MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cantnflem1aOLD Structured version   Unicode version

Theorem cantnflem1aOLD 8144
Description: Lemma for cantnfOLD 8151. (Contributed by Mario Carneiro, 4-Jun-2015.) Obsolete version of cantnflem1a 8121 as of 2-Jul-2019. (New usage is discouraged.) (Proof modification 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 8120 . . 3  |-  ( ph  ->  ( X  e.  B  /\  ( F `  X
)  e.  ( G `
 X )  /\  A. w  e.  B  ( X  e.  w  -> 
( F `  w
)  =  ( G `
 w ) ) ) )
109simp1d 1008 . 2  |-  ( ph  ->  X  e.  B )
119simp2d 1009 . . . 4  |-  ( ph  ->  ( F `  X
)  e.  ( G `
 X ) )
12 ne0i 3799 . . . 4  |-  ( ( F `  X )  e.  ( G `  X )  ->  ( G `  X )  =/=  (/) )
1311, 12syl 16 . . 3  |-  ( ph  ->  ( G `  X
)  =/=  (/) )
14 fvex 5882 . . . 4  |-  ( G `
 X )  e. 
_V
15 dif1o 7168 . . . 4  |-  ( ( G `  X )  e.  ( _V  \  1o )  <->  ( ( G `
 X )  e. 
_V  /\  ( G `  X )  =/=  (/) ) )
1614, 15mpbiran 918 . . 3  |-  ( ( G `  X )  e.  ( _V  \  1o )  <->  ( G `  X )  =/=  (/) )
1713, 16sylibr 212 . 2  |-  ( ph  ->  ( G `  X
)  e.  ( _V 
\  1o ) )
181, 2, 3cantnfsOLD 8132 . . . . 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 5737 . . 3  |-  ( G : B --> A  ->  G  Fn  B )
22 elpreima 6008 . . 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 922 1  |-  ( ph  ->  X  e.  ( `' G " ( _V 
\  1o ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819    =/= wne 2652   A.wral 2807   E.wrex 2808   {crab 2811   _Vcvv 3109    \ cdif 3468   (/)c0 3793   U.cuni 4251   class class class wbr 4456   {copab 4514   Oncon0 4887   `'ccnv 5007   dom cdm 5008   "cima 5011    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6296   1oc1o 7141   Fincfn 7535   CNF ccnf 8095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-recs 7060  df-rdg 7094  df-seqom 7131  df-1o 7148  df-er 7329  df-map 7440  df-en 7536  df-fin 7539  df-fsupp 7848  df-oi 7953  df-cnf 8096
This theorem is referenced by:  cantnflem1bOLD  8145  cantnflem1dOLD  8147  cantnflem1OLD  8148
  Copyright terms: Public domain W3C validator