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Theorem cantnflem1a 7996
Description: Lemma for cantnf 8004. (Contributed by Mario Carneiro, 4-Jun-2015.) (Revised by AV, 2-Jul-2019.)
Hypotheses
Ref Expression
cantnfs.s  |-  S  =  dom  ( A CNF  B
)
cantnfs.a  |-  ( ph  ->  A  e.  On )
cantnfs.b  |-  ( 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.f  |-  ( ph  ->  F  e.  S )
oemapval.g  |-  ( ph  ->  G  e.  S )
oemapvali.r  |-  ( ph  ->  F T G )
oemapvali.x  |-  X  = 
U. { c  e.  B  |  ( F `
 c )  e.  ( G `  c
) }
Assertion
Ref Expression
cantnflem1a  |-  ( ph  ->  X  e.  ( G supp  (/) ) )
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.s . . . 4  |-  S  =  dom  ( A CNF  B
)
2 cantnfs.a . . . 4  |-  ( ph  ->  A  e.  On )
3 cantnfs.b . . . 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.f . . . 4  |-  ( ph  ->  F  e.  S )
6 oemapval.g . . . 4  |-  ( ph  ->  G  e.  S )
7 oemapvali.r . . . 4  |-  ( ph  ->  F T G )
8 oemapvali.x . . . 4  |-  X  = 
U. { c  e.  B  |  ( F `
 c )  e.  ( G `  c
) }
91, 2, 3, 4, 5, 6, 7, 8oemapvali 7995 . . 3  |-  ( ph  ->  ( X  e.  B  /\  ( F `  X
)  e.  ( G `
 X )  /\  A. w  e.  B  ( X  e.  w  -> 
( F `  w
)  =  ( G `
 w ) ) ) )
109simp1d 1000 . 2  |-  ( ph  ->  X  e.  B )
119simp2d 1001 . . 3  |-  ( ph  ->  ( F `  X
)  e.  ( G `
 X ) )
12 ne0i 3743 . . 3  |-  ( ( F `  X )  e.  ( G `  X )  ->  ( G `  X )  =/=  (/) )
1311, 12syl 16 . 2  |-  ( ph  ->  ( G `  X
)  =/=  (/) )
141, 2, 3cantnfs 7977 . . . . . 6  |-  ( ph  ->  ( G  e.  S  <->  ( G : B --> A  /\  G finSupp 
(/) ) ) )
156, 14mpbid 210 . . . . 5  |-  ( ph  ->  ( G : B --> A  /\  G finSupp  (/) ) )
1615simpld 459 . . . 4  |-  ( ph  ->  G : B --> A )
17 ffn 5659 . . . 4  |-  ( G : B --> A  ->  G  Fn  B )
1816, 17syl 16 . . 3  |-  ( ph  ->  G  Fn  B )
19 0ex 4522 . . . 4  |-  (/)  e.  _V
2019a1i 11 . . 3  |-  ( ph  -> 
(/)  e.  _V )
21 elsuppfn 6800 . . 3  |-  ( ( G  Fn  B  /\  B  e.  On  /\  (/)  e.  _V )  ->  ( X  e.  ( G supp  (/) )  <->  ( X  e.  B  /\  ( G `  X )  =/=  (/) ) ) )
2218, 3, 20, 21syl3anc 1219 . 2  |-  ( ph  ->  ( X  e.  ( G supp  (/) )  <->  ( X  e.  B  /\  ( G `  X )  =/=  (/) ) ) )
2310, 13, 22mpbir2and 913 1  |-  ( ph  ->  X  e.  ( G supp  (/) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2644   A.wral 2795   E.wrex 2796   {crab 2799   _Vcvv 3070   (/)c0 3737   U.cuni 4191   class class class wbr 4392   {copab 4449   Oncon0 4819   dom cdm 4940    Fn wfn 5513   -->wf 5514   ` cfv 5518  (class class class)co 6192   supp csupp 6792   finSupp cfsupp 7723   CNF ccnf 7970
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-om 6579  df-supp 6793  df-recs 6934  df-rdg 6968  df-seqom 7005  df-1o 7022  df-er 7203  df-map 7318  df-en 7413  df-fin 7416  df-fsupp 7724  df-cnf 7971
This theorem is referenced by:  cantnflem1b  7997  cantnflem1d  7999  cantnflem1  8000
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