Metamath Proof Explorer

Theorem bnj1121

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

Proof

Step Hyp Ref Expression
1 bnj1121.1 
 |-  ( th <-> ( R _FrSe A /\ X e. A ) )
2 bnj1121.2 
 |-  ( ta <-> ( B e. _V /\ _TrFo ( B , A , R ) /\ _pred ( X , A , R ) C_ B ) )
3 bnj1121.3 
 |-  ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
4 bnj1121.4 
 |-  ( ze <-> ( i e. n /\ z e. ( f ` i ) ) )
5 bnj1121.5 
 |-  ( et <-> ( ( f e. K /\ i e. dom f ) -> ( f ` i ) C_ B ) )
6 bnj1121.6 
 |-  ( ( th /\ ta /\ ch /\ ze ) -> A. i e. n et )
7 bnj1121.7 
 |-  K = { f | E. n e. D ( f Fn n /\ ph /\ ps ) }
8 19.8a 
 |-  ( ch -> E. n ch )
9 8 bnj707 
 |-  ( ( th /\ ta /\ ch /\ ze ) -> E. n ch )
10 3 7 bnj1083 
 |-  ( f e. K <-> E. n ch )
11 9 10 sylibr 
 |-  ( ( th /\ ta /\ ch /\ ze ) -> f e. K )
12 4 simplbi 
 |-  ( ze -> i e. n )
13 12 bnj708 
 |-  ( ( th /\ ta /\ ch /\ ze ) -> i e. n )
14 3 bnj1235 
 |-  ( ch -> f Fn n )
15 14 bnj707 
 |-  ( ( th /\ ta /\ ch /\ ze ) -> f Fn n )
16 15 fndmd 
 |-  ( ( th /\ ta /\ ch /\ ze ) -> dom f = n )
17 13 16 eleqtrrd 
 |-  ( ( th /\ ta /\ ch /\ ze ) -> i e. dom f )
18 6 13 bnj1294 
 |-  ( ( th /\ ta /\ ch /\ ze ) -> et )
19 18 5 sylib 
 |-  ( ( th /\ ta /\ ch /\ ze ) -> ( ( f e. K /\ i e. dom f ) -> ( f ` i ) C_ B ) )
20 11 17 19 mp2and 
 |-  ( ( th /\ ta /\ ch /\ ze ) -> ( f ` i ) C_ B )
21 4 simprbi 
 |-  ( ze -> z e. ( f ` i ) )
22 21 bnj708 
 |-  ( ( th /\ ta /\ ch /\ ze ) -> z e. ( f ` i ) )
23 20 22 sseldd 
 |-  ( ( th /\ ta /\ ch /\ ze ) -> z e. B )