MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cantnfp1lem2 Structured version   Unicode version

Theorem cantnfp1lem2 8089
Description: Lemma for cantnfp1 8091. (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 30-Jun-2019.)
Hypotheses
Ref Expression
cantnfs.s  |-  S  =  dom  ( A CNF  B
)
cantnfs.a  |-  ( ph  ->  A  e.  On )
cantnfs.b  |-  ( ph  ->  B  e.  On )
cantnfp1.g  |-  ( ph  ->  G  e.  S )
cantnfp1.x  |-  ( ph  ->  X  e.  B )
cantnfp1.y  |-  ( ph  ->  Y  e.  A )
cantnfp1.s  |-  ( ph  ->  ( G supp  (/) )  C_  X )
cantnfp1.f  |-  F  =  ( t  e.  B  |->  if ( t  =  X ,  Y , 
( G `  t
) ) )
cantnfp1.e  |-  ( ph  -> 
(/)  e.  Y )
cantnfp1.o  |-  O  = OrdIso
(  _E  ,  ( F supp  (/) ) )
Assertion
Ref Expression
cantnfp1lem2  |-  ( ph  ->  dom  O  =  suc  U.
dom  O )
Distinct variable groups:    t, B    t, A    t, S    t, G    ph, t    t, Y   
t, X
Allowed substitution hints:    F( t)    O( t)

Proof of Theorem cantnfp1lem2
StepHypRef Expression
1 cantnfp1.x . . . . . . 7  |-  ( ph  ->  X  e.  B )
2 cantnfp1.y . . . . . . . . 9  |-  ( ph  ->  Y  e.  A )
3 iftrue 3935 . . . . . . . . . 10  |-  ( t  =  X  ->  if ( t  =  X ,  Y ,  ( G `  t ) )  =  Y )
4 cantnfp1.f . . . . . . . . . 10  |-  F  =  ( t  e.  B  |->  if ( t  =  X ,  Y , 
( G `  t
) ) )
53, 4fvmptg 5929 . . . . . . . . 9  |-  ( ( X  e.  B  /\  Y  e.  A )  ->  ( F `  X
)  =  Y )
61, 2, 5syl2anc 659 . . . . . . . 8  |-  ( ph  ->  ( F `  X
)  =  Y )
7 cantnfp1.e . . . . . . . . 9  |-  ( ph  -> 
(/)  e.  Y )
8 cantnfs.a . . . . . . . . . . 11  |-  ( ph  ->  A  e.  On )
9 onelon 4892 . . . . . . . . . . 11  |-  ( ( A  e.  On  /\  Y  e.  A )  ->  Y  e.  On )
108, 2, 9syl2anc 659 . . . . . . . . . 10  |-  ( ph  ->  Y  e.  On )
11 on0eln0 4922 . . . . . . . . . 10  |-  ( Y  e.  On  ->  ( (/) 
e.  Y  <->  Y  =/=  (/) ) )
1210, 11syl 16 . . . . . . . . 9  |-  ( ph  ->  ( (/)  e.  Y  <->  Y  =/=  (/) ) )
137, 12mpbid 210 . . . . . . . 8  |-  ( ph  ->  Y  =/=  (/) )
146, 13eqnetrd 2747 . . . . . . 7  |-  ( ph  ->  ( F `  X
)  =/=  (/) )
152adantr 463 . . . . . . . . . . 11  |-  ( (
ph  /\  t  e.  B )  ->  Y  e.  A )
16 cantnfp1.g . . . . . . . . . . . . . 14  |-  ( ph  ->  G  e.  S )
17 cantnfs.s . . . . . . . . . . . . . . 15  |-  S  =  dom  ( A CNF  B
)
18 cantnfs.b . . . . . . . . . . . . . . 15  |-  ( ph  ->  B  e.  On )
1917, 8, 18cantnfs 8076 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( G  e.  S  <->  ( G : B --> A  /\  G finSupp 
(/) ) ) )
2016, 19mpbid 210 . . . . . . . . . . . . 13  |-  ( ph  ->  ( G : B --> A  /\  G finSupp  (/) ) )
2120simpld 457 . . . . . . . . . . . 12  |-  ( ph  ->  G : B --> A )
2221ffvelrnda 6007 . . . . . . . . . . 11  |-  ( (
ph  /\  t  e.  B )  ->  ( G `  t )  e.  A )
2315, 22ifcld 3972 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  B )  ->  if ( t  =  X ,  Y ,  ( G `  t ) )  e.  A )
2423, 4fmptd 6031 . . . . . . . . 9  |-  ( ph  ->  F : B --> A )
25 ffn 5713 . . . . . . . . 9  |-  ( F : B --> A  ->  F  Fn  B )
2624, 25syl 16 . . . . . . . 8  |-  ( ph  ->  F  Fn  B )
27 0ex 4569 . . . . . . . . 9  |-  (/)  e.  _V
2827a1i 11 . . . . . . . 8  |-  ( ph  -> 
(/)  e.  _V )
29 elsuppfn 6899 . . . . . . . 8  |-  ( ( F  Fn  B  /\  B  e.  On  /\  (/)  e.  _V )  ->  ( X  e.  ( F supp  (/) )  <->  ( X  e.  B  /\  ( F `  X )  =/=  (/) ) ) )
3026, 18, 28, 29syl3anc 1226 . . . . . . 7  |-  ( ph  ->  ( X  e.  ( F supp  (/) )  <->  ( X  e.  B  /\  ( F `  X )  =/=  (/) ) ) )
311, 14, 30mpbir2and 920 . . . . . 6  |-  ( ph  ->  X  e.  ( F supp  (/) ) )
32 n0i 3788 . . . . . 6  |-  ( X  e.  ( F supp  (/) )  ->  -.  ( F supp  (/) )  =  (/) )
3331, 32syl 16 . . . . 5  |-  ( ph  ->  -.  ( F supp  (/) )  =  (/) )
34 suppssdm 6904 . . . . . . . . 9  |-  ( F supp  (/) )  C_  dom  F
354, 23dmmptd 5693 . . . . . . . . 9  |-  ( ph  ->  dom  F  =  B )
3634, 35syl5sseq 3537 . . . . . . . 8  |-  ( ph  ->  ( F supp  (/) )  C_  B )
3718, 36ssexd 4584 . . . . . . 7  |-  ( ph  ->  ( F supp  (/) )  e. 
_V )
38 cantnfp1.o . . . . . . . . 9  |-  O  = OrdIso
(  _E  ,  ( F supp  (/) ) )
39 cantnfp1.s . . . . . . . . . 10  |-  ( ph  ->  ( G supp  (/) )  C_  X )
4017, 8, 18, 16, 1, 2, 39, 4cantnfp1lem1 8088 . . . . . . . . 9  |-  ( ph  ->  F  e.  S )
4117, 8, 18, 38, 40cantnfcl 8077 . . . . . . . 8  |-  ( ph  ->  (  _E  We  ( F supp 
(/) )  /\  dom  O  e.  om ) )
4241simpld 457 . . . . . . 7  |-  ( ph  ->  _E  We  ( F supp  (/) ) )
4338oien 7955 . . . . . . 7  |-  ( ( ( F supp  (/) )  e. 
_V  /\  _E  We  ( F supp  (/) ) )  ->  dom  O  ~~  ( F supp  (/) ) )
4437, 42, 43syl2anc 659 . . . . . 6  |-  ( ph  ->  dom  O  ~~  ( F supp 
(/) ) )
45 breq1 4442 . . . . . . 7  |-  ( dom 
O  =  (/)  ->  ( dom  O  ~~  ( F supp  (/) )  <->  (/)  ~~  ( F supp  (/) ) ) )
46 ensymb 7556 . . . . . . . 8  |-  ( (/)  ~~  ( F supp  (/) )  <->  ( F supp  (/) )  ~~  (/) )
47 en0 7571 . . . . . . . 8  |-  ( ( F supp  (/) )  ~~  (/)  <->  ( F supp  (/) )  =  (/) )
4846, 47bitri 249 . . . . . . 7  |-  ( (/)  ~~  ( F supp  (/) )  <->  ( F supp  (/) )  =  (/) )
4945, 48syl6bb 261 . . . . . 6  |-  ( dom 
O  =  (/)  ->  ( dom  O  ~~  ( F supp  (/) )  <->  ( F supp  (/) )  =  (/) ) )
5044, 49syl5ibcom 220 . . . . 5  |-  ( ph  ->  ( dom  O  =  (/)  ->  ( F supp  (/) )  =  (/) ) )
5133, 50mtod 177 . . . 4  |-  ( ph  ->  -.  dom  O  =  (/) )
5241simprd 461 . . . . 5  |-  ( ph  ->  dom  O  e.  om )
53 nnlim 6686 . . . . 5  |-  ( dom 
O  e.  om  ->  -. 
Lim  dom  O )
5452, 53syl 16 . . . 4  |-  ( ph  ->  -.  Lim  dom  O
)
55 ioran 488 . . . 4  |-  ( -.  ( dom  O  =  (/)  \/  Lim  dom  O
)  <->  ( -.  dom  O  =  (/)  /\  -.  Lim  dom 
O ) )
5651, 54, 55sylanbrc 662 . . 3  |-  ( ph  ->  -.  ( dom  O  =  (/)  \/  Lim  dom  O ) )
57 nnord 6681 . . . 4  |-  ( dom 
O  e.  om  ->  Ord 
dom  O )
58 unizlim 4983 . . . 4  |-  ( Ord 
dom  O  ->  ( dom 
O  =  U. dom  O  <-> 
( dom  O  =  (/) 
\/  Lim  dom  O ) ) )
5952, 57, 583syl 20 . . 3  |-  ( ph  ->  ( dom  O  = 
U. dom  O  <->  ( dom  O  =  (/)  \/  Lim  dom 
O ) ) )
6056, 59mtbird 299 . 2  |-  ( ph  ->  -.  dom  O  = 
U. dom  O )
61 orduniorsuc 6638 . . . 4  |-  ( Ord 
dom  O  ->  ( dom 
O  =  U. dom  O  \/  dom  O  =  suc  U. dom  O
) )
6252, 57, 613syl 20 . . 3  |-  ( ph  ->  ( dom  O  = 
U. dom  O  \/  dom  O  =  suc  U. dom  O ) )
6362ord 375 . 2  |-  ( ph  ->  ( -.  dom  O  =  U. dom  O  ->  dom  O  =  suc  U. dom  O ) )
6460, 63mpd 15 1  |-  ( ph  ->  dom  O  =  suc  U.
dom  O )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 366    /\ wa 367    = wceq 1398    e. wcel 1823    =/= wne 2649   _Vcvv 3106    C_ wss 3461   (/)c0 3783   ifcif 3929   U.cuni 4235   class class class wbr 4439    |-> cmpt 4497    _E cep 4778    We wwe 4826   Ord word 4866   Oncon0 4867   Lim wlim 4868   suc csuc 4869   dom cdm 4988    Fn wfn 5565   -->wf 5566   ` cfv 5570  (class class class)co 6270   omcom 6673   supp csupp 6891    ~~ cen 7506   finSupp cfsupp 7821  OrdIsocoi 7926   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-rmo 2812  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-int 4272  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-se 4828  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-isom 5579  df-riota 6232  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-oadd 7126  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fsupp 7822  df-oi 7927  df-cnf 8070
This theorem is referenced by:  cantnfp1lem3  8090
  Copyright terms: Public domain W3C validator