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Theorem zorn2lem5 8879
Description: Lemma for zorn2 8885. (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 7066 . . . . 5  |-  F  Fn  On
3 fnfun 5677 . . . . 5  |-  ( F  Fn  On  ->  Fun  F )
42, 3ax-mp 5 . . . 4  |-  Fun  F
5 fvelima 5918 . . . 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 1683 . . . . 5  |-  F/ y ( w  We  A  /\  x  e.  On )
8 nfra1 2845 . . . . 5  |-  F/ y A. y  e.  x  H  =/=  (/)
97, 8nfan 1875 . . . 4  |-  F/ y ( ( w  We  A  /\  x  e.  On )  /\  A. y  e.  x  H  =/=  (/) )
10 nfv 1683 . . . 4  |-  F/ y  s  e.  A
11 df-ral 2819 . . . . . 6  |-  ( A. y  e.  x  H  =/=  (/)  <->  A. y ( y  e.  x  ->  H  =/=  (/) ) )
12 onelon 4903 . . . . . . . . . . . . 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 3585 . . . . . . . . . . . . . . . 16  |-  { z  e.  A  |  A. g  e.  ( F " y ) g R z }  C_  A
1513, 14eqsstri 3534 . . . . . . . . . . . . . . 15  |-  H  C_  A
16 zorn2lem.4 . . . . . . . . . . . . . . . 16  |-  C  =  { z  e.  A  |  A. g  e.  ran  f  g R z }
171, 16, 13zorn2lem1 8875 . . . . . . . . . . . . . . 15  |-  ( ( y  e.  On  /\  ( w  We  A  /\  H  =/=  (/) ) )  ->  ( F `  y )  e.  H
)
1815, 17sseldi 3502 . . . . . . . . . . . . . 14  |-  ( ( y  e.  On  /\  ( w  We  A  /\  H  =/=  (/) ) )  ->  ( F `  y )  e.  A
)
19 eleq1 2539 . . . . . . . . . . . . . 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 1816 . . . . . 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 2947 . . 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 3510 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 1377    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   E.wrex 2815   {crab 2818   _Vcvv 3113    C_ wss 3476   (/)c0 3785   class class class wbr 4447    |-> cmpt 4505    We wwe 4837   Oncon0 4878   ran crn 5000   "cima 5002   Fun wfun 5581    Fn wfn 5582   ` cfv 5587   iota_crio 6243  recscrecs 7041
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-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575
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-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  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-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-riota 6244  df-recs 7042
This theorem is referenced by:  zorn2lem6  8880  zorn2lem7  8881
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