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Theorem cantnflem1a 8095
Description: Lemma for cantnf 8103. (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 8094 . . 3  |-  ( ph  ->  ( X  e.  B  /\  ( F `  X
)  e.  ( G `
 X )  /\  A. w  e.  B  ( X  e.  w  -> 
( F `  w
)  =  ( G `
 w ) ) ) )
109simp1d 1006 . 2  |-  ( ph  ->  X  e.  B )
119simp2d 1007 . . 3  |-  ( ph  ->  ( F `  X
)  e.  ( G `
 X ) )
12 ne0i 3789 . . 3  |-  ( ( F `  X )  e.  ( G `  X )  ->  ( G `  X )  =/=  (/) )
1311, 12syl 16 . 2  |-  ( ph  ->  ( G `  X
)  =/=  (/) )
141, 2, 3cantnfs 8076 . . . . . 6  |-  ( ph  ->  ( G  e.  S  <->  ( G : B --> A  /\  G finSupp 
(/) ) ) )
156, 14mpbid 210 . . . . 5  |-  ( ph  ->  ( G : B --> A  /\  G finSupp  (/) ) )
1615simpld 457 . . . 4  |-  ( ph  ->  G : B --> A )
17 ffn 5713 . . . 4  |-  ( G : B --> A  ->  G  Fn  B )
1816, 17syl 16 . . 3  |-  ( ph  ->  G  Fn  B )
19 0ex 4569 . . . 4  |-  (/)  e.  _V
2019a1i 11 . . 3  |-  ( ph  -> 
(/)  e.  _V )
21 elsuppfn 6899 . . 3  |-  ( ( G  Fn  B  /\  B  e.  On  /\  (/)  e.  _V )  ->  ( X  e.  ( G supp  (/) )  <->  ( X  e.  B  /\  ( G `  X )  =/=  (/) ) ) )
2218, 3, 20, 21syl3anc 1226 . 2  |-  ( ph  ->  ( X  e.  ( G supp  (/) )  <->  ( X  e.  B  /\  ( G `  X )  =/=  (/) ) ) )
2310, 13, 22mpbir2and 920 1  |-  ( ph  ->  X  e.  ( G supp  (/) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823    =/= wne 2649   A.wral 2804   E.wrex 2805   {crab 2808   _Vcvv 3106   (/)c0 3783   U.cuni 4235   class class class wbr 4439   {copab 4496   Oncon0 4867   dom cdm 4988    Fn wfn 5565   -->wf 5566   ` cfv 5570  (class class class)co 6270   supp csupp 6891   finSupp cfsupp 7821   CNF ccnf 8069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-supp 6892  df-recs 7034  df-rdg 7068  df-seqom 7105  df-1o 7122  df-er 7303  df-map 7414  df-en 7510  df-fin 7513  df-fsupp 7822  df-cnf 8070
This theorem is referenced by:  cantnflem1b  8096  cantnflem1d  8098  cantnflem1  8099
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