Theorem mpodifsnif 6038

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 3760	. . . . 5 |- ( i e. ( A \ { X } ) <-> ( i e. A /\ i =/= X ) )
2		neneq 2398	. . . . 5 |- ( i =/= X -> -. i = X )
3	1, 2	simplbiim 387	. . . 4 |- ( i e. ( A \ { X } ) -> -. i = X )
4	3	adantr 276	. . 3 |- ( ( i e. ( A \ { X } ) /\ j e. B ) -> -. i = X )
5	4	iffalsed 3581	. 2 |- ( ( i e. ( A \ { X } ) /\ j e. B ) -> if ( i = X , C , D ) = D )
6	5	mpoeq3ia 6010	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 104   = wceq 1373   e. wcel 2176   =/= wne 2376   \ cdif 3163   ifcif 3571   {csn 3633   e. cmpo 5946

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-in1 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-v 2774  df-dif 3168  df-if 3572  df-sn 3639  df-oprab 5948  df-mpo 5949

This theorem is referenced by: (None)