Theorem bnj900 30999

Description: Technical lemma for bnj69 31078. 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.)

Hypotheses

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

Assertion

Ref	Expression
bnj900	|- ( f e. B -> (/) e. dom f )

Distinct variable group:   f, n

Allowed substitution hints:   ph( f, n)   ps( f, n)   B( f, n)   D( f, n)

Proof of Theorem bnj900

Step	Hyp	Ref	Expression
1		bnj900.4	. . . . . 6 |- B = { f | E. n e. D ( f Fn n /\ ph /\ ps ) }
2	1	bnj1436 30910	. . . . 5 |- ( f e. B -> E. n e. D ( f Fn n /\ ph /\ ps ) )
3		simp1 1061	. . . . . 6 |- ( ( f Fn n /\ ph /\ ps ) -> f Fn n )
4	3	reximi 3011	. . . . 5 |- ( E. n e. D ( f Fn n /\ ph /\ ps ) -> E. n e. D f Fn n )
5		fndm 5990	. . . . . 6 |- ( f Fn n -> dom f = n )
6	5	reximi 3011	. . . . 5 |- ( E. n e. D f Fn n -> E. n e. D dom f = n )
7	2, 4, 6	3syl 18	. . . 4 |- ( f e. B -> E. n e. D dom f = n )
8	7	bnj1196 30865	. . 3 |- ( f e. B -> E. n ( n e. D /\ dom f = n ) )
9		nfre1 3005	. . . . . . 7 |- F/ n E. n e. D ( f Fn n /\ ph /\ ps )
10	9	nfab 2769	. . . . . 6 |- F/_ n { f | E. n e. D ( f Fn n /\ ph /\ ps ) }
11	1, 10	nfcxfr 2762	. . . . 5 |- F/_ n B
12	11	nfcri 2758	. . . 4 |- F/ n f e. B
13	12	19.37 2100	. . 3 |- ( E. n ( f e. B -> ( n e. D /\ dom f = n ) ) <-> ( f e. B -> E. n ( n e. D /\ dom f = n ) ) )
14	8, 13	mpbir 221	. 2 |- E. n ( f e. B -> ( n e. D /\ dom f = n ) )
15		nfv 1843	. . . 4 |- F/ n (/) e. dom f
16	12, 15	nfim 1825	. . 3 |- F/ n ( f e. B -> (/) e. dom f )
17		bnj900.3	. . . . . 6 |- D = ( om \ { (/) } )
18	17	bnj529 30811	. . . . 5 |- ( n e. D -> (/) e. n )
19		eleq2 2690	. . . . . 6 |- ( dom f = n -> ( (/) e. dom f <-> (/) e. n ) )
20	19	biimparc 504	. . . . 5 |- ( ( (/) e. n /\ dom f = n ) -> (/) e. dom f )
21	18, 20	sylan 488	. . . 4 |- ( ( n e. D /\ dom f = n ) -> (/) e. dom f )
22	21	imim2i 16	. . 3 |- ( ( f e. B -> ( n e. D /\ dom f = n ) ) -> ( f e. B -> (/) e. dom f ) )
23	16, 22	exlimi 2086	. 2 |- ( E. n ( f e. B -> ( n e. D /\ dom f = n ) ) -> ( f e. B -> (/) e. dom f ) )
24	14, 23	ax-mp 5	1 |- ( f e. B -> (/) e. dom f )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   E.wrex 2913   \ cdif 3571   (/)c0 3915   {csn 4177   dom cdm 5114   Fn wfn 5883   omcom 7065

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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-fn 5891  df-om 7066

This theorem is referenced by:  bnj906  31000