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Metamath Proof Explorer

Theorem bnj601

Description: Technical lemma for bnj852 . 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 bnj601.1 
|- ( ph <-> ( f ` (/) ) = _pred ( x , A , R ) )
bnj601.2 
|- ( ps <-> A. i e. _om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) )
bnj601.3 
|- D = ( _om \ { (/) } )
bnj601.4 
|- ( ch <-> ( ( R _FrSe A /\ x e. A ) -> E! f ( f Fn n /\ ph /\ ps ) ) )
bnj601.5 
|- ( th <-> A. m e. D ( m _E n -> [. m / n ]. ch ) )
Assertion bnj601 
|- ( n =/= 1o -> ( ( n e. D /\ th ) -> ch ) )

Proof

Step Hyp Ref Expression
1 bnj601.1 
 |-  ( ph <-> ( f ` (/) ) = _pred ( x , A , R ) )
2 bnj601.2 
 |-  ( ps <-> A. i e. _om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) _pred ( y , A , R ) ) )
3 bnj601.3 
 |-  D = ( _om \ { (/) } )
4 bnj601.4 
 |-  ( ch <-> ( ( R _FrSe A /\ x e. A ) -> E! f ( f Fn n /\ ph /\ ps ) ) )
5 bnj601.5 
 |-  ( th <-> A. m e. D ( m _E n -> [. m / n ]. ch ) )
6 biid 
 |-  ( [. m / n ]. ph <-> [. m / n ]. ph )
7 biid 
 |-  ( [. m / n ]. ps <-> [. m / n ]. ps )
8 biid 
 |-  ( [. m / n ]. ch <-> [. m / n ]. ch )
9 bnj602 
 |-  ( y = z -> _pred ( y , A , R ) = _pred ( z , A , R ) )
10 9 cbviunv 
 |-  U_ y e. ( f ` p ) _pred ( y , A , R ) = U_ z e. ( f ` p ) _pred ( z , A , R )
11 10 opeq2i 
 |-  <. m , U_ y e. ( f ` p ) _pred ( y , A , R ) >. = <. m , U_ z e. ( f ` p ) _pred ( z , A , R ) >.
12 11 sneqi 
 |-  { <. m , U_ y e. ( f ` p ) _pred ( y , A , R ) >. } = { <. m , U_ z e. ( f ` p ) _pred ( z , A , R ) >. }
13 12 uneq2i 
 |-  ( f u. { <. m , U_ y e. ( f ` p ) _pred ( y , A , R ) >. } ) = ( f u. { <. m , U_ z e. ( f ` p ) _pred ( z , A , R ) >. } )
14 dfsbcq 
 |-  ( ( f u. { <. m , U_ y e. ( f ` p ) _pred ( y , A , R ) >. } ) = ( f u. { <. m , U_ z e. ( f ` p ) _pred ( z , A , R ) >. } ) -> ( [. ( f u. { <. m , U_ y e. ( f ` p ) _pred ( y , A , R ) >. } ) / f ]. ph <-> [. ( f u. { <. m , U_ z e. ( f ` p ) _pred ( z , A , R ) >. } ) / f ]. ph ) )
15 13 14 ax-mp 
 |-  ( [. ( f u. { <. m , U_ y e. ( f ` p ) _pred ( y , A , R ) >. } ) / f ]. ph <-> [. ( f u. { <. m , U_ z e. ( f ` p ) _pred ( z , A , R ) >. } ) / f ]. ph )
16 dfsbcq 
 |-  ( ( f u. { <. m , U_ y e. ( f ` p ) _pred ( y , A , R ) >. } ) = ( f u. { <. m , U_ z e. ( f ` p ) _pred ( z , A , R ) >. } ) -> ( [. ( f u. { <. m , U_ y e. ( f ` p ) _pred ( y , A , R ) >. } ) / f ]. ps <-> [. ( f u. { <. m , U_ z e. ( f ` p ) _pred ( z , A , R ) >. } ) / f ]. ps ) )
17 13 16 ax-mp 
 |-  ( [. ( f u. { <. m , U_ y e. ( f ` p ) _pred ( y , A , R ) >. } ) / f ]. ps <-> [. ( f u. { <. m , U_ z e. ( f ` p ) _pred ( z , A , R ) >. } ) / f ]. ps )
18 dfsbcq 
 |-  ( ( f u. { <. m , U_ y e. ( f ` p ) _pred ( y , A , R ) >. } ) = ( f u. { <. m , U_ z e. ( f ` p ) _pred ( z , A , R ) >. } ) -> ( [. ( f u. { <. m , U_ y e. ( f ` p ) _pred ( y , A , R ) >. } ) / f ]. ch <-> [. ( f u. { <. m , U_ z e. ( f ` p ) _pred ( z , A , R ) >. } ) / f ]. ch ) )
19 13 18 ax-mp 
 |-  ( [. ( f u. { <. m , U_ y e. ( f ` p ) _pred ( y , A , R ) >. } ) / f ]. ch <-> [. ( f u. { <. m , U_ z e. ( f ` p ) _pred ( z , A , R ) >. } ) / f ]. ch )
20 13 eqcomi 
 |-  ( f u. { <. m , U_ z e. ( f ` p ) _pred ( z , A , R ) >. } ) = ( f u. { <. m , U_ y e. ( f ` p ) _pred ( y , A , R ) >. } )
21 biid 
 |-  ( ( f Fn m /\ [. m / n ]. ph /\ [. m / n ]. ps ) <-> ( f Fn m /\ [. m / n ]. ph /\ [. m / n ]. ps ) )
22 biid 
 |-  ( ( m e. D /\ n = suc m /\ p e. m ) <-> ( m e. D /\ n = suc m /\ p e. m ) )
23 biid 
 |-  ( ( m e. D /\ n = suc m /\ p e. _om /\ m = suc p ) <-> ( m e. D /\ n = suc m /\ p e. _om /\ m = suc p ) )
24 biid 
 |-  ( ( i e. _om /\ suc i e. n /\ m = suc i ) <-> ( i e. _om /\ suc i e. n /\ m = suc i ) )
25 biid 
 |-  ( ( i e. _om /\ suc i e. n /\ m =/= suc i ) <-> ( i e. _om /\ suc i e. n /\ m =/= suc i ) )
26 eqid 
 |-  U_ y e. ( f ` i ) _pred ( y , A , R ) = U_ y e. ( f ` i ) _pred ( y , A , R )
27 eqid 
 |-  U_ y e. ( f ` p ) _pred ( y , A , R ) = U_ y e. ( f ` p ) _pred ( y , A , R )
28 eqid 
 |-  U_ y e. ( ( f u. { <. m , U_ z e. ( f ` p ) _pred ( z , A , R ) >. } ) ` i ) _pred ( y , A , R ) = U_ y e. ( ( f u. { <. m , U_ z e. ( f ` p ) _pred ( z , A , R ) >. } ) ` i ) _pred ( y , A , R )
29 eqid 
 |-  U_ y e. ( ( f u. { <. m , U_ z e. ( f ` p ) _pred ( z , A , R ) >. } ) ` p ) _pred ( y , A , R ) = U_ y e. ( ( f u. { <. m , U_ z e. ( f ` p ) _pred ( z , A , R ) >. } ) ` p ) _pred ( y , A , R )
30 1 2 3 4 5 6 7 8 15 17 19 20 21 22 23 24 25 26 27 28 29 20 bnj600 
 |-  ( n =/= 1o -> ( ( n e. D /\ th ) -> ch ) )