Theorem bnj900

Description: Technical lemma for bnj69 . This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

	Ref	Expression
Hypotheses	bnj900.3	|- D = ( _om \ { (/) } )
	bnj900.4	|- B = { f | E. n e. D ( f Fn n /\ ph /\ ps ) }
Assertion	bnj900	|- ( f e. B -> (/) e. dom f )

Proof

Step	Hyp	Ref	Expression
1		bnj900.3	|- D = ( _om \ { (/) } )
2		bnj900.4	|- B = { f | E. n e. D ( f Fn n /\ ph /\ ps ) }
3	2	bnj1436	|- ( f e. B -> E. n e. D ( f Fn n /\ ph /\ ps ) )
4		simp1	|- ( ( f Fn n /\ ph /\ ps ) -> f Fn n )
5	4	reximi	|- ( E. n e. D ( f Fn n /\ ph /\ ps ) -> E. n e. D f Fn n )
6		fndm	|- ( f Fn n -> dom f = n )
7	6	reximi	|- ( E. n e. D f Fn n -> E. n e. D dom f = n )
8	3 5 7	3syl	|- ( f e. B -> E. n e. D dom f = n )
9	8	bnj1196	|- ( f e. B -> E. n ( n e. D /\ dom f = n ) )
10		nfre1	|- F/ n E. n e. D ( f Fn n /\ ph /\ ps )
11	10	nfab	|- F/_ n { f | E. n e. D ( f Fn n /\ ph /\ ps ) }
12	2 11	nfcxfr	|- F/_ n B
13	12	nfcri	|- F/ n f e. B
14	13	19.37	|- ( E. n ( f e. B -> ( n e. D /\ dom f = n ) ) <-> ( f e. B -> E. n ( n e. D /\ dom f = n ) ) )
15	9 14	mpbir	|- E. n ( f e. B -> ( n e. D /\ dom f = n ) )
16		nfv	|- F/ n (/) e. dom f
17	13 16	nfim	|- F/ n ( f e. B -> (/) e. dom f )
18	1	bnj529	|- ( n e. D -> (/) e. n )
19		eleq2	|- ( dom f = n -> ( (/) e. dom f <-> (/) e. n ) )
20	19	biimparc	|- ( ( (/) e. n /\ dom f = n ) -> (/) e. dom f )
21	18 20	sylan	|- ( ( n e. D /\ dom f = n ) -> (/) e. dom f )
22	21	imim2i	|- ( ( f e. B -> ( n e. D /\ dom f = n ) ) -> ( f e. B -> (/) e. dom f ) )
23	17 22	exlimi	|- ( E. n ( f e. B -> ( n e. D /\ dom f = n ) ) -> ( f e. B -> (/) e. dom f ) )
24	15 23	ax-mp	|- ( f e. B -> (/) e. dom f )