Theorem mposnif 5959

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 3607	. . . 4 |- ( i e. { X } -> i = X )
2	1	adantr 276	. . 3 |- ( ( i e. { X } /\ j e. B ) -> i = X )
3	2	iftrued 3539	. 2 |- ( ( i e. { X } /\ j e. B ) -> if ( i = X , C , D ) = C )
4	3	mpoeq3ia 5930	1 |- ( i e. { X } , j e. B |-> if ( i = X , C , D ) ) = ( i e. { X } , j e. B |-> C )

Syntax hints:   /\ wa 104   = wceq 1353   e. wcel 2146   ifcif 3532   {csn 3589   e. cmpo 5867

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-v 2737  df-if 3533  df-sn 3595  df-oprab 5869  df-mpo 5870

This theorem is referenced by: (None)