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Theorem zorn2lem5 8385
Description: Lemma for zorn2 8391. (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 6661 . . . . 5  |-  F  Fn  On
3 fnfun 5545 . . . . 5  |-  ( F  Fn  On  ->  Fun  F )
42, 3ax-mp 5 . . . 4  |-  Fun  F
5 fvelima 5781 . . . 4  |-  ( ( Fun  F  /\  s  e.  ( F " x
) )  ->  E. y  e.  x  ( F `  y )  =  s )
64, 5mpan 653 . . 3  |-  ( s  e.  ( F "
x )  ->  E. y  e.  x  ( F `  y )  =  s )
7 nfv 1630 . . . . 5  |-  F/ y ( w  We  A  /\  x  e.  On )
8 nfra1 2758 . . . . 5  |-  F/ y A. y  e.  x  H  =/=  (/)
97, 8nfan 1847 . . . 4  |-  F/ y ( ( w  We  A  /\  x  e.  On )  /\  A. y  e.  x  H  =/=  (/) )
10 nfv 1630 . . . 4  |-  F/ y  s  e.  A
11 df-ral 2712 . . . . . 6  |-  ( A. y  e.  x  H  =/=  (/)  <->  A. y ( y  e.  x  ->  H  =/=  (/) ) )
12 onelon 4609 . . . . . . . . . . . . 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 3430 . . . . . . . . . . . . . . . 16  |-  { z  e.  A  |  A. g  e.  ( F " y ) g R z }  C_  A
1513, 14eqsstri 3380 . . . . . . . . . . . . . . 15  |-  H  C_  A
16 zorn2lem.4 . . . . . . . . . . . . . . . 16  |-  C  =  { z  e.  A  |  A. g  e.  ran  f  g R z }
171, 16, 13zorn2lem1 8381 . . . . . . . . . . . . . . 15  |-  ( ( y  e.  On  /\  ( w  We  A  /\  H  =/=  (/) ) )  ->  ( F `  y )  e.  H
)
1815, 17sseldi 3348 . . . . . . . . . . . . . 14  |-  ( ( y  e.  On  /\  ( w  We  A  /\  H  =/=  (/) ) )  ->  ( F `  y )  e.  A
)
19 eleq1 2498 . . . . . . . . . . . . . 14  |-  ( ( F `  y )  =  s  ->  (
( F `  y
)  e.  A  <->  s  e.  A ) )
2018, 19syl5ib 212 . . . . . . . . . . . . 13  |-  ( ( F `  y )  =  s  ->  (
( y  e.  On  /\  ( w  We  A  /\  H  =/=  (/) ) )  ->  s  e.  A
) )
2112, 20sylani 637 . . . . . . . . . . . 12  |-  ( ( F `  y )  =  s  ->  (
( ( x  e.  On  /\  y  e.  x )  /\  (
w  We  A  /\  H  =/=  (/) ) )  -> 
s  e.  A ) )
2221com12 30 . . . . . . . . . . 11  |-  ( ( ( x  e.  On  /\  y  e.  x )  /\  ( w  We  A  /\  H  =/=  (/) ) )  ->  (
( F `  y
)  =  s  -> 
s  e.  A ) )
2322exp43 597 . . . . . . . . . 10  |-  ( x  e.  On  ->  (
y  e.  x  -> 
( w  We  A  ->  ( H  =/=  (/)  ->  (
( F `  y
)  =  s  -> 
s  e.  A ) ) ) ) )
2423com3r 76 . . . . . . . . 9  |-  ( w  We  A  ->  (
x  e.  On  ->  ( y  e.  x  -> 
( H  =/=  (/)  ->  (
( F `  y
)  =  s  -> 
s  e.  A ) ) ) ) )
2524imp 420 . . . . . . . 8  |-  ( ( w  We  A  /\  x  e.  On )  ->  ( y  e.  x  ->  ( H  =/=  (/)  ->  (
( F `  y
)  =  s  -> 
s  e.  A ) ) ) )
2625a2d 25 . . . . . . 7  |-  ( ( w  We  A  /\  x  e.  On )  ->  ( ( y  e.  x  ->  H  =/=  (/) )  ->  ( y  e.  x  ->  ( ( F `  y )  =  s  ->  s  e.  A ) ) ) )
2726spsd 1772 . . . . . 6  |-  ( ( w  We  A  /\  x  e.  On )  ->  ( A. y ( y  e.  x  ->  H  =/=  (/) )  ->  (
y  e.  x  -> 
( ( F `  y )  =  s  ->  s  e.  A
) ) ) )
2811, 27syl5bi 210 . . . . 5  |-  ( ( w  We  A  /\  x  e.  On )  ->  ( A. y  e.  x  H  =/=  (/)  ->  (
y  e.  x  -> 
( ( F `  y )  =  s  ->  s  e.  A
) ) ) )
2928imp 420 . . . 4  |-  ( ( ( w  We  A  /\  x  e.  On )  /\  A. y  e.  x  H  =/=  (/) )  -> 
( y  e.  x  ->  ( ( F `  y )  =  s  ->  s  e.  A
) ) )
309, 10, 29rexlimd 2829 . . 3  |-  ( ( ( w  We  A  /\  x  e.  On )  /\  A. y  e.  x  H  =/=  (/) )  -> 
( E. y  e.  x  ( F `  y )  =  s  ->  s  e.  A
) )
316, 30syl5 31 . 2  |-  ( ( ( w  We  A  /\  x  e.  On )  /\  A. y  e.  x  H  =/=  (/) )  -> 
( s  e.  ( F " x )  ->  s  e.  A
) )
3231ssrdv 3356 1  |-  ( ( ( w  We  A  /\  x  e.  On )  /\  A. y  e.  x  H  =/=  (/) )  -> 
( F " x
)  C_  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360   A.wal 1550    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707   E.wrex 2708   {crab 2711   _Vcvv 2958    C_ wss 3322   (/)c0 3630   class class class wbr 4215    e. cmpt 4269    We wwe 4543   Oncon0 4584   ran crn 4882   "cima 4884   Fun wfun 5451    Fn wfn 5452   ` cfv 5457   iota_crio 6545  recscrecs 6635
This theorem is referenced by:  zorn2lem6  8386  zorn2lem7  8387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-suc 4590  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-riota 6552  df-recs 6636
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