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Theorem bnj125 28904
Description: Technical lemma for bnj150 28908. 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
bnj125.1  |-  ( ph  <->  ( f `  (/) )  = 
pred ( x ,  A ,  R ) )
bnj125.2  |-  ( ph'  <->  [. 1o  /  n ]. ph )
bnj125.3  |-  ( ph"  <->  [. F  / 
f ]. ph' )
bnj125.4  |-  F  =  { <. (/) ,  pred (
x ,  A ,  R ) >. }
Assertion
Ref Expression
bnj125  |-  ( ph"  <->  ( F `  (/) )  =  pred ( x ,  A ,  R ) )
Distinct variable groups:    A, f, n    f, F    R, f, n    x, f, n
Allowed substitution hints:    ph( x, f, n)    A( x)    R( x)    F( x, n)    ph'( x, f, n)   
ph"( x, f, n)

Proof of Theorem bnj125
StepHypRef Expression
1 bnj125.3 . 2  |-  ( ph"  <->  [. F  / 
f ]. ph' )
2 bnj125.2 . . . 4  |-  ( ph'  <->  [. 1o  /  n ]. ph )
3 bnj125.4 . . . . 5  |-  F  =  { <. (/) ,  pred (
x ,  A ,  R ) >. }
43bnj95 28896 . . . 4  |-  F  e. 
_V
52, 4bnj524 28766 . . 3  |-  ( [. F  /  f ]. ph'  <->  [. F  / 
f ]. [. 1o  /  n ]. ph )
6 bnj125.1 . . . . . 6  |-  ( ph  <->  ( f `  (/) )  = 
pred ( x ,  A ,  R ) )
7 bnj105 28750 . . . . . 6  |-  1o  e.  _V
86, 7bnj91 28893 . . . . 5  |-  ( [. 1o  /  n ]. ph  <->  ( f `  (/) )  =  pred ( x ,  A ,  R ) )
98, 4bnj524 28766 . . . 4  |-  ( [. F  /  f ]. [. 1o  /  n ]. ph  <->  [. F  / 
f ]. ( f `  (/) )  =  pred (
x ,  A ,  R ) )
10 fveq1 5524 . . . . . 6  |-  ( f  =  F  ->  (
f `  (/) )  =  ( F `  (/) ) )
1110eqeq1d 2291 . . . . 5  |-  ( f  =  F  ->  (
( f `  (/) )  = 
pred ( x ,  A ,  R )  <-> 
( F `  (/) )  = 
pred ( x ,  A ,  R ) ) )
124, 11sbcie 3025 . . . 4  |-  ( [. F  /  f ]. (
f `  (/) )  = 
pred ( x ,  A ,  R )  <-> 
( F `  (/) )  = 
pred ( x ,  A ,  R ) )
139, 12bitri 240 . . 3  |-  ( [. F  /  f ]. [. 1o  /  n ]. ph  <->  ( F `  (/) )  =  pred ( x ,  A ,  R ) )
145, 13bitri 240 . 2  |-  ( [. F  /  f ]. ph'  <->  ( F `  (/) )  =  pred ( x ,  A ,  R ) )
151, 14bitri 240 1  |-  ( ph"  <->  ( F `  (/) )  =  pred ( x ,  A ,  R ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1623   [.wsbc 2991   (/)c0 3455   {csn 3640   <.cop 3643   ` cfv 5255   1oc1o 6472    predc-bnj14 28713
This theorem is referenced by:  bnj150  28908  bnj153  28912
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rex 2549  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-pw 3627  df-sn 3646  df-pr 3647  df-uni 3828  df-br 4024  df-suc 4398  df-iota 5219  df-fv 5263  df-1o 6479
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