Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj517 Unicode version

Theorem bnj517 28917
Description: Technical lemma for bnj518 28918. 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.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj517.1  |-  ( ph  <->  ( F `  (/) )  = 
pred ( X ,  A ,  R )
)
bnj517.2  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  N  ->  ( F `  suc  i )  =  U_ y  e.  ( F `  i )  pred (
y ,  A ,  R ) ) )
Assertion
Ref Expression
bnj517  |-  ( ( N  e.  om  /\  ph 
/\  ps )  ->  A. n  e.  N  ( F `  n )  C_  A
)
Distinct variable groups:    i, n, y, A    i, F, n   
i, N, n
Allowed substitution hints:    ph( y, i, n)    ps( y, i, n)    R( y, i, n)    F( y)    N( y)    X( y, i, n)

Proof of Theorem bnj517
Dummy variable  m is distinct from all other variables.
StepHypRef Expression
1 bnj213 28914 . . . . 5  |-  pred ( X ,  A ,  R )  C_  A
2 fveq2 5525 . . . . . . 7  |-  ( m  =  (/)  ->  ( F `
 m )  =  ( F `  (/) ) )
3 simpl2 959 . . . . . . . 8  |-  ( ( ( N  e.  om  /\ 
ph  /\  ps )  /\  m  e.  N
)  ->  ph )
4 bnj517.1 . . . . . . . 8  |-  ( ph  <->  ( F `  (/) )  = 
pred ( X ,  A ,  R )
)
53, 4sylib 188 . . . . . . 7  |-  ( ( ( N  e.  om  /\ 
ph  /\  ps )  /\  m  e.  N
)  ->  ( F `  (/) )  =  pred ( X ,  A ,  R ) )
62, 5sylan9eqr 2337 . . . . . 6  |-  ( ( ( ( N  e. 
om  /\  ph  /\  ps )  /\  m  e.  N
)  /\  m  =  (/) )  ->  ( F `  m )  =  pred ( X ,  A ,  R ) )
76sseq1d 3205 . . . . 5  |-  ( ( ( ( N  e. 
om  /\  ph  /\  ps )  /\  m  e.  N
)  /\  m  =  (/) )  ->  ( ( F `  m )  C_  A  <->  pred ( X ,  A ,  R )  C_  A ) )
81, 7mpbiri 224 . . . 4  |-  ( ( ( ( N  e. 
om  /\  ph  /\  ps )  /\  m  e.  N
)  /\  m  =  (/) )  ->  ( F `  m )  C_  A
)
9 bnj517.2 . . . . . . 7  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  N  ->  ( F `  suc  i )  =  U_ y  e.  ( F `  i )  pred (
y ,  A ,  R ) ) )
10 r19.29r 2684 . . . . . . . . . 10  |-  ( ( E. i  e.  om  m  =  suc  i  /\  A. i  e.  om  ( suc  i  e.  N  ->  ( F `  suc  i )  =  U_ y  e.  ( F `  i )  pred (
y ,  A ,  R ) ) )  ->  E. i  e.  om  ( m  =  suc  i  /\  ( suc  i  e.  N  ->  ( F `
 suc  i )  =  U_ y  e.  ( F `  i ) 
pred ( y ,  A ,  R ) ) ) )
11 eleq1 2343 . . . . . . . . . . . . . 14  |-  ( m  =  suc  i  -> 
( m  e.  N  <->  suc  i  e.  N ) )
1211biimpd 198 . . . . . . . . . . . . 13  |-  ( m  =  suc  i  -> 
( m  e.  N  ->  suc  i  e.  N
) )
13 fveq2 5525 . . . . . . . . . . . . . . 15  |-  ( m  =  suc  i  -> 
( F `  m
)  =  ( F `
 suc  i )
)
1413eqeq1d 2291 . . . . . . . . . . . . . 14  |-  ( m  =  suc  i  -> 
( ( F `  m )  =  U_ y  e.  ( F `  i )  pred (
y ,  A ,  R )  <->  ( F `  suc  i )  = 
U_ y  e.  ( F `  i ) 
pred ( y ,  A ,  R ) ) )
15 bnj213 28914 . . . . . . . . . . . . . . . . 17  |-  pred (
y ,  A ,  R )  C_  A
1615rgenw 2610 . . . . . . . . . . . . . . . 16  |-  A. y  e.  ( F `  i
)  pred ( y ,  A ,  R ) 
C_  A
17 iunss 3943 . . . . . . . . . . . . . . . 16  |-  ( U_ y  e.  ( F `  i )  pred (
y ,  A ,  R )  C_  A  <->  A. y  e.  ( F `
 i )  pred ( y ,  A ,  R )  C_  A
)
1816, 17mpbir 200 . . . . . . . . . . . . . . 15  |-  U_ y  e.  ( F `  i
)  pred ( y ,  A ,  R ) 
C_  A
19 sseq1 3199 . . . . . . . . . . . . . . 15  |-  ( ( F `  m )  =  U_ y  e.  ( F `  i
)  pred ( y ,  A ,  R )  ->  ( ( F `
 m )  C_  A 
<-> 
U_ y  e.  ( F `  i ) 
pred ( y ,  A ,  R ) 
C_  A ) )
2018, 19mpbiri 224 . . . . . . . . . . . . . 14  |-  ( ( F `  m )  =  U_ y  e.  ( F `  i
)  pred ( y ,  A ,  R )  ->  ( F `  m )  C_  A
)
2114, 20syl6bir 220 . . . . . . . . . . . . 13  |-  ( m  =  suc  i  -> 
( ( F `  suc  i )  =  U_ y  e.  ( F `  i )  pred (
y ,  A ,  R )  ->  ( F `  m )  C_  A ) )
2212, 21imim12d 68 . . . . . . . . . . . 12  |-  ( m  =  suc  i  -> 
( ( suc  i  e.  N  ->  ( F `
 suc  i )  =  U_ y  e.  ( F `  i ) 
pred ( y ,  A ,  R ) )  ->  ( m  e.  N  ->  ( F `
 m )  C_  A ) ) )
2322imp 418 . . . . . . . . . . 11  |-  ( ( m  =  suc  i  /\  ( suc  i  e.  N  ->  ( F `  suc  i )  = 
U_ y  e.  ( F `  i ) 
pred ( y ,  A ,  R ) ) )  ->  (
m  e.  N  -> 
( F `  m
)  C_  A )
)
2423rexlimivw 2663 . . . . . . . . . 10  |-  ( E. i  e.  om  (
m  =  suc  i  /\  ( suc  i  e.  N  ->  ( F `  suc  i )  = 
U_ y  e.  ( F `  i ) 
pred ( y ,  A ,  R ) ) )  ->  (
m  e.  N  -> 
( F `  m
)  C_  A )
)
2510, 24syl 15 . . . . . . . . 9  |-  ( ( E. i  e.  om  m  =  suc  i  /\  A. i  e.  om  ( suc  i  e.  N  ->  ( F `  suc  i )  =  U_ y  e.  ( F `  i )  pred (
y ,  A ,  R ) ) )  ->  ( m  e.  N  ->  ( F `  m )  C_  A
) )
2625ex 423 . . . . . . . 8  |-  ( E. i  e.  om  m  =  suc  i  ->  ( A. i  e.  om  ( suc  i  e.  N  ->  ( F `  suc  i )  =  U_ y  e.  ( F `  i )  pred (
y ,  A ,  R ) )  -> 
( m  e.  N  ->  ( F `  m
)  C_  A )
) )
2726com3l 75 . . . . . . 7  |-  ( A. i  e.  om  ( suc  i  e.  N  ->  ( F `  suc  i )  =  U_ y  e.  ( F `  i )  pred (
y ,  A ,  R ) )  -> 
( m  e.  N  ->  ( E. i  e. 
om  m  =  suc  i  ->  ( F `  m )  C_  A
) ) )
289, 27sylbi 187 . . . . . 6  |-  ( ps 
->  ( m  e.  N  ->  ( E. i  e. 
om  m  =  suc  i  ->  ( F `  m )  C_  A
) ) )
29283ad2ant3 978 . . . . 5  |-  ( ( N  e.  om  /\  ph 
/\  ps )  ->  (
m  e.  N  -> 
( E. i  e. 
om  m  =  suc  i  ->  ( F `  m )  C_  A
) ) )
3029imp31 421 . . . 4  |-  ( ( ( ( N  e. 
om  /\  ph  /\  ps )  /\  m  e.  N
)  /\  E. i  e.  om  m  =  suc  i )  ->  ( F `  m )  C_  A )
31 simpr 447 . . . . . 6  |-  ( ( ( N  e.  om  /\ 
ph  /\  ps )  /\  m  e.  N
)  ->  m  e.  N )
32 simpl1 958 . . . . . 6  |-  ( ( ( N  e.  om  /\ 
ph  /\  ps )  /\  m  e.  N
)  ->  N  e.  om )
33 elnn 4666 . . . . . 6  |-  ( ( m  e.  N  /\  N  e.  om )  ->  m  e.  om )
3431, 32, 33syl2anc 642 . . . . 5  |-  ( ( ( N  e.  om  /\ 
ph  /\  ps )  /\  m  e.  N
)  ->  m  e.  om )
35 nn0suc 4680 . . . . 5  |-  ( m  e.  om  ->  (
m  =  (/)  \/  E. i  e.  om  m  =  suc  i ) )
3634, 35syl 15 . . . 4  |-  ( ( ( N  e.  om  /\ 
ph  /\  ps )  /\  m  e.  N
)  ->  ( m  =  (/)  \/  E. i  e.  om  m  =  suc  i ) )
378, 30, 36mpjaodan 761 . . 3  |-  ( ( ( N  e.  om  /\ 
ph  /\  ps )  /\  m  e.  N
)  ->  ( F `  m )  C_  A
)
3837ralrimiva 2626 . 2  |-  ( ( N  e.  om  /\  ph 
/\  ps )  ->  A. m  e.  N  ( F `  m )  C_  A
)
39 fveq2 5525 . . . 4  |-  ( m  =  n  ->  ( F `  m )  =  ( F `  n ) )
4039sseq1d 3205 . . 3  |-  ( m  =  n  ->  (
( F `  m
)  C_  A  <->  ( F `  n )  C_  A
) )
4140cbvralv 2764 . 2  |-  ( A. m  e.  N  ( F `  m )  C_  A  <->  A. n  e.  N  ( F `  n ) 
C_  A )
4238, 41sylib 188 1  |-  ( ( N  e.  om  /\  ph 
/\  ps )  ->  A. n  e.  N  ( F `  n )  C_  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    C_ wss 3152   (/)c0 3455   U_ciun 3905   suc csuc 4394   omcom 4656   ` cfv 5255    predc-bnj14 28713
This theorem is referenced by:  bnj518  28918
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-13 1686  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-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-iota 5219  df-fv 5263  df-bnj14 28714
  Copyright terms: Public domain W3C validator