e.  { X } ,  j  e.  B  |->  C )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1369    e. wcel 1756   ifcif 3796   {csn 3882    e. cmpt2 6098
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-v 2979  df-if 3797  df-sn 3883  df-oprab 6100  df-mpt2 6101
This theorem is referenced by:  smadiadetglem2  18483
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