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Theorem bnj229 31875
Description: Technical lemma for bnj517 31876. 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 31873 . . 3  |-  pred (
y ,  A ,  R )  C_  A
21bnj226 31723 . 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 31874 . . . . . . 7  |-  ( ps  <->  A. m  e.  om  ( suc  m  e.  N  -> 
( F `  suc  m )  =  U_ y  e.  ( F `  m )  pred (
y ,  A ,  R ) ) )
54bnj228 31724 . . . . . 6  |-  ( ( m  e.  om  /\  ps )  ->  ( suc  m  e.  N  -> 
( F `  suc  m )  =  U_ y  e.  ( F `  m )  pred (
y ,  A ,  R ) ) )
65adantl 466 . . . . 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 2502 . . . . . . 7  |-  ( suc  m  =  n  -> 
( suc  m  e.  N 
<->  n  e.  N ) )
8 fveq2 5690 . . . . . . . 8  |-  ( suc  m  =  n  -> 
( F `  suc  m )  =  ( F `  n ) )
98eqeq1d 2450 . . . . . . 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 320 . . . . . 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 465 . . . . 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 210 . . . 4  |-  ( ( suc  m  =  n  /\  ( m  e. 
om  /\  ps )
)  ->  ( n  e.  N  ->  ( F `
 n )  = 
U_ y  e.  ( F `  m ) 
pred ( y ,  A ,  R ) ) )
13123impb 1183 . . 3  |-  ( ( suc  m  =  n  /\  m  e.  om  /\ 
ps )  ->  (
n  e.  N  -> 
( F `  n
)  =  U_ y  e.  ( F `  m
)  pred ( y ,  A ,  R ) ) )
1413impcom 430 . 2  |-  ( ( n  e.  N  /\  ( suc  m  =  n  /\  m  e.  om  /\ 
ps ) )  -> 
( F `  n
)  =  U_ y  e.  ( F `  m
)  pred ( y ,  A ,  R ) )
152, 14bnj1262 31802 1  |-  ( ( n  e.  N  /\  ( suc  m  =  n  /\  m  e.  om  /\ 
ps ) )  -> 
( F `  n
)  C_  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2714    C_ wss 3327   U_ciun 4170   suc csuc 4720   ` cfv 5417   omcom 6475    predc-bnj14 31674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-iun 4172  df-br 4292  df-suc 4724  df-iota 5380  df-fv 5425  df-bnj14 31675
This theorem is referenced by: (None)
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