Metamath Proof Explorer

Theorem bnj229

Description: Technical lemma for bnj517 . 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
Hypothesis bnj229.1 
|- ( ps <-> A. i e. _om ( suc i e. N -> ( F ` suc i ) = U_ y e. ( F ` i ) _pred ( y , A , R ) ) )
Assertion bnj229 
|- ( ( n e. N /\ ( suc m = n /\ m e. _om /\ ps ) ) -> ( F ` n ) C_ A )

Proof

Step Hyp Ref Expression
1 bnj229.1 
 |-  ( ps <-> A. i e. _om ( suc i e. N -> ( F ` suc i ) = U_ y e. ( F ` i ) _pred ( y , A , R ) ) )
2 bnj213 
 |-  _pred ( y , A , R ) C_ A
3 2 bnj226 
 |-  U_ y e. ( F ` m ) _pred ( y , A , R ) C_ A
4 1 bnj222 
 |-  ( ps <-> A. m e. _om ( suc m e. N -> ( F ` suc m ) = U_ y e. ( F ` m ) _pred ( y , A , R ) ) )
5 4 bnj228 
 |-  ( ( m e. _om /\ ps ) -> ( suc m e. N -> ( F ` suc m ) = U_ y e. ( F ` m ) _pred ( y , A , R ) ) )
6 5 adantl 
 |-  ( ( suc m = n /\ ( m e. _om /\ ps ) ) -> ( suc m e. N -> ( F ` suc m ) = U_ y e. ( F ` m ) _pred ( y , A , R ) ) )
7 eleq1 
 |-  ( suc m = n -> ( suc m e. N <-> n e. N ) )
8 fveqeq2 
 |-  ( suc m = n -> ( ( F ` suc m ) = U_ y e. ( F ` m ) _pred ( y , A , R ) <-> ( F ` n ) = U_ y e. ( F ` m ) _pred ( y , A , R ) ) )
9 7 8 imbi12d 
 |-  ( suc m = n -> ( ( suc m e. N -> ( F ` suc m ) = U_ y e. ( F ` m ) _pred ( y , A , R ) ) <-> ( n e. N -> ( F ` n ) = U_ y e. ( F ` m ) _pred ( y , A , R ) ) ) )
10 9 adantr 
 |-  ( ( suc m = n /\ ( m e. _om /\ ps ) ) -> ( ( suc m e. N -> ( F ` suc m ) = U_ y e. ( F ` m ) _pred ( y , A , R ) ) <-> ( n e. N -> ( F ` n ) = U_ y e. ( F ` m ) _pred ( y , A , R ) ) ) )
11 6 10 mpbid 
 |-  ( ( suc m = n /\ ( m e. _om /\ ps ) ) -> ( n e. N -> ( F ` n ) = U_ y e. ( F ` m ) _pred ( y , A , R ) ) )
12 11 3impb 
 |-  ( ( suc m = n /\ m e. _om /\ ps ) -> ( n e. N -> ( F ` n ) = U_ y e. ( F ` m ) _pred ( y , A , R ) ) )
13 12 impcom 
 |-  ( ( n e. N /\ ( suc m = n /\ m e. _om /\ ps ) ) -> ( F ` n ) = U_ y e. ( F ` m ) _pred ( y , A , R ) )
14 3 13 bnj1262 
 |-  ( ( n e. N /\ ( suc m = n /\ m e. _om /\ ps ) ) -> ( F ` n ) C_ A )