Metamath Proof Explorer

Theorem ndfatafv2

Description: The alternate function value at a class A if the function is not defined at this set A . (Contributed by AV, 2-Sep-2022)

		Ref	Expression
	Assertion	ndfatafv2	\|- ( -. F defAt A -> ( F '''' A ) = ~P U. ran F )

Step	Hyp	Ref	Expression
1		df-afv2	\|- ( F '''' A ) = if ( F defAt A , ( iota x A F x ) , ~P U. ran F )
2		iffalse	\|- ( -. F defAt A -> if ( F defAt A , ( iota x A F x ) , ~P U. ran F ) = ~P U. ran F )
3	1 2	syl5eq	\|- ( -. F defAt A -> ( F '''' A ) = ~P U. ran F )

Step

Hyp

Ref

Expression

 |-  ( F '''' A ) = if ( F defAt A , ( iota x A F x ) , ~P U. ran F )

 |-  ( -. F defAt A -> if ( F defAt A , ( iota x A F x ) , ~P U. ran F ) = ~P U. ran F )

1 2

 |-  ( -. F defAt A -> ( F '''' A ) = ~P U. ran F )