Theorem mpodifsnif 6015

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 3749	. . . . 5 |- ( i e. ( A \ { X } ) <-> ( i e. A /\ i =/= X ) )
2		neneq 2389	. . . . 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 3571	. 2 |- ( ( i e. ( A \ { X } ) /\ j e. B ) -> if ( i = X , C , D ) = D )
6	5	mpoeq3ia 5987	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 1364   e. wcel 2167   =/= wne 2367   \ cdif 3154   ifcif 3561   {csn 3622   e. cmpo 5924

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-v 2765  df-dif 3159  df-if 3562  df-sn 3628  df-oprab 5926  df-mpo 5927

This theorem is referenced by: (None)