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Theorem cantnfp1lem2 7991
Description: Lemma for cantnfp1 7993. (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 3898 . . . . . . . . . 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 5874 . . . . . . . . 9  |-  ( ( X  e.  B  /\  Y  e.  A )  ->  ( F `  X
)  =  Y )
61, 2, 5syl2anc 661 . . . . . . . 8  |-  ( ph  ->  ( F `  X
)  =  Y )
7 cantnfp1.e . . . . . . . . 9  |-  ( ph  -> 
(/)  e.  Y )
8 cantnfs.a . . . . . . . . . . 11  |-  ( ph  ->  A  e.  On )
9 onelon 4845 . . . . . . . . . . 11  |-  ( ( A  e.  On  /\  Y  e.  A )  ->  Y  e.  On )
108, 2, 9syl2anc 661 . . . . . . . . . 10  |-  ( ph  ->  Y  e.  On )
11 on0eln0 4875 . . . . . . . . . 10  |-  ( Y  e.  On  ->  ( (/) 
e.  Y  <->  Y  =/=  (/) ) )
1210, 11syl 16 . . . . . . . . 9  |-  ( ph  ->  ( (/)  e.  Y  <->  Y  =/=  (/) ) )
137, 12mpbid 210 . . . . . . . 8  |-  ( ph  ->  Y  =/=  (/) )
146, 13eqnetrd 2741 . . . . . . 7  |-  ( ph  ->  ( F `  X
)  =/=  (/) )
152adantr 465 . . . . . . . . . . 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 7978 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( G  e.  S  <->  ( G : B --> A  /\  G finSupp 
(/) ) ) )
2016, 19mpbid 210 . . . . . . . . . . . . 13  |-  ( ph  ->  ( G : B --> A  /\  G finSupp  (/) ) )
2120simpld 459 . . . . . . . . . . . 12  |-  ( ph  ->  G : B --> A )
2221ffvelrnda 5945 . . . . . . . . . . 11  |-  ( (
ph  /\  t  e.  B )  ->  ( G `  t )  e.  A )
2315, 22ifcld 3933 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  B )  ->  if ( t  =  X ,  Y ,  ( G `  t ) )  e.  A )
2423, 4fmptd 5969 . . . . . . . . 9  |-  ( ph  ->  F : B --> A )
25 ffn 5660 . . . . . . . . 9  |-  ( F : B --> A  ->  F  Fn  B )
2624, 25syl 16 . . . . . . . 8  |-  ( ph  ->  F  Fn  B )
27 0ex 4523 . . . . . . . . 9  |-  (/)  e.  _V
2827a1i 11 . . . . . . . 8  |-  ( ph  -> 
(/)  e.  _V )
29 elsuppfn 6801 . . . . . . . 8  |-  ( ( F  Fn  B  /\  B  e.  On  /\  (/)  e.  _V )  ->  ( X  e.  ( F supp  (/) )  <->  ( X  e.  B  /\  ( F `  X )  =/=  (/) ) ) )
3026, 18, 28, 29syl3anc 1219 . . . . . . 7  |-  ( ph  ->  ( X  e.  ( F supp  (/) )  <->  ( X  e.  B  /\  ( F `  X )  =/=  (/) ) ) )
311, 14, 30mpbir2and 913 . . . . . 6  |-  ( ph  ->  X  e.  ( F supp  (/) ) )
32 n0i 3743 . . . . . 6  |-  ( X  e.  ( F supp  (/) )  ->  -.  ( F supp  (/) )  =  (/) )
3331, 32syl 16 . . . . 5  |-  ( ph  ->  -.  ( F supp  (/) )  =  (/) )
34 suppssdm 6806 . . . . . . . . 9  |-  ( F supp  (/) )  C_  dom  F
35 fdm 5664 . . . . . . . . . 10  |-  ( F : B --> A  ->  dom  F  =  B )
3624, 35syl 16 . . . . . . . . 9  |-  ( ph  ->  dom  F  =  B )
3734, 36syl5sseq 3505 . . . . . . . 8  |-  ( ph  ->  ( F supp  (/) )  C_  B )
3818, 37ssexd 4540 . . . . . . 7  |-  ( ph  ->  ( F supp  (/) )  e. 
_V )
39 cantnfp1.o . . . . . . . . 9  |-  O  = OrdIso
(  _E  ,  ( F supp  (/) ) )
40 cantnfp1.s . . . . . . . . . 10  |-  ( ph  ->  ( G supp  (/) )  C_  X )
4117, 8, 18, 16, 1, 2, 40, 4cantnfp1lem1 7990 . . . . . . . . 9  |-  ( ph  ->  F  e.  S )
4217, 8, 18, 39, 41cantnfcl 7979 . . . . . . . 8  |-  ( ph  ->  (  _E  We  ( F supp 
(/) )  /\  dom  O  e.  om ) )
4342simpld 459 . . . . . . 7  |-  ( ph  ->  _E  We  ( F supp  (/) ) )
4439oien 7856 . . . . . . 7  |-  ( ( ( F supp  (/) )  e. 
_V  /\  _E  We  ( F supp  (/) ) )  ->  dom  O  ~~  ( F supp  (/) ) )
4538, 43, 44syl2anc 661 . . . . . 6  |-  ( ph  ->  dom  O  ~~  ( F supp 
(/) ) )
46 breq1 4396 . . . . . . 7  |-  ( dom 
O  =  (/)  ->  ( dom  O  ~~  ( F supp  (/) )  <->  (/)  ~~  ( F supp  (/) ) ) )
47 ensymb 7460 . . . . . . . 8  |-  ( (/)  ~~  ( F supp  (/) )  <->  ( F supp  (/) )  ~~  (/) )
48 en0 7475 . . . . . . . 8  |-  ( ( F supp  (/) )  ~~  (/)  <->  ( F supp  (/) )  =  (/) )
4947, 48bitri 249 . . . . . . 7  |-  ( (/)  ~~  ( F supp  (/) )  <->  ( F supp  (/) )  =  (/) )
5046, 49syl6bb 261 . . . . . 6  |-  ( dom 
O  =  (/)  ->  ( dom  O  ~~  ( F supp  (/) )  <->  ( F supp  (/) )  =  (/) ) )
5145, 50syl5ibcom 220 . . . . 5  |-  ( ph  ->  ( dom  O  =  (/)  ->  ( F supp  (/) )  =  (/) ) )
5233, 51mtod 177 . . . 4  |-  ( ph  ->  -.  dom  O  =  (/) )
5342simprd 463 . . . . 5  |-  ( ph  ->  dom  O  e.  om )
54 nnlim 6592 . . . . 5  |-  ( dom 
O  e.  om  ->  -. 
Lim  dom  O )
5553, 54syl 16 . . . 4  |-  ( ph  ->  -.  Lim  dom  O
)
56 ioran 490 . . . 4  |-  ( -.  ( dom  O  =  (/)  \/  Lim  dom  O
)  <->  ( -.  dom  O  =  (/)  /\  -.  Lim  dom 
O ) )
5752, 55, 56sylanbrc 664 . . 3  |-  ( ph  ->  -.  ( dom  O  =  (/)  \/  Lim  dom  O ) )
58 nnord 6587 . . . 4  |-  ( dom 
O  e.  om  ->  Ord 
dom  O )
59 unizlim 4936 . . . 4  |-  ( Ord 
dom  O  ->  ( dom 
O  =  U. dom  O  <-> 
( dom  O  =  (/) 
\/  Lim  dom  O ) ) )
6053, 58, 593syl 20 . . 3  |-  ( ph  ->  ( dom  O  = 
U. dom  O  <->  ( dom  O  =  (/)  \/  Lim  dom 
O ) ) )
6157, 60mtbird 301 . 2  |-  ( ph  ->  -.  dom  O  = 
U. dom  O )
62 orduniorsuc 6544 . . . 4  |-  ( Ord 
dom  O  ->  ( dom 
O  =  U. dom  O  \/  dom  O  =  suc  U. dom  O
) )
6353, 58, 623syl 20 . . 3  |-  ( ph  ->  ( dom  O  = 
U. dom  O  \/  dom  O  =  suc  U. dom  O ) )
6463ord 377 . 2  |-  ( ph  ->  ( -.  dom  O  =  U. dom  O  ->  dom  O  =  suc  U. dom  O ) )
6561, 64mpd 15 1  |-  ( ph  ->  dom  O  =  suc  U.
dom  O )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2644   _Vcvv 3071    C_ wss 3429   (/)c0 3738   ifcif 3892   U.cuni 4192   class class class wbr 4393    |-> cmpt 4451    _E cep 4731    We wwe 4779   Ord word 4819   Oncon0 4820   Lim wlim 4821   suc csuc 4822   dom cdm 4941    Fn wfn 5514   -->wf 5515   ` cfv 5519  (class class class)co 6193   omcom 6579   supp csupp 6793    ~~ cen 7410   finSupp cfsupp 7724  OrdIsocoi 7827   CNF ccnf 7971
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 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475
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-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-se 4781  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-isom 5528  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-supp 6794  df-recs 6935  df-rdg 6969  df-seqom 7006  df-1o 7023  df-oadd 7027  df-er 7204  df-map 7319  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-fsupp 7725  df-oi 7828  df-cnf 7972
This theorem is referenced by:  cantnfp1lem3  7992
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