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Theorem cantnfp1lem2OLD 8025
Description: Lemma for cantnfp1OLD 8027. (Contributed by Mario Carneiro, 28-May-2015.) Obsolete version of cantnfp1lem2 7999 as of 30-Jun-2019. (New usage is discouraged.)
Hypotheses
Ref Expression
cantnfsOLD.1  |-  S  =  dom  ( A CNF  B
)
cantnfsOLD.2  |-  ( ph  ->  A  e.  On )
cantnfsOLD.3  |-  ( ph  ->  B  e.  On )
cantnfp1OLD.4  |-  ( ph  ->  G  e.  S )
cantnfp1OLD.5  |-  ( ph  ->  X  e.  B )
cantnfp1OLD.6  |-  ( ph  ->  Y  e.  A )
cantnfp1OLD.7  |-  ( ph  ->  ( `' G "
( _V  \  1o ) )  C_  X
)
cantnfp1OLD.f  |-  F  =  ( t  e.  B  |->  if ( t  =  X ,  Y , 
( G `  t
) ) )
cantnfp1OLD.8  |-  ( ph  -> 
(/)  e.  Y )
cantnfp1OLD.o  |-  O  = OrdIso
(  _E  ,  ( `' F " ( _V 
\  1o ) ) )
Assertion
Ref Expression
cantnfp1lem2OLD  |-  ( 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 cantnfp1lem2OLD
StepHypRef Expression
1 cantnfp1OLD.5 . . . . . . 7  |-  ( ph  ->  X  e.  B )
2 cantnfp1OLD.6 . . . . . . . . . 10  |-  ( ph  ->  Y  e.  A )
3 iftrue 3906 . . . . . . . . . . 11  |-  ( t  =  X  ->  if ( t  =  X ,  Y ,  ( G `  t ) )  =  Y )
4 cantnfp1OLD.f . . . . . . . . . . 11  |-  F  =  ( t  e.  B  |->  if ( t  =  X ,  Y , 
( G `  t
) ) )
53, 4fvmptg 5882 . . . . . . . . . 10  |-  ( ( X  e.  B  /\  Y  e.  A )  ->  ( F `  X
)  =  Y )
61, 2, 5syl2anc 661 . . . . . . . . 9  |-  ( ph  ->  ( F `  X
)  =  Y )
7 cantnfp1OLD.8 . . . . . . . . . 10  |-  ( ph  -> 
(/)  e.  Y )
8 cantnfsOLD.2 . . . . . . . . . . . 12  |-  ( ph  ->  A  e.  On )
9 onelon 4853 . . . . . . . . . . . 12  |-  ( ( A  e.  On  /\  Y  e.  A )  ->  Y  e.  On )
108, 2, 9syl2anc 661 . . . . . . . . . . 11  |-  ( ph  ->  Y  e.  On )
11 on0eln0 4883 . . . . . . . . . . 11  |-  ( Y  e.  On  ->  ( (/) 
e.  Y  <->  Y  =/=  (/) ) )
1210, 11syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( (/)  e.  Y  <->  Y  =/=  (/) ) )
137, 12mpbid 210 . . . . . . . . 9  |-  ( ph  ->  Y  =/=  (/) )
146, 13eqnetrd 2745 . . . . . . . 8  |-  ( ph  ->  ( F `  X
)  =/=  (/) )
15 fvex 5810 . . . . . . . . 9  |-  ( F `
 X )  e. 
_V
16 dif1o 7051 . . . . . . . . 9  |-  ( ( F `  X )  e.  ( _V  \  1o )  <->  ( ( F `
 X )  e. 
_V  /\  ( F `  X )  =/=  (/) ) )
1715, 16mpbiran 909 . . . . . . . 8  |-  ( ( F `  X )  e.  ( _V  \  1o )  <->  ( F `  X )  =/=  (/) )
1814, 17sylibr 212 . . . . . . 7  |-  ( ph  ->  ( F `  X
)  e.  ( _V 
\  1o ) )
192adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  B )  ->  Y  e.  A )
20 cantnfp1OLD.4 . . . . . . . . . . . . 13  |-  ( ph  ->  G  e.  S )
21 cantnfsOLD.1 . . . . . . . . . . . . . 14  |-  S  =  dom  ( A CNF  B
)
22 cantnfsOLD.3 . . . . . . . . . . . . . 14  |-  ( ph  ->  B  e.  On )
2321, 8, 22cantnfsOLD 8016 . . . . . . . . . . . . 13  |-  ( ph  ->  ( G  e.  S  <->  ( G : B --> A  /\  ( `' G " ( _V 
\  1o ) )  e.  Fin ) ) )
2420, 23mpbid 210 . . . . . . . . . . . 12  |-  ( ph  ->  ( G : B --> A  /\  ( `' G " ( _V  \  1o ) )  e.  Fin ) )
2524simpld 459 . . . . . . . . . . 11  |-  ( ph  ->  G : B --> A )
2625ffvelrnda 5953 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  B )  ->  ( G `  t )  e.  A )
27 ifcl 3940 . . . . . . . . . 10  |-  ( ( Y  e.  A  /\  ( G `  t )  e.  A )  ->  if ( t  =  X ,  Y ,  ( G `  t ) )  e.  A )
2819, 26, 27syl2anc 661 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  B )  ->  if ( t  =  X ,  Y ,  ( G `  t ) )  e.  A )
2928, 4fmptd 5977 . . . . . . . 8  |-  ( ph  ->  F : B --> A )
30 ffn 5668 . . . . . . . 8  |-  ( F : B --> A  ->  F  Fn  B )
31 elpreima 5933 . . . . . . . 8  |-  ( F  Fn  B  ->  ( X  e.  ( `' F " ( _V  \  1o ) )  <->  ( X  e.  B  /\  ( F `  X )  e.  ( _V  \  1o ) ) ) )
3229, 30, 313syl 20 . . . . . . 7  |-  ( ph  ->  ( X  e.  ( `' F " ( _V 
\  1o ) )  <-> 
( X  e.  B  /\  ( F `  X
)  e.  ( _V 
\  1o ) ) ) )
331, 18, 32mpbir2and 913 . . . . . 6  |-  ( ph  ->  X  e.  ( `' F " ( _V 
\  1o ) ) )
34 n0i 3751 . . . . . 6  |-  ( X  e.  ( `' F " ( _V  \  1o ) )  ->  -.  ( `' F " ( _V 
\  1o ) )  =  (/) )
3533, 34syl 16 . . . . 5  |-  ( ph  ->  -.  ( `' F " ( _V  \  1o ) )  =  (/) )
36 cnvimass 5298 . . . . . . . . 9  |-  ( `' F " ( _V 
\  1o ) ) 
C_  dom  F
37 fdm 5672 . . . . . . . . . 10  |-  ( F : B --> A  ->  dom  F  =  B )
3829, 37syl 16 . . . . . . . . 9  |-  ( ph  ->  dom  F  =  B )
3936, 38syl5sseq 3513 . . . . . . . 8  |-  ( ph  ->  ( `' F "
( _V  \  1o ) )  C_  B
)
4022, 39ssexd 4548 . . . . . . 7  |-  ( ph  ->  ( `' F "
( _V  \  1o ) )  e.  _V )
41 cantnfp1OLD.o . . . . . . . . 9  |-  O  = OrdIso
(  _E  ,  ( `' F " ( _V 
\  1o ) ) )
42 cantnfp1OLD.7 . . . . . . . . . 10  |-  ( ph  ->  ( `' G "
( _V  \  1o ) )  C_  X
)
4321, 8, 22, 20, 1, 2, 42, 4cantnfp1lem1OLD 8024 . . . . . . . . 9  |-  ( ph  ->  F  e.  S )
4421, 8, 22, 41, 43cantnfclOLD 8017 . . . . . . . 8  |-  ( ph  ->  (  _E  We  ( `' F " ( _V 
\  1o ) )  /\  dom  O  e. 
om ) )
4544simpld 459 . . . . . . 7  |-  ( ph  ->  _E  We  ( `' F " ( _V 
\  1o ) ) )
4641oien 7864 . . . . . . 7  |-  ( ( ( `' F "
( _V  \  1o ) )  e.  _V  /\  _E  We  ( `' F " ( _V 
\  1o ) ) )  ->  dom  O  ~~  ( `' F " ( _V 
\  1o ) ) )
4740, 45, 46syl2anc 661 . . . . . 6  |-  ( ph  ->  dom  O  ~~  ( `' F " ( _V 
\  1o ) ) )
48 breq1 4404 . . . . . . 7  |-  ( dom 
O  =  (/)  ->  ( dom  O  ~~  ( `' F " ( _V 
\  1o ) )  <->  (/)  ~~  ( `' F "
( _V  \  1o ) ) ) )
49 ensymb 7468 . . . . . . . 8  |-  ( (/)  ~~  ( `' F "
( _V  \  1o ) )  <->  ( `' F " ( _V  \  1o ) )  ~~  (/) )
50 en0 7483 . . . . . . . 8  |-  ( ( `' F " ( _V 
\  1o ) ) 
~~  (/)  <->  ( `' F " ( _V  \  1o ) )  =  (/) )
5149, 50bitri 249 . . . . . . 7  |-  ( (/)  ~~  ( `' F "
( _V  \  1o ) )  <->  ( `' F " ( _V  \  1o ) )  =  (/) )
5248, 51syl6bb 261 . . . . . 6  |-  ( dom 
O  =  (/)  ->  ( dom  O  ~~  ( `' F " ( _V 
\  1o ) )  <-> 
( `' F "
( _V  \  1o ) )  =  (/) ) )
5347, 52syl5ibcom 220 . . . . 5  |-  ( ph  ->  ( dom  O  =  (/)  ->  ( `' F " ( _V  \  1o ) )  =  (/) ) )
5435, 53mtod 177 . . . 4  |-  ( ph  ->  -.  dom  O  =  (/) )
5544simprd 463 . . . . 5  |-  ( ph  ->  dom  O  e.  om )
56 nnlim 6600 . . . . 5  |-  ( dom 
O  e.  om  ->  -. 
Lim  dom  O )
5755, 56syl 16 . . . 4  |-  ( ph  ->  -.  Lim  dom  O
)
58 ioran 490 . . . 4  |-  ( -.  ( dom  O  =  (/)  \/  Lim  dom  O
)  <->  ( -.  dom  O  =  (/)  /\  -.  Lim  dom 
O ) )
5954, 57, 58sylanbrc 664 . . 3  |-  ( ph  ->  -.  ( dom  O  =  (/)  \/  Lim  dom  O ) )
60 nnord 6595 . . . 4  |-  ( dom 
O  e.  om  ->  Ord 
dom  O )
61 unizlim 4944 . . . 4  |-  ( Ord 
dom  O  ->  ( dom 
O  =  U. dom  O  <-> 
( dom  O  =  (/) 
\/  Lim  dom  O ) ) )
6255, 60, 613syl 20 . . 3  |-  ( ph  ->  ( dom  O  = 
U. dom  O  <->  ( dom  O  =  (/)  \/  Lim  dom 
O ) ) )
6359, 62mtbird 301 . 2  |-  ( ph  ->  -.  dom  O  = 
U. dom  O )
64 orduniorsuc 6552 . . . 4  |-  ( Ord 
dom  O  ->  ( dom 
O  =  U. dom  O  \/  dom  O  =  suc  U. dom  O
) )
6555, 60, 643syl 20 . . 3  |-  ( ph  ->  ( dom  O  = 
U. dom  O  \/  dom  O  =  suc  U. dom  O ) )
6665ord 377 . 2  |-  ( ph  ->  ( -.  dom  O  =  U. dom  O  ->  dom  O  =  suc  U. dom  O ) )
6763, 66mpd 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 2648   _Vcvv 3078    \ cdif 3434    C_ wss 3437   (/)c0 3746   ifcif 3900   U.cuni 4200   class class class wbr 4401    |-> cmpt 4459    _E cep 4739    We wwe 4787   Ord word 4827   Oncon0 4828   Lim wlim 4829   suc csuc 4830   `'ccnv 4948   dom cdm 4949   "cima 4952    Fn wfn 5522   -->wf 5523   ` cfv 5527  (class class class)co 6201   omcom 6587   1oc1o 7024    ~~ cen 7418   Fincfn 7421  OrdIsocoi 7835   CNF ccnf 7979
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 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-se 4789  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-isom 5536  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-1st 6688  df-2nd 6689  df-supp 6802  df-recs 6943  df-rdg 6977  df-seqom 7014  df-1o 7031  df-oadd 7035  df-er 7212  df-map 7327  df-en 7422  df-dom 7423  df-sdom 7424  df-fin 7425  df-fsupp 7733  df-oi 7836  df-cnf 7980
This theorem is referenced by:  cantnfp1lem3OLD  8026
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