Theorem nelrnmpt

Description: Non-membership in the range of a function in maps-to notaion. (Contributed by Glauco Siliprandi, 3-Mar-2021)

	Ref	Expression
Hypotheses	nelrnmpt.x	|- F/ x ph
	nelrnmpt.f	|- F = ( x e. A |-> B )
	nelrnmpt.c	|- ( ph -> C e. V )
	nelrnmpt.n	|- ( ( ph /\ x e. A ) -> C =/= B )
Assertion	nelrnmpt	|- ( ph -> -. C e. ran F )

Proof

Step	Hyp	Ref	Expression
1		nelrnmpt.x	|- F/ x ph
2		nelrnmpt.f	|- F = ( x e. A |-> B )
3		nelrnmpt.c	|- ( ph -> C e. V )
4		nelrnmpt.n	|- ( ( ph /\ x e. A ) -> C =/= B )
5	4	neneqd	|- ( ( ph /\ x e. A ) -> -. C = B )
6	5	ex	|- ( ph -> ( x e. A -> -. C = B ) )
7	1 6	ralrimi	|- ( ph -> A. x e. A -. C = B )
8		ralnex	|- ( A. x e. A -. C = B <-> -. E. x e. A C = B )
9	7 8	sylib	|- ( ph -> -. E. x e. A C = B )
10	2	elrnmpt	|- ( C e. V -> ( C e. ran F <-> E. x e. A C = B ) )
11	3 10	syl	|- ( ph -> ( C e. ran F <-> E. x e. A C = B ) )
12	9 11	mtbird	|- ( ph -> -. C e. ran F )