Theorem mposnif 6114

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 3687	. . . 4 |- ( i e. { X } -> i = X )
2	1	adantr 276	. . 3 |- ( ( i e. { X } /\ j e. B ) -> i = X )
3	2	iftrued 3612	. 2 |- ( ( i e. { X } /\ j e. B ) -> if ( i = X , C , D ) = C )
4	3	mpoeq3ia 6085	1 |- ( i e. { X } , j e. B |-> if ( i = X , C , D ) ) = ( i e. { X } , j e. B |-> C )

Syntax hints:   /\ wa 104   = wceq 1397   e. wcel 2202   ifcif 3605   {csn 3669   e. cmpo 6019

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 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-if 3606  df-sn 3675  df-oprab 6021  df-mpo 6022

This theorem is referenced by: (None)