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Theorem cantnflem1a 7574
Description: Lemma for cantnf 7582. (Contributed by Mario Carneiro, 4-Jun-2015.)
Hypotheses
Ref Expression
cantnfs.1  |-  S  =  dom  ( A CNF  B
)
cantnfs.2  |-  ( ph  ->  A  e.  On )
cantnfs.3  |-  ( ph  ->  B  e.  On )
oemapval.t  |-  T  =  { <. x ,  y
>.  |  E. z  e.  B  ( (
x `  z )  e.  ( y `  z
)  /\  A. w  e.  B  ( z  e.  w  ->  ( x `
 w )  =  ( y `  w
) ) ) }
oemapval.3  |-  ( ph  ->  F  e.  S )
oemapval.4  |-  ( ph  ->  G  e.  S )
oemapvali.5  |-  ( ph  ->  F T G )
oemapvali.6  |-  X  = 
U. { c  e.  B  |  ( F `
 c )  e.  ( G `  c
) }
Assertion
Ref Expression
cantnflem1a  |-  ( 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 cantnflem1a
StepHypRef Expression
1 cantnfs.1 . . . 4  |-  S  =  dom  ( A CNF  B
)
2 cantnfs.2 . . . 4  |-  ( ph  ->  A  e.  On )
3 cantnfs.3 . . . 4  |-  ( ph  ->  B  e.  On )
4 oemapval.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 oemapval.3 . . . 4  |-  ( ph  ->  F  e.  S )
6 oemapval.4 . . . 4  |-  ( ph  ->  G  e.  S )
7 oemapvali.5 . . . 4  |-  ( ph  ->  F T G )
8 oemapvali.6 . . . 4  |-  X  = 
U. { c  e.  B  |  ( F `
 c )  e.  ( G `  c
) }
91, 2, 3, 4, 5, 6, 7, 8oemapvali 7573 . . 3  |-  ( ph  ->  ( X  e.  B  /\  ( F `  X
)  e.  ( G `
 X )  /\  A. w  e.  B  ( X  e.  w  -> 
( F `  w
)  =  ( G `
 w ) ) ) )
109simp1d 969 . 2  |-  ( ph  ->  X  e.  B )
119simp2d 970 . . . 4  |-  ( ph  ->  ( F `  X
)  e.  ( G `
 X ) )
12 ne0i 3577 . . . 4  |-  ( ( F `  X )  e.  ( G `  X )  ->  ( G `  X )  =/=  (/) )
1311, 12syl 16 . . 3  |-  ( ph  ->  ( G `  X
)  =/=  (/) )
14 fvex 5682 . . . 4  |-  ( G `
 X )  e. 
_V
15 dif1o 6680 . . . 4  |-  ( ( G `  X )  e.  ( _V  \  1o )  <->  ( ( G `
 X )  e. 
_V  /\  ( G `  X )  =/=  (/) ) )
1614, 15mpbiran 885 . . 3  |-  ( ( G `  X )  e.  ( _V  \  1o )  <->  ( G `  X )  =/=  (/) )
1713, 16sylibr 204 . 2  |-  ( ph  ->  ( G `  X
)  e.  ( _V 
\  1o ) )
181, 2, 3cantnfs 7554 . . . . 5  |-  ( ph  ->  ( G  e.  S  <->  ( G : B --> A  /\  ( `' G " ( _V 
\  1o ) )  e.  Fin ) ) )
196, 18mpbid 202 . . . 4  |-  ( ph  ->  ( G : B --> A  /\  ( `' G " ( _V  \  1o ) )  e.  Fin ) )
2019simpld 446 . . 3  |-  ( ph  ->  G : B --> A )
21 ffn 5531 . . 3  |-  ( G : B --> A  ->  G  Fn  B )
22 elpreima 5789 . . 3  |-  ( G  Fn  B  ->  ( X  e.  ( `' G " ( _V  \  1o ) )  <->  ( X  e.  B  /\  ( G `  X )  e.  ( _V  \  1o ) ) ) )
2320, 21, 223syl 19 . 2  |-  ( ph  ->  ( X  e.  ( `' G " ( _V 
\  1o ) )  <-> 
( X  e.  B  /\  ( G `  X
)  e.  ( _V 
\  1o ) ) ) )
2410, 17, 23mpbir2and 889 1  |-  ( ph  ->  X  e.  ( `' G " ( _V 
\  1o ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2550   A.wral 2649   E.wrex 2650   {crab 2653   _Vcvv 2899    \ cdif 3260   (/)c0 3571   U.cuni 3957   class class class wbr 4153   {copab 4206   Oncon0 4522   `'ccnv 4817   dom cdm 4818   "cima 4821    Fn wfn 5389   -->wf 5390   ` cfv 5394  (class class class)co 6020   1oc1o 6653   Fincfn 7045   CNF ccnf 7549
This theorem is referenced by:  cantnflem1b  7575  cantnflem1d  7577  cantnflem1  7578
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-se 4483  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-isom 5403  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-riota 6485  df-recs 6569  df-rdg 6604  df-seqom 6641  df-1o 6660  df-er 6841  df-map 6956  df-en 7046  df-fin 7049  df-oi 7412  df-cnf 7550
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