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Theorem zorn2lem3 8929
Description: Lemma for zorn2 8937. (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
zorn2lem3  |-  ( ( R  Po  A  /\  ( x  e.  On  /\  ( w  We  A  /\  D  =/=  (/) ) ) )  ->  ( y  e.  x  ->  -.  ( F `  x )  =  ( F `  y ) ) )
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 zorn2lem3
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, 3zorn2lem2 8928 . . 3  |-  ( ( x  e.  On  /\  ( w  We  A  /\  D  =/=  (/) ) )  ->  ( y  e.  x  ->  ( F `  y ) R ( F `  x ) ) )
54adantl 467 . 2  |-  ( ( R  Po  A  /\  ( x  e.  On  /\  ( w  We  A  /\  D  =/=  (/) ) ) )  ->  ( y  e.  x  ->  ( F `
 y ) R ( F `  x
) ) )
6 ssrab2 3546 . . . . 5  |-  { z  e.  A  |  A. g  e.  ( F " x ) g R z }  C_  A
73, 6eqsstri 3494 . . . 4  |-  D  C_  A
81, 2, 3zorn2lem1 8927 . . . 4  |-  ( ( x  e.  On  /\  ( w  We  A  /\  D  =/=  (/) ) )  ->  ( F `  x )  e.  D
)
97, 8sseldi 3462 . . 3  |-  ( ( x  e.  On  /\  ( w  We  A  /\  D  =/=  (/) ) )  ->  ( F `  x )  e.  A
)
10 breq1 4423 . . . . . 6  |-  ( ( F `  x )  =  ( F `  y )  ->  (
( F `  x
) R ( F `
 x )  <->  ( F `  y ) R ( F `  x ) ) )
1110biimprcd 228 . . . . 5  |-  ( ( F `  y ) R ( F `  x )  ->  (
( F `  x
)  =  ( F `
 y )  -> 
( F `  x
) R ( F `
 x ) ) )
12 poirr 4782 . . . . 5  |-  ( ( R  Po  A  /\  ( F `  x )  e.  A )  ->  -.  ( F `  x
) R ( F `
 x ) )
1311, 12nsyli 146 . . . 4  |-  ( ( F `  y ) R ( F `  x )  ->  (
( R  Po  A  /\  ( F `  x
)  e.  A )  ->  -.  ( F `  x )  =  ( F `  y ) ) )
1413com12 32 . . 3  |-  ( ( R  Po  A  /\  ( F `  x )  e.  A )  -> 
( ( F `  y ) R ( F `  x )  ->  -.  ( F `  x )  =  ( F `  y ) ) )
159, 14sylan2 476 . 2  |-  ( ( R  Po  A  /\  ( x  e.  On  /\  ( w  We  A  /\  D  =/=  (/) ) ) )  ->  ( ( F `  y ) R ( F `  x )  ->  -.  ( F `  x )  =  ( F `  y ) ) )
165, 15syld 45 1  |-  ( ( R  Po  A  /\  ( x  e.  On  /\  ( w  We  A  /\  D  =/=  (/) ) ) )  ->  ( y  e.  x  ->  -.  ( F `  x )  =  ( F `  y ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1868    =/= wne 2618   A.wral 2775   {crab 2779   _Vcvv 3081   (/)c0 3761   class class class wbr 4420    |-> cmpt 4479    Po wpo 4769    We wwe 4808   ran crn 4851   "cima 4853   Oncon0 5439   ` cfv 5598   iota_crio 6263  recscrecs 7094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4761  df-id 4765  df-po 4771  df-so 4772  df-fr 4809  df-we 4811  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-pred 5396  df-ord 5442  df-on 5443  df-suc 5445  df-iota 5562  df-fun 5600  df-fn 5601  df-f 5602  df-f1 5603  df-fo 5604  df-f1o 5605  df-fv 5606  df-riota 6264  df-wrecs 7033  df-recs 7095
This theorem is referenced by:  zorn2lem4  8930
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