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Theorem bnj1097 29068
Description: Technical lemma for bnj69 29097. 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 187 . . . . . . . 8  |-  ( ph  ->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
41, 3bnj771 28851 . . . . . . 7  |-  ( ch 
->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
543ad2ant3 980 . . . . . 6  |-  ( ( th  /\  ta  /\  ch )  ->  ( f `
 (/) )  =  pred ( X ,  A ,  R ) )
65adantr 452 . . . . 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 974 . . . . . . 7  |-  ( ta 
->  pred ( X ,  A ,  R )  C_  B )
983ad2ant2 979 . . . . . 6  |-  ( ( th  /\  ta  /\  ch )  ->  pred ( X ,  A ,  R )  C_  B
)
109adantr 452 . . . . 5  |-  ( ( ( th  /\  ta  /\ 
ch )  /\  ph0 )  ->  pred ( X ,  A ,  R )  C_  B )
116, 10jca 519 . . . 4  |-  ( ( ( th  /\  ta  /\ 
ch )  /\  ph0 )  ->  ( ( f `  (/) )  =  pred ( X ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B
) )
1211anim2i 553 . . 3  |-  ( ( i  =  (/)  /\  (
( th  /\  ta  /\ 
ch )  /\  ph0 )
)  ->  ( i  =  (/)  /\  ( ( f `  (/) )  = 
pred ( X ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B ) ) )
13 3anass 940 . . 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 204 . 2  |-  ( ( i  =  (/)  /\  (
( th  /\  ta  /\ 
ch )  /\  ph0 )
)  ->  ( i  =  (/)  /\  ( f `
 (/) )  =  pred ( X ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B
) )
15 fveq2 5695 . . . . . . 7  |-  ( i  =  (/)  ->  ( f `
 i )  =  ( f `  (/) ) )
1615eqeq1d 2420 . . . . . 6  |-  ( i  =  (/)  ->  ( ( f `  i )  =  pred ( X ,  A ,  R )  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
) )
1716biimpar 472 . . . . 5  |-  ( ( i  =  (/)  /\  (
f `  (/) )  = 
pred ( X ,  A ,  R )
)  ->  ( f `  i )  =  pred ( X ,  A ,  R ) )
1817adantr 452 . . . 4  |-  ( ( ( i  =  (/)  /\  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)  /\  pred ( X ,  A ,  R
)  C_  B )  ->  ( f `  i
)  =  pred ( X ,  A ,  R ) )
19 simpr 448 . . . 4  |-  ( ( ( i  =  (/)  /\  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)  /\  pred ( X ,  A ,  R
)  C_  B )  ->  pred ( X ,  A ,  R )  C_  B )
2018, 19eqsstrd 3350 . . 3  |-  ( ( ( i  =  (/)  /\  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)  /\  pred ( X ,  A ,  R
)  C_  B )  ->  ( f `  i
)  C_  B )
21203impa 1148 . 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   _Vcvv 2924    C_ wss 3288   (/)c0 3596    Fn wfn 5416   ` cfv 5421    /\ w-bnj17 28768    predc-bnj14 28770    TrFow-bnj19 28778
This theorem is referenced by:  bnj1030  29074
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-rex 2680  df-rab 2683  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-iota 5385  df-fv 5429  df-bnj17 28769
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