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Theorem mpt2snif 6378
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 4052 . . . 4  |-  ( i  e.  { X }  ->  i  =  X )
21adantr 465 . . 3  |-  ( ( i  e.  { X }  /\  j  e.  B
)  ->  i  =  X )
3 iftrue 3945 . . 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 6344 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 1379    e. wcel 1767   ifcif 3939   {csn 4027    |-> cmpt2 6284
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-v 3115  df-if 3940  df-sn 4028  df-oprab 6286  df-mpt2 6287
This theorem is referenced by:  smadiadetglem2  18938
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