Theorem mposnif 5951

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 3602	. . . 4 |- ( i e. { X } -> i = X )
2	1	adantr 274	. . 3 |- ( ( i e. { X } /\ j e. B ) -> i = X )
3	2	iftrued 3534	. 2 |- ( ( i e. { X } /\ j e. B ) -> if ( i = X , C , D ) = C )
4	3	mpoeq3ia 5922	1 |- ( i e. { X } , j e. B |-> if ( i = X , C , D ) ) = ( i e. { X } , j e. B |-> C )

Syntax hints:   /\ wa 103   = wceq 1349   e. wcel 2142   ifcif 3527   {csn 3584   e. cmpo 5859

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 611  ax-io 705  ax-5 1441  ax-7 1442  ax-gen 1443  ax-ie1 1487  ax-ie2 1488  ax-8 1498  ax-10 1499  ax-11 1500  ax-i12 1501  ax-bndl 1503  ax-4 1504  ax-17 1520  ax-i9 1524  ax-ial 1528  ax-i5r 1529  ax-ext 2153

This theorem depends on definitions:  df-bi 116  df-3an 976  df-tru 1352  df-nf 1455  df-sb 1757  df-clab 2158  df-cleq 2164  df-clel 2167  df-nfc 2302  df-v 2733  df-if 3528  df-sn 3590  df-oprab 5861  df-mpo 5862

This theorem is referenced by: (None)