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Theorem zorn2lem2 8666
Description: Lemma for zorn2 8675. (Contributed by NM, 3-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 }
Assertion
Ref Expression
zorn2lem2  |-  ( ( x  e.  On  /\  ( w  We  A  /\  D  =/=  (/) ) )  ->  ( y  e.  x  ->  ( F `  y ) R ( F `  x ) ) )
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
Allowed substitution hints:    C( x, y, z, w, u, f, g)    D( x, z, w, g)    F( w)

Proof of Theorem zorn2lem2
StepHypRef Expression
1 zorn2lem.3 . . . 4  |-  F  = recs ( ( f  e. 
_V  |->  ( iota_ v  e.  C  A. u  e.  C  -.  u w v ) ) )
2 zorn2lem.4 . . . 4  |-  C  =  { z  e.  A  |  A. g  e.  ran  f  g R z }
3 zorn2lem.5 . . . 4  |-  D  =  { z  e.  A  |  A. g  e.  ( F " x ) g R z }
41, 2, 3zorn2lem1 8665 . . 3  |-  ( ( x  e.  On  /\  ( w  We  A  /\  D  =/=  (/) ) )  ->  ( F `  x )  e.  D
)
5 breq2 4296 . . . . . 6  |-  ( z  =  ( F `  x )  ->  (
g R z  <->  g R
( F `  x
) ) )
65ralbidv 2735 . . . . 5  |-  ( z  =  ( F `  x )  ->  ( A. g  e.  ( F " x ) g R z  <->  A. g  e.  ( F " x
) g R ( F `  x ) ) )
76, 3elrab2 3119 . . . 4  |-  ( ( F `  x )  e.  D  <->  ( ( F `  x )  e.  A  /\  A. g  e.  ( F " x
) g R ( F `  x ) ) )
87simprbi 464 . . 3  |-  ( ( F `  x )  e.  D  ->  A. g  e.  ( F " x
) g R ( F `  x ) )
94, 8syl 16 . 2  |-  ( ( x  e.  On  /\  ( w  We  A  /\  D  =/=  (/) ) )  ->  A. g  e.  ( F " x ) g R ( F `
 x ) )
101tfr1 6856 . . . 4  |-  F  Fn  On
11 onss 6402 . . . 4  |-  ( x  e.  On  ->  x  C_  On )
12 fnfvima 5955 . . . . 5  |-  ( ( F  Fn  On  /\  x  C_  On  /\  y  e.  x )  ->  ( F `  y )  e.  ( F " x
) )
13123expia 1189 . . . 4  |-  ( ( F  Fn  On  /\  x  C_  On )  -> 
( y  e.  x  ->  ( F `  y
)  e.  ( F
" x ) ) )
1410, 11, 13sylancr 663 . . 3  |-  ( x  e.  On  ->  (
y  e.  x  -> 
( F `  y
)  e.  ( F
" x ) ) )
1514adantr 465 . 2  |-  ( ( x  e.  On  /\  ( w  We  A  /\  D  =/=  (/) ) )  ->  ( y  e.  x  ->  ( F `  y )  e.  ( F " x ) ) )
16 breq1 4295 . . 3  |-  ( g  =  ( F `  y )  ->  (
g R ( F `
 x )  <->  ( F `  y ) R ( F `  x ) ) )
1716rspccv 3070 . 2  |-  ( A. g  e.  ( F " x ) g R ( F `  x
)  ->  ( ( F `  y )  e.  ( F " x
)  ->  ( F `  y ) R ( F `  x ) ) )
189, 15, 17sylsyld 56 1  |-  ( ( x  e.  On  /\  ( w  We  A  /\  D  =/=  (/) ) )  ->  ( y  e.  x  ->  ( F `  y ) R ( F `  x ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2606   A.wral 2715   {crab 2719   _Vcvv 2972    C_ wss 3328   (/)c0 3637   class class class wbr 4292    e. cmpt 4350    We wwe 4678   Oncon0 4719   ran crn 4841   "cima 4843    Fn wfn 5413   ` cfv 5418   iota_crio 6051  recscrecs 6831
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 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
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 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-recs 6832
This theorem is referenced by:  zorn2lem3  8667  zorn2lem6  8670
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