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Theorem bnj1097 29524
Description: Technical lemma for bnj69 29553. 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
bnj1097.1  |-  ( ph  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
bnj1097.3  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
bnj1097.5  |-  ( ta  <->  ( B  e.  _V  /\  TrFo ( B ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B
) )
Assertion
Ref Expression
bnj1097  |-  ( ( i  =  (/)  /\  (
( th  /\  ta  /\ 
ch )  /\  ph0 )
)  ->  ( f `  i )  C_  B
)

Proof of Theorem bnj1097
StepHypRef Expression
1 bnj1097.3 . . . . . . . 8  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
2 bnj1097.1 . . . . . . . . 9  |-  ( ph  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
32biimpi 188 . . . . . . . 8  |-  ( ph  ->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
41, 3bnj771 29307 . . . . . . 7  |-  ( ch 
->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
543ad2ant3 981 . . . . . 6  |-  ( ( th  /\  ta  /\  ch )  ->  ( f `
 (/) )  =  pred ( X ,  A ,  R ) )
65adantr 453 . . . . 5  |-  ( ( ( th  /\  ta  /\ 
ch )  /\  ph0 )  ->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
7 bnj1097.5 . . . . . . . 8  |-  ( ta  <->  ( B  e.  _V  /\  TrFo ( B ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B
) )
87simp3bi 975 . . . . . . 7  |-  ( ta 
->  pred ( X ,  A ,  R )  C_  B )
983ad2ant2 980 . . . . . 6  |-  ( ( th  /\  ta  /\  ch )  ->  pred ( X ,  A ,  R )  C_  B
)
109adantr 453 . . . . 5  |-  ( ( ( th  /\  ta  /\ 
ch )  /\  ph0 )  ->  pred ( X ,  A ,  R )  C_  B )
116, 10jca 520 . . . 4  |-  ( ( ( th  /\  ta  /\ 
ch )  /\  ph0 )  ->  ( ( f `  (/) )  =  pred ( X ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B
) )
1211anim2i 554 . . 3  |-  ( ( i  =  (/)  /\  (
( th  /\  ta  /\ 
ch )  /\  ph0 )
)  ->  ( i  =  (/)  /\  ( ( f `  (/) )  = 
pred ( X ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B ) ) )
13 3anass 941 . . 3  |-  ( ( i  =  (/)  /\  (
f `  (/) )  = 
pred ( X ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B )  <->  ( i  =  (/)  /\  ( ( f `  (/) )  = 
pred ( X ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B ) ) )
1412, 13sylibr 205 . 2  |-  ( ( i  =  (/)  /\  (
( th  /\  ta  /\ 
ch )  /\  ph0 )
)  ->  ( i  =  (/)  /\  ( f `
 (/) )  =  pred ( X ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B
) )
15 fveq2 5763 . . . . . . 7  |-  ( i  =  (/)  ->  ( f `
 i )  =  ( f `  (/) ) )
1615eqeq1d 2451 . . . . . 6  |-  ( i  =  (/)  ->  ( ( f `  i )  =  pred ( X ,  A ,  R )  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
) )
1716biimpar 473 . . . . 5  |-  ( ( i  =  (/)  /\  (
f `  (/) )  = 
pred ( X ,  A ,  R )
)  ->  ( f `  i )  =  pred ( X ,  A ,  R ) )
1817adantr 453 . . . 4  |-  ( ( ( i  =  (/)  /\  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)  /\  pred ( X ,  A ,  R
)  C_  B )  ->  ( f `  i
)  =  pred ( X ,  A ,  R ) )
19 simpr 449 . . . 4  |-  ( ( ( i  =  (/)  /\  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)  /\  pred ( X ,  A ,  R
)  C_  B )  ->  pred ( X ,  A ,  R )  C_  B )
2018, 19eqsstrd 3371 . . 3  |-  ( ( ( i  =  (/)  /\  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)  /\  pred ( X ,  A ,  R
)  C_  B )  ->  ( f `  i
)  C_  B )
21203impa 1149 . 2  |-  ( ( i  =  (/)  /\  (
f `  (/) )  = 
pred ( X ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B )  ->  (
f `  i )  C_  B )
2214, 21syl 16 1  |-  ( ( i  =  (/)  /\  (
( th  /\  ta  /\ 
ch )  /\  ph0 )
)  ->  ( f `  i )  C_  B
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1654    e. wcel 1728   _Vcvv 2965    C_ wss 3309   (/)c0 3616    Fn wfn 5484   ` cfv 5489    /\ w-bnj17 29224    predc-bnj14 29226    TrFow-bnj19 29234
This theorem is referenced by:  bnj1030  29530
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-rex 2718  df-rab 2721  df-v 2967  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-nul 3617  df-if 3768  df-sn 3849  df-pr 3850  df-op 3852  df-uni 4045  df-br 4244  df-iota 5453  df-fv 5497  df-bnj17 29225
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