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Theorem cantnflem1aOLD 8118
Description: Lemma for cantnfOLD 8125. (Contributed by Mario Carneiro, 4-Jun-2015.) Obsolete version of cantnflem1a 8095 as of 2-Jul-2019. (New usage is discouraged.)
Hypotheses
Ref Expression
cantnfsOLD.1  |-  S  =  dom  ( A CNF  B
)
cantnfsOLD.2  |-  ( ph  ->  A  e.  On )
cantnfsOLD.3  |-  ( ph  ->  B  e.  On )
oemapvalOLD.t  |-  T  =  { <. x ,  y
>.  |  E. z  e.  B  ( (
x `  z )  e.  ( y `  z
)  /\  A. w  e.  B  ( z  e.  w  ->  ( x `
 w )  =  ( y `  w
) ) ) }
oemapvalOLD.3  |-  ( ph  ->  F  e.  S )
oemapvalOLD.4  |-  ( ph  ->  G  e.  S )
oemapvalOLD.5  |-  ( ph  ->  F T G )
oemapvalOLD.6  |-  X  = 
U. { c  e.  B  |  ( F `
 c )  e.  ( G `  c
) }
Assertion
Ref Expression
cantnflem1aOLD  |-  ( ph  ->  X  e.  ( `' G " ( _V 
\  1o ) ) )
Distinct variable groups:    w, c, x, y, z, B    A, c, w, x, y, z    T, c    w, F, x, y, z    S, c, x, y, z    G, c, w, x, y, z    ph, x, y, z    w, X, x, y, z    F, c    ph, c
Allowed substitution hints:    ph( w)    S( w)    T( x, y, z, w)    X( c)

Proof of Theorem cantnflem1aOLD
StepHypRef Expression
1 cantnfsOLD.1 . . . 4  |-  S  =  dom  ( A CNF  B
)
2 cantnfsOLD.2 . . . 4  |-  ( ph  ->  A  e.  On )
3 cantnfsOLD.3 . . . 4  |-  ( ph  ->  B  e.  On )
4 oemapvalOLD.t . . . 4  |-  T  =  { <. x ,  y
>.  |  E. z  e.  B  ( (
x `  z )  e.  ( y `  z
)  /\  A. w  e.  B  ( z  e.  w  ->  ( x `
 w )  =  ( y `  w
) ) ) }
5 oemapvalOLD.3 . . . 4  |-  ( ph  ->  F  e.  S )
6 oemapvalOLD.4 . . . 4  |-  ( ph  ->  G  e.  S )
7 oemapvalOLD.5 . . . 4  |-  ( ph  ->  F T G )
8 oemapvalOLD.6 . . . 4  |-  X  = 
U. { c  e.  B  |  ( F `
 c )  e.  ( G `  c
) }
91, 2, 3, 4, 5, 6, 7, 8oemapvali 8094 . . 3  |-  ( ph  ->  ( X  e.  B  /\  ( F `  X
)  e.  ( G `
 X )  /\  A. w  e.  B  ( X  e.  w  -> 
( F `  w
)  =  ( G `
 w ) ) ) )
109simp1d 1003 . 2  |-  ( ph  ->  X  e.  B )
119simp2d 1004 . . . 4  |-  ( ph  ->  ( F `  X
)  e.  ( G `
 X ) )
12 ne0i 3786 . . . 4  |-  ( ( F `  X )  e.  ( G `  X )  ->  ( G `  X )  =/=  (/) )
1311, 12syl 16 . . 3  |-  ( ph  ->  ( G `  X
)  =/=  (/) )
14 fvex 5869 . . . 4  |-  ( G `
 X )  e. 
_V
15 dif1o 7142 . . . 4  |-  ( ( G `  X )  e.  ( _V  \  1o )  <->  ( ( G `
 X )  e. 
_V  /\  ( G `  X )  =/=  (/) ) )
1614, 15mpbiran 911 . . 3  |-  ( ( G `  X )  e.  ( _V  \  1o )  <->  ( G `  X )  =/=  (/) )
1713, 16sylibr 212 . 2  |-  ( ph  ->  ( G `  X
)  e.  ( _V 
\  1o ) )
181, 2, 3cantnfsOLD 8106 . . . . 5  |-  ( ph  ->  ( G  e.  S  <->  ( G : B --> A  /\  ( `' G " ( _V 
\  1o ) )  e.  Fin ) ) )
196, 18mpbid 210 . . . 4  |-  ( ph  ->  ( G : B --> A  /\  ( `' G " ( _V  \  1o ) )  e.  Fin ) )
2019simpld 459 . . 3  |-  ( ph  ->  G : B --> A )
21 ffn 5724 . . 3  |-  ( G : B --> A  ->  G  Fn  B )
22 elpreima 5994 . . 3  |-  ( G  Fn  B  ->  ( X  e.  ( `' G " ( _V  \  1o ) )  <->  ( X  e.  B  /\  ( G `  X )  e.  ( _V  \  1o ) ) ) )
2320, 21, 223syl 20 . 2  |-  ( ph  ->  ( X  e.  ( `' G " ( _V 
\  1o ) )  <-> 
( X  e.  B  /\  ( G `  X
)  e.  ( _V 
\  1o ) ) ) )
2410, 17, 23mpbir2and 915 1  |-  ( ph  ->  X  e.  ( `' G " ( _V 
\  1o ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1374    e. wcel 1762    =/= wne 2657   A.wral 2809   E.wrex 2810   {crab 2813   _Vcvv 3108    \ cdif 3468   (/)c0 3780   U.cuni 4240   class class class wbr 4442   {copab 4499   Oncon0 4873   `'ccnv 4993   dom cdm 4994   "cima 4997    Fn wfn 5576   -->wf 5577   ` cfv 5581  (class class class)co 6277   1oc1o 7115   Fincfn 7508   CNF ccnf 8069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-se 4834  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-1st 6776  df-2nd 6777  df-supp 6894  df-recs 7034  df-rdg 7068  df-seqom 7105  df-1o 7122  df-er 7303  df-map 7414  df-en 7509  df-fin 7512  df-fsupp 7821  df-oi 7926  df-cnf 8070
This theorem is referenced by:  cantnflem1bOLD  8119  cantnflem1dOLD  8121  cantnflem1OLD  8122
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