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Theorem bnj517 34065
Description: Technical lemma for bnj518 34066. 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 5872 . . . . . 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 2520 . . . . 5  |-  ( ( ( ( N  e. 
om  /\  ph  /\  ps )  /\  m  e.  N
)  /\  m  =  (/) )  ->  ( F `  m )  =  pred ( X ,  A ,  R ) )
6 bnj213 34062 . . . . 5  |-  pred ( X ,  A ,  R )  C_  A
75, 6syl6eqss 3549 . . . 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 2993 . . . . . . . . . 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 2529 . . . . . . . . . . . . . 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 5872 . . . . . . . . . . . . . . 15  |-  ( m  =  suc  i  -> 
( F `  m
)  =  ( F `
 suc  i )
)
1312eqeq1d 2459 . . . . . . . . . . . . . 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 34062 . . . . . . . . . . . . . . . . 17  |-  pred (
y ,  A ,  R )  C_  A
1514rgenw 2818 . . . . . . . . . . . . . . . 16  |-  A. y  e.  ( F `  i
)  pred ( y ,  A ,  R ) 
C_  A
16 iunss 4373 . . . . . . . . . . . . . . . 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 3520 . . . . . . . . . . . . . . 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 2946 . . . . . . . . . 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 6709 . . . . . 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 6723 . . . . 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 786 . . 3  |-  ( ( ( N  e.  om  /\ 
ph  /\  ps )  /\  m  e.  N
)  ->  ( F `  m )  C_  A
)
3736ralrimiva 2871 . 2  |-  ( ( N  e.  om  /\  ph 
/\  ps )  ->  A. m  e.  N  ( F `  m )  C_  A
)
38 fveq2 5872 . . . 4  |-  ( m  =  n  ->  ( F `  m )  =  ( F `  n ) )
3938sseq1d 3526 . . 3  |-  ( m  =  n  ->  (
( F `  m
)  C_  A  <->  ( F `  n )  C_  A
) )
4039cbvralv 3084 . 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 1395    e. wcel 1819   A.wral 2807   E.wrex 2808    C_ wss 3471   (/)c0 3793   U_ciun 4332   suc csuc 4889   ` cfv 5594   omcom 6699    predc-bnj14 33862
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-tr 4551  df-eprel 4800  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-iota 5557  df-fv 5602  df-om 6700  df-bnj14 33863
This theorem is referenced by:  bnj518  34066
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