MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cantnflem1a Unicode version

Theorem cantnflem1a 7387
Description: Lemma for cantnf 7395. (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 7386 . . 3  |-  ( ph  ->  ( X  e.  B  /\  ( F `  X
)  e.  ( G `
 X )  /\  A. w  e.  B  ( X  e.  w  -> 
( F `  w
)  =  ( G `
 w ) ) ) )
109simp1d 967 . 2  |-  ( ph  ->  X  e.  B )
119simp2d 968 . . . 4  |-  ( ph  ->  ( F `  X
)  e.  ( G `
 X ) )
12 ne0i 3461 . . . 4  |-  ( ( F `  X )  e.  ( G `  X )  ->  ( G `  X )  =/=  (/) )
1311, 12syl 15 . . 3  |-  ( ph  ->  ( G `  X
)  =/=  (/) )
14 fvex 5539 . . . 4  |-  ( G `
 X )  e. 
_V
15 dif1o 6499 . . . 4  |-  ( ( G `  X )  e.  ( _V  \  1o )  <->  ( ( G `
 X )  e. 
_V  /\  ( G `  X )  =/=  (/) ) )
1614, 15mpbiran 884 . . 3  |-  ( ( G `  X )  e.  ( _V  \  1o )  <->  ( G `  X )  =/=  (/) )
1713, 16sylibr 203 . 2  |-  ( ph  ->  ( G `  X
)  e.  ( _V 
\  1o ) )
181, 2, 3cantnfs 7367 . . . . 5  |-  ( ph  ->  ( G  e.  S  <->  ( G : B --> A  /\  ( `' G " ( _V 
\  1o ) )  e.  Fin ) ) )
196, 18mpbid 201 . . . 4  |-  ( ph  ->  ( G : B --> A  /\  ( `' G " ( _V  \  1o ) )  e.  Fin ) )
2019simpld 445 . . 3  |-  ( ph  ->  G : B --> A )
21 ffn 5389 . . 3  |-  ( G : B --> A  ->  G  Fn  B )
22 elpreima 5645 . . 3  |-  ( G  Fn  B  ->  ( X  e.  ( `' G " ( _V  \  1o ) )  <->  ( X  e.  B  /\  ( G `  X )  e.  ( _V  \  1o ) ) ) )
2320, 21, 223syl 18 . 2  |-  ( ph  ->  ( X  e.  ( `' G " ( _V 
\  1o ) )  <-> 
( X  e.  B  /\  ( G `  X
)  e.  ( _V 
\  1o ) ) ) )
2410, 17, 23mpbir2and 888 1  |-  ( ph  ->  X  e.  ( `' G " ( _V 
\  1o ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544   {crab 2547   _Vcvv 2788    \ cdif 3149   (/)c0 3455   U.cuni 3827   class class class wbr 4023   {copab 4076   Oncon0 4392   `'ccnv 4688   dom cdm 4689   "cima 4692    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   1oc1o 6472   Fincfn 6863   CNF ccnf 7362
This theorem is referenced by:  cantnflem1b  7388  cantnflem1d  7390  cantnflem1  7391
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-recs 6388  df-rdg 6423  df-seqom 6460  df-1o 6479  df-er 6660  df-map 6774  df-en 6864  df-fin 6867  df-oi 7225  df-cnf 7363
  Copyright terms: Public domain W3C validator