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Theorem cantnfp1lem2 8210
Description: Lemma for cantnfp1 8212. (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 3899 . . . . . . . . . 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 5969 . . . . . . . . 9  |-  ( ( X  e.  B  /\  Y  e.  A )  ->  ( F `  X
)  =  Y )
61, 2, 5syl2anc 671 . . . . . . . 8  |-  ( ph  ->  ( F `  X
)  =  Y )
7 cantnfp1.e . . . . . . . . 9  |-  ( ph  -> 
(/)  e.  Y )
8 cantnfs.a . . . . . . . . . . 11  |-  ( ph  ->  A  e.  On )
9 onelon 5467 . . . . . . . . . . 11  |-  ( ( A  e.  On  /\  Y  e.  A )  ->  Y  e.  On )
108, 2, 9syl2anc 671 . . . . . . . . . 10  |-  ( ph  ->  Y  e.  On )
11 on0eln0 5497 . . . . . . . . . 10  |-  ( Y  e.  On  ->  ( (/) 
e.  Y  <->  Y  =/=  (/) ) )
1210, 11syl 17 . . . . . . . . 9  |-  ( ph  ->  ( (/)  e.  Y  <->  Y  =/=  (/) ) )
137, 12mpbid 215 . . . . . . . 8  |-  ( ph  ->  Y  =/=  (/) )
146, 13eqnetrd 2703 . . . . . . 7  |-  ( ph  ->  ( F `  X
)  =/=  (/) )
152adantr 471 . . . . . . . . . . 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 8197 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( G  e.  S  <->  ( G : B --> A  /\  G finSupp 
(/) ) ) )
2016, 19mpbid 215 . . . . . . . . . . . . 13  |-  ( ph  ->  ( G : B --> A  /\  G finSupp  (/) ) )
2120simpld 465 . . . . . . . . . . . 12  |-  ( ph  ->  G : B --> A )
2221ffvelrnda 6045 . . . . . . . . . . 11  |-  ( (
ph  /\  t  e.  B )  ->  ( G `  t )  e.  A )
2315, 22ifcld 3936 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  B )  ->  if ( t  =  X ,  Y ,  ( G `  t ) )  e.  A )
2423, 4fmptd 6069 . . . . . . . . 9  |-  ( ph  ->  F : B --> A )
25 ffn 5751 . . . . . . . . 9  |-  ( F : B --> A  ->  F  Fn  B )
2624, 25syl 17 . . . . . . . 8  |-  ( ph  ->  F  Fn  B )
27 0ex 4549 . . . . . . . . 9  |-  (/)  e.  _V
2827a1i 11 . . . . . . . 8  |-  ( ph  -> 
(/)  e.  _V )
29 elsuppfn 6949 . . . . . . . 8  |-  ( ( F  Fn  B  /\  B  e.  On  /\  (/)  e.  _V )  ->  ( X  e.  ( F supp  (/) )  <->  ( X  e.  B  /\  ( F `  X )  =/=  (/) ) ) )
3026, 18, 28, 29syl3anc 1276 . . . . . . 7  |-  ( ph  ->  ( X  e.  ( F supp  (/) )  <->  ( X  e.  B  /\  ( F `  X )  =/=  (/) ) ) )
311, 14, 30mpbir2and 938 . . . . . 6  |-  ( ph  ->  X  e.  ( F supp  (/) ) )
32 n0i 3748 . . . . . 6  |-  ( X  e.  ( F supp  (/) )  ->  -.  ( F supp  (/) )  =  (/) )
3331, 32syl 17 . . . . 5  |-  ( ph  ->  -.  ( F supp  (/) )  =  (/) )
34 suppssdm 6954 . . . . . . . . 9  |-  ( F supp  (/) )  C_  dom  F
354, 23dmmptd 5730 . . . . . . . . 9  |-  ( ph  ->  dom  F  =  B )
3634, 35syl5sseq 3492 . . . . . . . 8  |-  ( ph  ->  ( F supp  (/) )  C_  B )
3718, 36ssexd 4564 . . . . . . 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 8209 . . . . . . . . 9  |-  ( ph  ->  F  e.  S )
4117, 8, 18, 38, 40cantnfcl 8198 . . . . . . . 8  |-  ( ph  ->  (  _E  We  ( F supp 
(/) )  /\  dom  O  e.  om ) )
4241simpld 465 . . . . . . 7  |-  ( ph  ->  _E  We  ( F supp  (/) ) )
4338oien 8079 . . . . . . 7  |-  ( ( ( F supp  (/) )  e. 
_V  /\  _E  We  ( F supp  (/) ) )  ->  dom  O  ~~  ( F supp  (/) ) )
4437, 42, 43syl2anc 671 . . . . . 6  |-  ( ph  ->  dom  O  ~~  ( F supp 
(/) ) )
45 breq1 4419 . . . . . . 7  |-  ( dom 
O  =  (/)  ->  ( dom  O  ~~  ( F supp  (/) )  <->  (/)  ~~  ( F supp  (/) ) ) )
46 ensymb 7643 . . . . . . . 8  |-  ( (/)  ~~  ( F supp  (/) )  <->  ( F supp  (/) )  ~~  (/) )
47 en0 7658 . . . . . . . 8  |-  ( ( F supp  (/) )  ~~  (/)  <->  ( F supp  (/) )  =  (/) )
4846, 47bitri 257 . . . . . . 7  |-  ( (/)  ~~  ( F supp  (/) )  <->  ( F supp  (/) )  =  (/) )
4945, 48syl6bb 269 . . . . . 6  |-  ( dom 
O  =  (/)  ->  ( dom  O  ~~  ( F supp  (/) )  <->  ( F supp  (/) )  =  (/) ) )
5044, 49syl5ibcom 228 . . . . 5  |-  ( ph  ->  ( dom  O  =  (/)  ->  ( F supp  (/) )  =  (/) ) )
5133, 50mtod 182 . . . 4  |-  ( ph  ->  -.  dom  O  =  (/) )
5241simprd 469 . . . . 5  |-  ( ph  ->  dom  O  e.  om )
53 nnlim 6732 . . . . 5  |-  ( dom 
O  e.  om  ->  -. 
Lim  dom  O )
5452, 53syl 17 . . . 4  |-  ( ph  ->  -.  Lim  dom  O
)
55 ioran 497 . . . 4  |-  ( -.  ( dom  O  =  (/)  \/  Lim  dom  O
)  <->  ( -.  dom  O  =  (/)  /\  -.  Lim  dom 
O ) )
5651, 54, 55sylanbrc 675 . . 3  |-  ( ph  ->  -.  ( dom  O  =  (/)  \/  Lim  dom  O ) )
57 nnord 6727 . . . 4  |-  ( dom 
O  e.  om  ->  Ord 
dom  O )
58 unizlim 5558 . . . 4  |-  ( Ord 
dom  O  ->  ( dom 
O  =  U. dom  O  <-> 
( dom  O  =  (/) 
\/  Lim  dom  O ) ) )
5952, 57, 583syl 18 . . 3  |-  ( ph  ->  ( dom  O  = 
U. dom  O  <->  ( dom  O  =  (/)  \/  Lim  dom 
O ) ) )
6056, 59mtbird 307 . 2  |-  ( ph  ->  -.  dom  O  = 
U. dom  O )
61 orduniorsuc 6684 . . . 4  |-  ( Ord 
dom  O  ->  ( dom 
O  =  U. dom  O  \/  dom  O  =  suc  U. dom  O
) )
6252, 57, 613syl 18 . . 3  |-  ( ph  ->  ( dom  O  = 
U. dom  O  \/  dom  O  =  suc  U. dom  O ) )
6362ord 383 . 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 189    \/ wo 374    /\ wa 375    = wceq 1455    e. wcel 1898    =/= wne 2633   _Vcvv 3057    C_ wss 3416   (/)c0 3743   ifcif 3893   U.cuni 4212   class class class wbr 4416    |-> cmpt 4475    _E cep 4762    We wwe 4811   dom cdm 4853   Ord word 5441   Oncon0 5442   Lim wlim 5443   suc csuc 5444    Fn wfn 5596   -->wf 5597   ` cfv 5601  (class class class)co 6315   omcom 6719   supp csupp 6941    ~~ cen 7592   finSupp cfsupp 7909  OrdIsocoi 8050   CNF ccnf 8192
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4529  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-fal 1461  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-int 4249  df-iun 4294  df-br 4417  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-se 4813  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6277  df-ov 6318  df-oprab 6319  df-mpt2 6320  df-om 6720  df-supp 6942  df-wrecs 7054  df-recs 7116  df-rdg 7154  df-seqom 7191  df-1o 7208  df-oadd 7212  df-er 7389  df-map 7500  df-en 7596  df-dom 7597  df-sdom 7598  df-fin 7599  df-fsupp 7910  df-oi 8051  df-cnf 8193
This theorem is referenced by:  cantnfp1lem3  8211
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