Theorem bnj1121 31053

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
bnj1121.1	|- ( th <-> ( R FrSe A /\ X e. A ) )
bnj1121.2	|- ( ta <-> ( B e. _V /\ TrFo ( B , A , R ) /\ pred ( X , A , R ) C_ B ) )
bnj1121.3	|- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
bnj1121.4	|- ( ze <-> ( i e. n /\ z e. ( f ` i ) ) )
bnj1121.5	|- ( et <-> ( ( f e. K /\ i e. dom f ) -> ( f ` i ) C_ B ) )
bnj1121.6	|- ( ( th /\ ta /\ ch /\ ze ) -> A. i e. n et )
bnj1121.7	|- K = { f | E. n e. D ( f Fn n /\ ph /\ ps ) }

Assertion

Ref	Expression
bnj1121	|- ( ( th /\ ta /\ ch /\ ze ) -> z e. B )

Proof of Theorem bnj1121

Step	Hyp	Ref	Expression
1		19.8a 2052	. . . . 5 |- ( ch -> E. n ch )
2	1	bnj707 30825	. . . 4 |- ( ( th /\ ta /\ ch /\ ze ) -> E. n ch )
3		bnj1121.3	. . . . 5 |- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
4		bnj1121.7	. . . . 5 |- K = { f | E. n e. D ( f Fn n /\ ph /\ ps ) }
5	3, 4	bnj1083 31046	. . . 4 |- ( f e. K <-> E. n ch )
6	2, 5	sylibr 224	. . 3 |- ( ( th /\ ta /\ ch /\ ze ) -> f e. K )
7		bnj1121.4	. . . . . 6 |- ( ze <-> ( i e. n /\ z e. ( f ` i ) ) )
8	7	simplbi 476	. . . . 5 |- ( ze -> i e. n )
9	8	bnj708 30826	. . . 4 |- ( ( th /\ ta /\ ch /\ ze ) -> i e. n )
10	3	bnj1235 30875	. . . . . 6 |- ( ch -> f Fn n )
11	10	bnj707 30825	. . . . 5 |- ( ( th /\ ta /\ ch /\ ze ) -> f Fn n )
12		fndm 5990	. . . . 5 |- ( f Fn n -> dom f = n )
13	11, 12	syl 17	. . . 4 |- ( ( th /\ ta /\ ch /\ ze ) -> dom f = n )
14	9, 13	eleqtrrd 2704	. . 3 |- ( ( th /\ ta /\ ch /\ ze ) -> i e. dom f )
15		bnj1121.6	. . . . 5 |- ( ( th /\ ta /\ ch /\ ze ) -> A. i e. n et )
16	15, 9	bnj1294 30888	. . . 4 |- ( ( th /\ ta /\ ch /\ ze ) -> et )
17		bnj1121.5	. . . 4 |- ( et <-> ( ( f e. K /\ i e. dom f ) -> ( f ` i ) C_ B ) )
18	16, 17	sylib 208	. . 3 |- ( ( th /\ ta /\ ch /\ ze ) -> ( ( f e. K /\ i e. dom f ) -> ( f ` i ) C_ B ) )
19	6, 14, 18	mp2and 715	. 2 |- ( ( th /\ ta /\ ch /\ ze ) -> ( f ` i ) C_ B )
20	7	simprbi 480	. . 3 |- ( ze -> z e. ( f ` i ) )
21	20	bnj708 30826	. 2 |- ( ( th /\ ta /\ ch /\ ze ) -> z e. ( f ` i ) )
22	19, 21	sseldd 3604	1 |- ( ( th /\ ta /\ ch /\ ze ) -> z e. B )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   A.wral 2912   E.wrex 2913   _Vcvv 3200   C_ wss 3574   dom cdm 5114   Fn wfn 5883   ` cfv 5888   /\ w-bnj17 30752   predc-bnj14 30754   FrSe w-bnj15 30758   TrFow-bnj19 30762

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-ral 2917  df-rex 2918  df-in 3581  df-ss 3588  df-fn 5891  df-bnj17 30753

This theorem is referenced by:  bnj1030  31055