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Theorem bnj517 28962
Description: Technical lemma for bnj518 28963. 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 5687 . . . . . 6  |-  ( m  =  (/)  ->  ( F `
 m )  =  ( F `  (/) ) )
2 simpl2 961 . . . . . . 7  |-  ( ( ( N  e.  om  /\ 
ph  /\  ps )  /\  m  e.  N
)  ->  ph )
3 bnj517.1 . . . . . . 7  |-  ( ph  <->  ( F `  (/) )  = 
pred ( X ,  A ,  R )
)
42, 3sylib 189 . . . . . 6  |-  ( ( ( N  e.  om  /\ 
ph  /\  ps )  /\  m  e.  N
)  ->  ( F `  (/) )  =  pred ( X ,  A ,  R ) )
51, 4sylan9eqr 2458 . . . . 5  |-  ( ( ( ( N  e. 
om  /\  ph  /\  ps )  /\  m  e.  N
)  /\  m  =  (/) )  ->  ( F `  m )  =  pred ( X ,  A ,  R ) )
6 bnj213 28959 . . . . 5  |-  pred ( X ,  A ,  R )  C_  A
75, 6syl6eqss 3358 . . . 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 2807 . . . . . . . . . 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 2464 . . . . . . . . . . . . . 14  |-  ( m  =  suc  i  -> 
( m  e.  N  <->  suc  i  e.  N ) )
1110biimpd 199 . . . . . . . . . . . . 13  |-  ( m  =  suc  i  -> 
( m  e.  N  ->  suc  i  e.  N
) )
12 fveq2 5687 . . . . . . . . . . . . . . 15  |-  ( m  =  suc  i  -> 
( F `  m
)  =  ( F `
 suc  i )
)
1312eqeq1d 2412 . . . . . . . . . . . . . 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 28959 . . . . . . . . . . . . . . . . 17  |-  pred (
y ,  A ,  R )  C_  A
1514rgenw 2733 . . . . . . . . . . . . . . . 16  |-  A. y  e.  ( F `  i
)  pred ( y ,  A ,  R ) 
C_  A
16 iunss 4092 . . . . . . . . . . . . . . . 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 201 . . . . . . . . . . . . . . 15  |-  U_ y  e.  ( F `  i
)  pred ( y ,  A ,  R ) 
C_  A
18 sseq1 3329 . . . . . . . . . . . . . . 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 225 . . . . . . . . . . . . . 14  |-  ( ( F `  m )  =  U_ y  e.  ( F `  i
)  pred ( y ,  A ,  R )  ->  ( F `  m )  C_  A
)
2013, 19syl6bir 221 . . . . . . . . . . . . 13  |-  ( m  =  suc  i  -> 
( ( F `  suc  i )  =  U_ y  e.  ( F `  i )  pred (
y ,  A ,  R )  ->  ( F `  m )  C_  A ) )
2111, 20imim12d 70 . . . . . . . . . . . 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 419 . . . . . . . . . . 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 2786 . . . . . . . . . 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 424 . . . . . . . 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 77 . . . . . . 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 188 . . . . . 6  |-  ( ps 
->  ( m  e.  N  ->  ( E. i  e. 
om  m  =  suc  i  ->  ( F `  m )  C_  A
) ) )
28273ad2ant3 980 . . . . 5  |-  ( ( N  e.  om  /\  ph 
/\  ps )  ->  (
m  e.  N  -> 
( E. i  e. 
om  m  =  suc  i  ->  ( F `  m )  C_  A
) ) )
2928imp31 422 . . . 4  |-  ( ( ( ( N  e. 
om  /\  ph  /\  ps )  /\  m  e.  N
)  /\  E. i  e.  om  m  =  suc  i )  ->  ( F `  m )  C_  A )
30 simpr 448 . . . . . 6  |-  ( ( ( N  e.  om  /\ 
ph  /\  ps )  /\  m  e.  N
)  ->  m  e.  N )
31 simpl1 960 . . . . . 6  |-  ( ( ( N  e.  om  /\ 
ph  /\  ps )  /\  m  e.  N
)  ->  N  e.  om )
32 elnn 4814 . . . . . 6  |-  ( ( m  e.  N  /\  N  e.  om )  ->  m  e.  om )
3330, 31, 32syl2anc 643 . . . . 5  |-  ( ( ( N  e.  om  /\ 
ph  /\  ps )  /\  m  e.  N
)  ->  m  e.  om )
34 nn0suc 4828 . . . . 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 762 . . 3  |-  ( ( ( N  e.  om  /\ 
ph  /\  ps )  /\  m  e.  N
)  ->  ( F `  m )  C_  A
)
3736ralrimiva 2749 . 2  |-  ( ( N  e.  om  /\  ph 
/\  ps )  ->  A. m  e.  N  ( F `  m )  C_  A
)
38 fveq2 5687 . . . 4  |-  ( m  =  n  ->  ( F `  m )  =  ( F `  n ) )
3938sseq1d 3335 . . 3  |-  ( m  =  n  ->  (
( F `  m
)  C_  A  <->  ( F `  n )  C_  A
) )
4039cbvralv 2892 . 2  |-  ( A. m  e.  N  ( F `  m )  C_  A  <->  A. n  e.  N  ( F `  n ) 
C_  A )
4137, 40sylib 189 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 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2666   E.wrex 2667    C_ wss 3280   (/)c0 3588   U_ciun 4053   suc csuc 4543   omcom 4804   ` cfv 5413    predc-bnj14 28758
This theorem is referenced by:  bnj518  28963
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-tr 4263  df-eprel 4454  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-iota 5377  df-fv 5421  df-bnj14 28759
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