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Theorem bnj229 28916
Description: Technical lemma for bnj517 28917. 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.)
Hypothesis
Ref Expression
bnj229.1  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  N  ->  ( F `  suc  i )  =  U_ y  e.  ( F `  i )  pred (
y ,  A ,  R ) ) )
Assertion
Ref Expression
bnj229  |-  ( ( n  e.  N  /\  ( suc  m  =  n  /\  m  e.  om  /\ 
ps ) )  -> 
( F `  n
)  C_  A )
Distinct variable groups:    A, i, m, y    i, F, m, y    i, N, m    R, i, m
Allowed substitution hints:    ps( y, i, m, n)    A( n)    R( y, n)    F( n)    N( y, n)

Proof of Theorem bnj229
StepHypRef Expression
1 bnj213 28914 . . 3  |-  pred (
y ,  A ,  R )  C_  A
21bnj226 28762 . 2  |-  U_ y  e.  ( F `  m
)  pred ( y ,  A ,  R ) 
C_  A
3 bnj229.1 . . . . . . . 8  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  N  ->  ( F `  suc  i )  =  U_ y  e.  ( F `  i )  pred (
y ,  A ,  R ) ) )
43bnj222 28915 . . . . . . 7  |-  ( ps  <->  A. m  e.  om  ( suc  m  e.  N  -> 
( F `  suc  m )  =  U_ y  e.  ( F `  m )  pred (
y ,  A ,  R ) ) )
54bnj228 28763 . . . . . 6  |-  ( ( m  e.  om  /\  ps )  ->  ( suc  m  e.  N  -> 
( F `  suc  m )  =  U_ y  e.  ( F `  m )  pred (
y ,  A ,  R ) ) )
65adantl 452 . . . . 5  |-  ( ( suc  m  =  n  /\  ( m  e. 
om  /\  ps )
)  ->  ( suc  m  e.  N  ->  ( F `  suc  m
)  =  U_ y  e.  ( F `  m
)  pred ( y ,  A ,  R ) ) )
7 eleq1 2343 . . . . . . 7  |-  ( suc  m  =  n  -> 
( suc  m  e.  N 
<->  n  e.  N ) )
8 fveq2 5525 . . . . . . . 8  |-  ( suc  m  =  n  -> 
( F `  suc  m )  =  ( F `  n ) )
98eqeq1d 2291 . . . . . . 7  |-  ( suc  m  =  n  -> 
( ( F `  suc  m )  =  U_ y  e.  ( F `  m )  pred (
y ,  A ,  R )  <->  ( F `  n )  =  U_ y  e.  ( F `  m )  pred (
y ,  A ,  R ) ) )
107, 9imbi12d 311 . . . . . 6  |-  ( suc  m  =  n  -> 
( ( suc  m  e.  N  ->  ( F `
 suc  m )  =  U_ y  e.  ( F `  m ) 
pred ( y ,  A ,  R ) )  <->  ( n  e.  N  ->  ( F `  n )  =  U_ y  e.  ( F `  m )  pred (
y ,  A ,  R ) ) ) )
1110adantr 451 . . . . 5  |-  ( ( suc  m  =  n  /\  ( m  e. 
om  /\  ps )
)  ->  ( ( suc  m  e.  N  -> 
( F `  suc  m )  =  U_ y  e.  ( F `  m )  pred (
y ,  A ,  R ) )  <->  ( n  e.  N  ->  ( F `
 n )  = 
U_ y  e.  ( F `  m ) 
pred ( y ,  A ,  R ) ) ) )
126, 11mpbid 201 . . . 4  |-  ( ( suc  m  =  n  /\  ( m  e. 
om  /\  ps )
)  ->  ( n  e.  N  ->  ( F `
 n )  = 
U_ y  e.  ( F `  m ) 
pred ( y ,  A ,  R ) ) )
13123impb 1147 . . 3  |-  ( ( suc  m  =  n  /\  m  e.  om  /\ 
ps )  ->  (
n  e.  N  -> 
( F `  n
)  =  U_ y  e.  ( F `  m
)  pred ( y ,  A ,  R ) ) )
1413impcom 419 . 2  |-  ( ( n  e.  N  /\  ( suc  m  =  n  /\  m  e.  om  /\ 
ps ) )  -> 
( F `  n
)  =  U_ y  e.  ( F `  m
)  pred ( y ,  A ,  R ) )
152, 14bnj1262 28843 1  |-  ( ( n  e.  N  /\  ( suc  m  =  n  /\  m  e.  om  /\ 
ps ) )  -> 
( F `  n
)  C_  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543    C_ wss 3152   U_ciun 3905   suc csuc 4394   omcom 4656   ` cfv 5255    predc-bnj14 28713
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
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-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-suc 4398  df-iota 5219  df-fv 5263  df-bnj14 28714
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