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Theorem mpt2snif 6376
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 3996 . . . 4  |-  ( i  e.  { X }  ->  i  =  X )
21adantr 463 . . 3  |-  ( ( i  e.  { X }  /\  j  e.  B
)  ->  i  =  X )
32iftrued 3892 . 2  |-  ( ( i  e.  { X }  /\  j  e.  B
)  ->  if (
i  =  X ,  C ,  D )  =  C )
43mpt2eq3ia 6342 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 367    = wceq 1405    e. wcel 1842   ifcif 3884   {csn 3971    |-> cmpt2 6279
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-v 3060  df-if 3885  df-sn 3972  df-oprab 6281  df-mpt2 6282
This theorem is referenced by:  mdetrsca2  19396  mdetrlin2  19399  mdetunilem5  19408  smadiadetglem2  19464
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