Theorem mposnif 6012

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 3636	. . . 4 |- ( i e. { X } -> i = X )
2	1	adantr 276	. . 3 |- ( ( i e. { X } /\ j e. B ) -> i = X )
3	2	iftrued 3564	. 2 |- ( ( i e. { X } /\ j e. B ) -> if ( i = X , C , D ) = C )
4	3	mpoeq3ia 5983	1 |- ( i e. { X } , j e. B |-> if ( i = X , C , D ) ) = ( i e. { X } , j e. B |-> C )

Syntax hints:   /\ wa 104   = wceq 1364   e. wcel 2164   ifcif 3557   {csn 3618   e. cmpo 5920

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 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-if 3558  df-sn 3624  df-oprab 5922  df-mpo 5923

This theorem is referenced by: (None)