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Theorem zorn2lem5 8871
Description: Lemma for zorn2 8877. (Contributed by NM, 4-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
Hypotheses
Ref Expression
zorn2lem.3  |-  F  = recs ( ( f  e. 
_V  |->  ( iota_ v  e.  C  A. u  e.  C  -.  u w v ) ) )
zorn2lem.4  |-  C  =  { z  e.  A  |  A. g  e.  ran  f  g R z }
zorn2lem.5  |-  D  =  { z  e.  A  |  A. g  e.  ( F " x ) g R z }
zorn2lem.7  |-  H  =  { z  e.  A  |  A. g  e.  ( F " y ) g R z }
Assertion
Ref Expression
zorn2lem5  |-  ( ( ( w  We  A  /\  x  e.  On )  /\  A. y  e.  x  H  =/=  (/) )  -> 
( F " x
)  C_  A )
Distinct variable groups:    f, g, u, v, w, x, y, z, A    D, f, u, v, y    f, F, g, u, v, x, y, z    R, f, g, u, v, w, x, y, z    v, C    x, H, u, v, f
Allowed substitution hints:    C( x, y, z, w, u, f, g)    D( x, z, w, g)    F( w)    H( y,
z, w, g)

Proof of Theorem zorn2lem5
Dummy variable  s is distinct from all other variables.
StepHypRef Expression
1 zorn2lem.3 . . . . . 6  |-  F  = recs ( ( f  e. 
_V  |->  ( iota_ v  e.  C  A. u  e.  C  -.  u w v ) ) )
21tfr1 7058 . . . . 5  |-  F  Fn  On
3 fnfun 5660 . . . . 5  |-  ( F  Fn  On  ->  Fun  F )
42, 3ax-mp 5 . . . 4  |-  Fun  F
5 fvelima 5900 . . . 4  |-  ( ( Fun  F  /\  s  e.  ( F " x
) )  ->  E. y  e.  x  ( F `  y )  =  s )
64, 5mpan 668 . . 3  |-  ( s  e.  ( F "
x )  ->  E. y  e.  x  ( F `  y )  =  s )
7 nfv 1712 . . . . 5  |-  F/ y ( w  We  A  /\  x  e.  On )
8 nfra1 2835 . . . . 5  |-  F/ y A. y  e.  x  H  =/=  (/)
97, 8nfan 1933 . . . 4  |-  F/ y ( ( w  We  A  /\  x  e.  On )  /\  A. y  e.  x  H  =/=  (/) )
10 nfv 1712 . . . 4  |-  F/ y  s  e.  A
11 df-ral 2809 . . . . . 6  |-  ( A. y  e.  x  H  =/=  (/)  <->  A. y ( y  e.  x  ->  H  =/=  (/) ) )
12 onelon 4892 . . . . . . . . . . . . 13  |-  ( ( x  e.  On  /\  y  e.  x )  ->  y  e.  On )
13 zorn2lem.7 . . . . . . . . . . . . . . . 16  |-  H  =  { z  e.  A  |  A. g  e.  ( F " y ) g R z }
14 ssrab2 3571 . . . . . . . . . . . . . . . 16  |-  { z  e.  A  |  A. g  e.  ( F " y ) g R z }  C_  A
1513, 14eqsstri 3519 . . . . . . . . . . . . . . 15  |-  H  C_  A
16 zorn2lem.4 . . . . . . . . . . . . . . . 16  |-  C  =  { z  e.  A  |  A. g  e.  ran  f  g R z }
171, 16, 13zorn2lem1 8867 . . . . . . . . . . . . . . 15  |-  ( ( y  e.  On  /\  ( w  We  A  /\  H  =/=  (/) ) )  ->  ( F `  y )  e.  H
)
1815, 17sseldi 3487 . . . . . . . . . . . . . 14  |-  ( ( y  e.  On  /\  ( w  We  A  /\  H  =/=  (/) ) )  ->  ( F `  y )  e.  A
)
19 eleq1 2526 . . . . . . . . . . . . . 14  |-  ( ( F `  y )  =  s  ->  (
( F `  y
)  e.  A  <->  s  e.  A ) )
2018, 19syl5ib 219 . . . . . . . . . . . . 13  |-  ( ( F `  y )  =  s  ->  (
( y  e.  On  /\  ( w  We  A  /\  H  =/=  (/) ) )  ->  s  e.  A
) )
2112, 20sylani 652 . . . . . . . . . . . 12  |-  ( ( F `  y )  =  s  ->  (
( ( x  e.  On  /\  y  e.  x )  /\  (
w  We  A  /\  H  =/=  (/) ) )  -> 
s  e.  A ) )
2221com12 31 . . . . . . . . . . 11  |-  ( ( ( x  e.  On  /\  y  e.  x )  /\  ( w  We  A  /\  H  =/=  (/) ) )  ->  (
( F `  y
)  =  s  -> 
s  e.  A ) )
2322exp43 610 . . . . . . . . . 10  |-  ( x  e.  On  ->  (
y  e.  x  -> 
( w  We  A  ->  ( H  =/=  (/)  ->  (
( F `  y
)  =  s  -> 
s  e.  A ) ) ) ) )
2423com3r 79 . . . . . . . . 9  |-  ( w  We  A  ->  (
x  e.  On  ->  ( y  e.  x  -> 
( H  =/=  (/)  ->  (
( F `  y
)  =  s  -> 
s  e.  A ) ) ) ) )
2524imp 427 . . . . . . . 8  |-  ( ( w  We  A  /\  x  e.  On )  ->  ( y  e.  x  ->  ( H  =/=  (/)  ->  (
( F `  y
)  =  s  -> 
s  e.  A ) ) ) )
2625a2d 26 . . . . . . 7  |-  ( ( w  We  A  /\  x  e.  On )  ->  ( ( y  e.  x  ->  H  =/=  (/) )  ->  ( y  e.  x  ->  ( ( F `  y )  =  s  ->  s  e.  A ) ) ) )
2726spsd 1872 . . . . . 6  |-  ( ( w  We  A  /\  x  e.  On )  ->  ( A. y ( y  e.  x  ->  H  =/=  (/) )  ->  (
y  e.  x  -> 
( ( F `  y )  =  s  ->  s  e.  A
) ) ) )
2811, 27syl5bi 217 . . . . 5  |-  ( ( w  We  A  /\  x  e.  On )  ->  ( A. y  e.  x  H  =/=  (/)  ->  (
y  e.  x  -> 
( ( F `  y )  =  s  ->  s  e.  A
) ) ) )
2928imp 427 . . . 4  |-  ( ( ( w  We  A  /\  x  e.  On )  /\  A. y  e.  x  H  =/=  (/) )  -> 
( y  e.  x  ->  ( ( F `  y )  =  s  ->  s  e.  A
) ) )
309, 10, 29rexlimd 2938 . . 3  |-  ( ( ( w  We  A  /\  x  e.  On )  /\  A. y  e.  x  H  =/=  (/) )  -> 
( E. y  e.  x  ( F `  y )  =  s  ->  s  e.  A
) )
316, 30syl5 32 . 2  |-  ( ( ( w  We  A  /\  x  e.  On )  /\  A. y  e.  x  H  =/=  (/) )  -> 
( s  e.  ( F " x )  ->  s  e.  A
) )
3231ssrdv 3495 1  |-  ( ( ( w  We  A  /\  x  e.  On )  /\  A. y  e.  x  H  =/=  (/) )  -> 
( F " x
)  C_  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367   A.wal 1396    = wceq 1398    e. wcel 1823    =/= wne 2649   A.wral 2804   E.wrex 2805   {crab 2808   _Vcvv 3106    C_ wss 3461   (/)c0 3783   class class class wbr 4439    |-> cmpt 4497    We wwe 4826   Oncon0 4867   ran crn 4989   "cima 4991   Fun wfun 5564    Fn wfn 5565   ` cfv 5570   iota_crio 6231  recscrecs 7033
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-recs 7034
This theorem is referenced by:  zorn2lem6  8872  zorn2lem7  8873
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