Theorem mpt2difsnif 6753

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
mpt2difsnif	|- ( i e. ( A \ { X } ) , j e. B |-> if ( i = X , C , D ) ) = ( i e. ( A \ { X } ) , j e. B |-> D )

Proof of Theorem mpt2difsnif

Step	Hyp	Ref	Expression
1		eldifsn 4317	. . . . 5 |- ( i e. ( A \ { X } ) <-> ( i e. A /\ i =/= X ) )
2		neneq 2800	. . . . 5 |- ( i =/= X -> -. i = X )
3	1, 2	simplbiim 659	. . . 4 |- ( i e. ( A \ { X } ) -> -. i = X )
4	3	adantr 481	. . 3 |- ( ( i e. ( A \ { X } ) /\ j e. B ) -> -. i = X )
5	4	iffalsed 4097	. 2 |- ( ( i e. ( A \ { X } ) /\ j e. B ) -> if ( i = X , C , D ) = D )
6	5	mpt2eq3ia 6720	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 384   = wceq 1483   e. wcel 1990   =/= wne 2794   \ cdif 3571   ifcif 4086   {csn 4177   |-> cmpt2 6652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-v 3202  df-dif 3577  df-if 4087  df-sn 4178  df-oprab 6654  df-mpt2 6655

This theorem is referenced by:  smadiadetglem1  20477