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Theorem zorn2lem5 8681
Description: Lemma for zorn2 8687. (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 6868 . . . . 5  |-  F  Fn  On
3 fnfun 5520 . . . . 5  |-  ( F  Fn  On  ->  Fun  F )
42, 3ax-mp 5 . . . 4  |-  Fun  F
5 fvelima 5755 . . . 4  |-  ( ( Fun  F  /\  s  e.  ( F " x
) )  ->  E. y  e.  x  ( F `  y )  =  s )
64, 5mpan 670 . . 3  |-  ( s  e.  ( F "
x )  ->  E. y  e.  x  ( F `  y )  =  s )
7 nfv 1673 . . . . 5  |-  F/ y ( w  We  A  /\  x  e.  On )
8 nfra1 2778 . . . . 5  |-  F/ y A. y  e.  x  H  =/=  (/)
97, 8nfan 1861 . . . 4  |-  F/ y ( ( w  We  A  /\  x  e.  On )  /\  A. y  e.  x  H  =/=  (/) )
10 nfv 1673 . . . 4  |-  F/ y  s  e.  A
11 df-ral 2732 . . . . . 6  |-  ( A. y  e.  x  H  =/=  (/)  <->  A. y ( y  e.  x  ->  H  =/=  (/) ) )
12 onelon 4756 . . . . . . . . . . . . 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 3449 . . . . . . . . . . . . . . . 16  |-  { z  e.  A  |  A. g  e.  ( F " y ) g R z }  C_  A
1513, 14eqsstri 3398 . . . . . . . . . . . . . . 15  |-  H  C_  A
16 zorn2lem.4 . . . . . . . . . . . . . . . 16  |-  C  =  { z  e.  A  |  A. g  e.  ran  f  g R z }
171, 16, 13zorn2lem1 8677 . . . . . . . . . . . . . . 15  |-  ( ( y  e.  On  /\  ( w  We  A  /\  H  =/=  (/) ) )  ->  ( F `  y )  e.  H
)
1815, 17sseldi 3366 . . . . . . . . . . . . . 14  |-  ( ( y  e.  On  /\  ( w  We  A  /\  H  =/=  (/) ) )  ->  ( F `  y )  e.  A
)
19 eleq1 2503 . . . . . . . . . . . . . 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 654 . . . . . . . . . . . 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 612 . . . . . . . . . 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 429 . . . . . . . 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 1802 . . . . . 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 429 . . . 4  |-  ( ( ( w  We  A  /\  x  e.  On )  /\  A. y  e.  x  H  =/=  (/) )  -> 
( y  e.  x  ->  ( ( F `  y )  =  s  ->  s  e.  A
) ) )
309, 10, 29rexlimd 2850 . . 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 3374 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 369   A.wal 1367    = wceq 1369    e. wcel 1756    =/= wne 2618   A.wral 2727   E.wrex 2728   {crab 2731   _Vcvv 2984    C_ wss 3340   (/)c0 3649   class class class wbr 4304    e. cmpt 4362    We wwe 4690   Oncon0 4731   ran crn 4853   "cima 4855   Fun wfun 5424    Fn wfn 5425   ` cfv 5430   iota_crio 6063  recscrecs 6843
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-recs 6844
This theorem is referenced by:  zorn2lem6  8682  zorn2lem7  8683
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