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Theorem bnj1097 34138
Description: Technical lemma for bnj69 34167. 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 194 . . . . . . . 8  |-  ( ph  ->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
41, 3bnj771 33923 . . . . . . 7  |-  ( ch 
->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
543ad2ant3 1019 . . . . . 6  |-  ( ( th  /\  ta  /\  ch )  ->  ( f `
 (/) )  =  pred ( X ,  A ,  R ) )
65adantr 465 . . . . 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 1013 . . . . . . 7  |-  ( ta 
->  pred ( X ,  A ,  R )  C_  B )
983ad2ant2 1018 . . . . . 6  |-  ( ( th  /\  ta  /\  ch )  ->  pred ( X ,  A ,  R )  C_  B
)
109adantr 465 . . . . 5  |-  ( ( ( th  /\  ta  /\ 
ch )  /\  ph0 )  ->  pred ( X ,  A ,  R )  C_  B )
116, 10jca 532 . . . 4  |-  ( ( ( th  /\  ta  /\ 
ch )  /\  ph0 )  ->  ( ( f `  (/) )  =  pred ( X ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B
) )
1211anim2i 569 . . 3  |-  ( ( i  =  (/)  /\  (
( th  /\  ta  /\ 
ch )  /\  ph0 )
)  ->  ( i  =  (/)  /\  ( ( f `  (/) )  = 
pred ( X ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B ) ) )
13 3anass 977 . . 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 212 . 2  |-  ( ( i  =  (/)  /\  (
( th  /\  ta  /\ 
ch )  /\  ph0 )
)  ->  ( i  =  (/)  /\  ( f `
 (/) )  =  pred ( X ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B
) )
15 fveq2 5872 . . . . . . 7  |-  ( i  =  (/)  ->  ( f `
 i )  =  ( f `  (/) ) )
1615eqeq1d 2459 . . . . . 6  |-  ( i  =  (/)  ->  ( ( f `  i )  =  pred ( X ,  A ,  R )  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
) )
1716biimpar 485 . . . . 5  |-  ( ( i  =  (/)  /\  (
f `  (/) )  = 
pred ( X ,  A ,  R )
)  ->  ( f `  i )  =  pred ( X ,  A ,  R ) )
1817adantr 465 . . . 4  |-  ( ( ( i  =  (/)  /\  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)  /\  pred ( X ,  A ,  R
)  C_  B )  ->  ( f `  i
)  =  pred ( X ,  A ,  R ) )
19 simpr 461 . . . 4  |-  ( ( ( i  =  (/)  /\  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)  /\  pred ( X ,  A ,  R
)  C_  B )  ->  pred ( X ,  A ,  R )  C_  B )
2018, 19eqsstrd 3533 . . 3  |-  ( ( ( i  =  (/)  /\  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)  /\  pred ( X ,  A ,  R
)  C_  B )  ->  ( f `  i
)  C_  B )
21203impa 1191 . 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   _Vcvv 3109    C_ wss 3471   (/)c0 3793    Fn wfn 5589   ` cfv 5594    /\ w-bnj17 33839    predc-bnj14 33841    TrFow-bnj19 33849
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-iota 5557  df-fv 5602  df-bnj17 33840
This theorem is referenced by:  bnj1030  34144
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