Theorem mpodifsnif 5946

Description: A mapping with two arguments with the first argument from a difference set with a singleton and a conditional as result. (Contributed by AV, 13-Feb-2019.)

Assertion

Ref	Expression
mpodifsnif	|- ( i e. ( A \ { X } ) , j e. B |-> if ( i = X , C , D ) ) = ( i e. ( A \ { X } ) , j e. B |-> D )

Proof of Theorem mpodifsnif

Step	Hyp	Ref	Expression
1		eldifsn 3710	. . . . 5 |- ( i e. ( A \ { X } ) <-> ( i e. A /\ i =/= X ) )
2		neneq 2362	. . . . 5 |- ( i =/= X -> -. i = X )
3	1, 2	simplbiim 385	. . . 4 |- ( i e. ( A \ { X } ) -> -. i = X )
4	3	adantr 274	. . 3 |- ( ( i e. ( A \ { X } ) /\ j e. B ) -> -. i = X )
5	4	iffalsed 3536	. 2 |- ( ( i e. ( A \ { X } ) /\ j e. B ) -> if ( i = X , C , D ) = D )
6	5	mpoeq3ia 5918	1 |- ( i e. ( A \ { X } ) , j e. B |-> if ( i = X , C , D ) ) = ( i e. ( A \ { X } ) , j e. B |-> D )

Syntax hints:   -. wn 3   /\ wa 103   = wceq 1348   e. wcel 2141   =/= wne 2340   \ cdif 3118   ifcif 3526   {csn 3583   e. cmpo 5855

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-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-v 2732  df-dif 3123  df-if 3527  df-sn 3589  df-oprab 5857  df-mpo 5858

This theorem is referenced by: (None)