Theorem mposnif 5912

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
mposnif	|- ( i e. { X } , j e. B |-> if ( i = X , C , D ) ) = ( i e. { X } , j e. B |-> C )

Proof of Theorem mposnif

Step	Hyp	Ref	Expression
1		elsni 3578	. . . 4 |- ( i e. { X } -> i = X )
2	1	adantr 274	. . 3 |- ( ( i e. { X } /\ j e. B ) -> i = X )
3	2	iftrued 3512	. 2 |- ( ( i e. { X } /\ j e. B ) -> if ( i = X , C , D ) = C )
4	3	mpoeq3ia 5883	1 |- ( i e. { X } , j e. B |-> if ( i = X , C , D ) ) = ( i e. { X } , j e. B |-> C )

Syntax hints:   /\ wa 103   = wceq 1335   e. wcel 2128   ifcif 3505   {csn 3560   e. cmpo 5823

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-if 3506  df-sn 3566  df-oprab 5825  df-mpo 5826

This theorem is referenced by: (None)