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Theorem bnj517 33022
Description: Technical lemma for bnj518 33023. 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 fveq2 5864 . . . . . 6  |-  ( m  =  (/)  ->  ( F `
 m )  =  ( F `  (/) ) )
2 simpl2 1000 . . . . . . 7  |-  ( ( ( N  e.  om  /\ 
ph  /\  ps )  /\  m  e.  N
)  ->  ph )
3 bnj517.1 . . . . . . 7  |-  ( ph  <->  ( F `  (/) )  = 
pred ( X ,  A ,  R )
)
42, 3sylib 196 . . . . . 6  |-  ( ( ( N  e.  om  /\ 
ph  /\  ps )  /\  m  e.  N
)  ->  ( F `  (/) )  =  pred ( X ,  A ,  R ) )
51, 4sylan9eqr 2530 . . . . 5  |-  ( ( ( ( N  e. 
om  /\  ph  /\  ps )  /\  m  e.  N
)  /\  m  =  (/) )  ->  ( F `  m )  =  pred ( X ,  A ,  R ) )
6 bnj213 33019 . . . . 5  |-  pred ( X ,  A ,  R )  C_  A
75, 6syl6eqss 3554 . . . 4  |-  ( ( ( ( N  e. 
om  /\  ph  /\  ps )  /\  m  e.  N
)  /\  m  =  (/) )  ->  ( F `  m )  C_  A
)
8 bnj517.2 . . . . . . 7  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  N  ->  ( F `  suc  i )  =  U_ y  e.  ( F `  i )  pred (
y ,  A ,  R ) ) )
9 r19.29r 2998 . . . . . . . . . 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 ) ) ) )
10 eleq1 2539 . . . . . . . . . . . . . 14  |-  ( m  =  suc  i  -> 
( m  e.  N  <->  suc  i  e.  N ) )
1110biimpd 207 . . . . . . . . . . . . 13  |-  ( m  =  suc  i  -> 
( m  e.  N  ->  suc  i  e.  N
) )
12 fveq2 5864 . . . . . . . . . . . . . . 15  |-  ( m  =  suc  i  -> 
( F `  m
)  =  ( F `
 suc  i )
)
1312eqeq1d 2469 . . . . . . . . . . . . . 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 ) ) )
14 bnj213 33019 . . . . . . . . . . . . . . . . 17  |-  pred (
y ,  A ,  R )  C_  A
1514rgenw 2825 . . . . . . . . . . . . . . . 16  |-  A. y  e.  ( F `  i
)  pred ( y ,  A ,  R ) 
C_  A
16 iunss 4366 . . . . . . . . . . . . . . . 16  |-  ( U_ y  e.  ( F `  i )  pred (
y ,  A ,  R )  C_  A  <->  A. y  e.  ( F `
 i )  pred ( y ,  A ,  R )  C_  A
)
1715, 16mpbir 209 . . . . . . . . . . . . . . 15  |-  U_ y  e.  ( F `  i
)  pred ( y ,  A ,  R ) 
C_  A
18 sseq1 3525 . . . . . . . . . . . . . . 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 ) )
1917, 18mpbiri 233 . . . . . . . . . . . . . 14  |-  ( ( F `  m )  =  U_ y  e.  ( F `  i
)  pred ( y ,  A ,  R )  ->  ( F `  m )  C_  A
)
2013, 19syl6bir 229 . . . . . . . . . . . . 13  |-  ( m  =  suc  i  -> 
( ( F `  suc  i )  =  U_ y  e.  ( F `  i )  pred (
y ,  A ,  R )  ->  ( F `  m )  C_  A ) )
2111, 20imim12d 74 . . . . . . . . . . . 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 ) ) )
2221imp 429 . . . . . . . . . . 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 )
)
2322rexlimivw 2952 . . . . . . . . . 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 )
)
249, 23syl 16 . . . . . . . . 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
) )
2524ex 434 . . . . . . . 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 )
) )
2625com3l 81 . . . . . . 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
) ) )
278, 26sylbi 195 . . . . . 6  |-  ( ps 
->  ( m  e.  N  ->  ( E. i  e. 
om  m  =  suc  i  ->  ( F `  m )  C_  A
) ) )
28273ad2ant3 1019 . . . . 5  |-  ( ( N  e.  om  /\  ph 
/\  ps )  ->  (
m  e.  N  -> 
( E. i  e. 
om  m  =  suc  i  ->  ( F `  m )  C_  A
) ) )
2928imp31 432 . . . 4  |-  ( ( ( ( N  e. 
om  /\  ph  /\  ps )  /\  m  e.  N
)  /\  E. i  e.  om  m  =  suc  i )  ->  ( F `  m )  C_  A )
30 simpr 461 . . . . . 6  |-  ( ( ( N  e.  om  /\ 
ph  /\  ps )  /\  m  e.  N
)  ->  m  e.  N )
31 simpl1 999 . . . . . 6  |-  ( ( ( N  e.  om  /\ 
ph  /\  ps )  /\  m  e.  N
)  ->  N  e.  om )
32 elnn 6688 . . . . . 6  |-  ( ( m  e.  N  /\  N  e.  om )  ->  m  e.  om )
3330, 31, 32syl2anc 661 . . . . 5  |-  ( ( ( N  e.  om  /\ 
ph  /\  ps )  /\  m  e.  N
)  ->  m  e.  om )
34 nn0suc 6702 . . . . 5  |-  ( m  e.  om  ->  (
m  =  (/)  \/  E. i  e.  om  m  =  suc  i ) )
3533, 34syl 16 . . . 4  |-  ( ( ( N  e.  om  /\ 
ph  /\  ps )  /\  m  e.  N
)  ->  ( m  =  (/)  \/  E. i  e.  om  m  =  suc  i ) )
367, 29, 35mpjaodan 784 . . 3  |-  ( ( ( N  e.  om  /\ 
ph  /\  ps )  /\  m  e.  N
)  ->  ( F `  m )  C_  A
)
3736ralrimiva 2878 . 2  |-  ( ( N  e.  om  /\  ph 
/\  ps )  ->  A. m  e.  N  ( F `  m )  C_  A
)
38 fveq2 5864 . . . 4  |-  ( m  =  n  ->  ( F `  m )  =  ( F `  n ) )
3938sseq1d 3531 . . 3  |-  ( m  =  n  ->  (
( F `  m
)  C_  A  <->  ( F `  n )  C_  A
) )
4039cbvralv 3088 . 2  |-  ( A. m  e.  N  ( F `  m )  C_  A  <->  A. n  e.  N  ( F `  n ) 
C_  A )
4137, 40sylib 196 1  |-  ( ( N  e.  om  /\  ph 
/\  ps )  ->  A. n  e.  N  ( F `  n )  C_  A
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2814   E.wrex 2815    C_ wss 3476   (/)c0 3785   U_ciun 4325   suc csuc 4880   ` cfv 5586   omcom 6678    predc-bnj14 32820
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-tr 4541  df-eprel 4791  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-iota 5549  df-fv 5594  df-om 6679  df-bnj14 32821
This theorem is referenced by:  bnj518  33023
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