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Theorem bnj229 28595
Description: Technical lemma for bnj517 28596. 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 28593 . . 3  |-  pred (
y ,  A ,  R )  C_  A
21bnj226 28441 . 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 28594 . . . . . . 7  |-  ( ps  <->  A. m  e.  om  ( suc  m  e.  N  -> 
( F `  suc  m )  =  U_ y  e.  ( F `  m )  pred (
y ,  A ,  R ) ) )
54bnj228 28442 . . . . . 6  |-  ( ( m  e.  om  /\  ps )  ->  ( suc  m  e.  N  -> 
( F `  suc  m )  =  U_ y  e.  ( F `  m )  pred (
y ,  A ,  R ) ) )
65adantl 453 . . . . 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 2449 . . . . . . 7  |-  ( suc  m  =  n  -> 
( suc  m  e.  N 
<->  n  e.  N ) )
8 fveq2 5670 . . . . . . . 8  |-  ( suc  m  =  n  -> 
( F `  suc  m )  =  ( F `  n ) )
98eqeq1d 2397 . . . . . . 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 312 . . . . . 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 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 ) )  <->  ( n  e.  N  ->  ( F `
 n )  = 
U_ y  e.  ( F `  m ) 
pred ( y ,  A ,  R ) ) ) )
126, 11mpbid 202 . . . 4  |-  ( ( suc  m  =  n  /\  ( m  e. 
om  /\  ps )
)  ->  ( n  e.  N  ->  ( F `
 n )  = 
U_ y  e.  ( F `  m ) 
pred ( y ,  A ,  R ) ) )
13123impb 1149 . . 3  |-  ( ( suc  m  =  n  /\  m  e.  om  /\ 
ps )  ->  (
n  e.  N  -> 
( F `  n
)  =  U_ y  e.  ( F `  m
)  pred ( y ,  A ,  R ) ) )
1413impcom 420 . 2  |-  ( ( n  e.  N  /\  ( suc  m  =  n  /\  m  e.  om  /\ 
ps ) )  -> 
( F `  n
)  =  U_ y  e.  ( F `  m
)  pred ( y ,  A ,  R ) )
152, 14bnj1262 28522 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   A.wral 2651    C_ wss 3265   U_ciun 4037   suc csuc 4526   omcom 4787   ` cfv 5396    predc-bnj14 28392
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 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370
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 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-suc 4530  df-iota 5360  df-fv 5404  df-bnj14 28393
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