Theorem mpodifsnif 5935

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 3703	. . . . 5 |- ( i e. ( A \ { X } ) <-> ( i e. A /\ i =/= X ) )
2		neneq 2358	. . . . 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 3530	. 2 |- ( ( i e. ( A \ { X } ) /\ j e. B ) -> if ( i = X , C , D ) = D )
6	5	mpoeq3ia 5907	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 1343   e. wcel 2136   =/= wne 2336   \ cdif 3113   ifcif 3520   {csn 3576   e. cmpo 5844

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-v 2728  df-dif 3118  df-if 3521  df-sn 3582  df-oprab 5846  df-mpo 5847

This theorem is referenced by: (None)