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Theorem cantnfp1lem2OLD 8059
Description: Lemma for cantnfp1OLD 8061. (Contributed by Mario Carneiro, 28-May-2015.) Obsolete version of cantnfp1lem2 8033 as of 30-Jun-2019. (New usage is discouraged.) (Proof modification 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 3880 . . . . . . . . . . 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 5872 . . . . . . . . . 10  |-  ( ( X  e.  B  /\  Y  e.  A )  ->  ( F `  X
)  =  Y )
61, 2, 5syl2anc 659 . . . . . . . . 9  |-  ( ph  ->  ( F `  X
)  =  Y )
7 cantnfp1OLD.8 . . . . . . . . . 10  |-  ( ph  -> 
(/)  e.  Y )
8 cantnfsOLD.2 . . . . . . . . . . . 12  |-  ( ph  ->  A  e.  On )
9 onelon 4834 . . . . . . . . . . . 12  |-  ( ( A  e.  On  /\  Y  e.  A )  ->  Y  e.  On )
108, 2, 9syl2anc 659 . . . . . . . . . . 11  |-  ( ph  ->  Y  e.  On )
11 on0eln0 4864 . . . . . . . . . . 11  |-  ( Y  e.  On  ->  ( (/) 
e.  Y  <->  Y  =/=  (/) ) )
1210, 11syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( (/)  e.  Y  <->  Y  =/=  (/) ) )
137, 12mpbid 210 . . . . . . . . 9  |-  ( ph  ->  Y  =/=  (/) )
146, 13eqnetrd 2689 . . . . . . . 8  |-  ( ph  ->  ( F `  X
)  =/=  (/) )
15 fvex 5801 . . . . . . . . 9  |-  ( F `
 X )  e. 
_V
16 dif1o 7090 . . . . . . . . 9  |-  ( ( F `  X )  e.  ( _V  \  1o )  <->  ( ( F `
 X )  e. 
_V  /\  ( F `  X )  =/=  (/) ) )
1715, 16mpbiran 916 . . . . . . . 8  |-  ( ( F `  X )  e.  ( _V  \  1o )  <->  ( F `  X )  =/=  (/) )
1814, 17sylibr 212 . . . . . . 7  |-  ( ph  ->  ( F `  X
)  e.  ( _V 
\  1o ) )
192adantr 463 . . . . . . . . . 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 8050 . . . . . . . . . . . . 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 457 . . . . . . . . . . 11  |-  ( ph  ->  G : B --> A )
2625ffvelrnda 5950 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  B )  ->  ( G `  t )  e.  A )
27 ifcl 3916 . . . . . . . . . 10  |-  ( ( Y  e.  A  /\  ( G `  t )  e.  A )  ->  if ( t  =  X ,  Y ,  ( G `  t ) )  e.  A )
2819, 26, 27syl2anc 659 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  B )  ->  if ( t  =  X ,  Y ,  ( G `  t ) )  e.  A )
2928, 4fmptd 5974 . . . . . . . 8  |-  ( ph  ->  F : B --> A )
30 ffn 5656 . . . . . . . 8  |-  ( F : B --> A  ->  F  Fn  B )
31 elpreima 5926 . . . . . . . 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 920 . . . . . 6  |-  ( ph  ->  X  e.  ( `' F " ( _V 
\  1o ) ) )
34 n0i 3733 . . . . . 6  |-  ( X  e.  ( `' F " ( _V  \  1o ) )  ->  -.  ( `' F " ( _V 
\  1o ) )  =  (/) )
3533, 34syl 16 . . . . 5  |-  ( ph  ->  -.  ( `' F " ( _V  \  1o ) )  =  (/) )
36 cnvimass 5286 . . . . . . . . 9  |-  ( `' F " ( _V 
\  1o ) ) 
C_  dom  F
37 fdm 5660 . . . . . . . . . 10  |-  ( F : B --> A  ->  dom  F  =  B )
3829, 37syl 16 . . . . . . . . 9  |-  ( ph  ->  dom  F  =  B )
3936, 38syl5sseq 3482 . . . . . . . 8  |-  ( ph  ->  ( `' F "
( _V  \  1o ) )  C_  B
)
4022, 39ssexd 4529 . . . . . . 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 8058 . . . . . . . . 9  |-  ( ph  ->  F  e.  S )
4421, 8, 22, 41, 43cantnfclOLD 8051 . . . . . . . 8  |-  ( ph  ->  (  _E  We  ( `' F " ( _V 
\  1o ) )  /\  dom  O  e. 
om ) )
4544simpld 457 . . . . . . 7  |-  ( ph  ->  _E  We  ( `' F " ( _V 
\  1o ) ) )
4641oien 7900 . . . . . . 7  |-  ( ( ( `' F "
( _V  \  1o ) )  e.  _V  /\  _E  We  ( `' F " ( _V 
\  1o ) ) )  ->  dom  O  ~~  ( `' F " ( _V 
\  1o ) ) )
4740, 45, 46syl2anc 659 . . . . . 6  |-  ( ph  ->  dom  O  ~~  ( `' F " ( _V 
\  1o ) ) )
48 breq1 4387 . . . . . . 7  |-  ( dom 
O  =  (/)  ->  ( dom  O  ~~  ( `' F " ( _V 
\  1o ) )  <->  (/)  ~~  ( `' F "
( _V  \  1o ) ) ) )
49 ensymb 7504 . . . . . . . 8  |-  ( (/)  ~~  ( `' F "
( _V  \  1o ) )  <->  ( `' F " ( _V  \  1o ) )  ~~  (/) )
50 en0 7519 . . . . . . . 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 461 . . . . 5  |-  ( ph  ->  dom  O  e.  om )
56 nnlim 6634 . . . . 5  |-  ( dom 
O  e.  om  ->  -. 
Lim  dom  O )
5755, 56syl 16 . . . 4  |-  ( ph  ->  -.  Lim  dom  O
)
58 ioran 488 . . . 4  |-  ( -.  ( dom  O  =  (/)  \/  Lim  dom  O
)  <->  ( -.  dom  O  =  (/)  /\  -.  Lim  dom 
O ) )
5954, 57, 58sylanbrc 662 . . 3  |-  ( ph  ->  -.  ( dom  O  =  (/)  \/  Lim  dom  O ) )
60 nnord 6629 . . . 4  |-  ( dom 
O  e.  om  ->  Ord 
dom  O )
61 unizlim 4925 . . . 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 299 . 2  |-  ( ph  ->  -.  dom  O  = 
U. dom  O )
64 orduniorsuc 6586 . . . 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 375 . 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 366    /\ wa 367    = wceq 1399    e. wcel 1836    =/= wne 2591   _Vcvv 3051    \ cdif 3403    C_ wss 3406   (/)c0 3728   ifcif 3874   U.cuni 4180   class class class wbr 4384    |-> cmpt 4442    _E cep 4720    We wwe 4768   Ord word 4808   Oncon0 4809   Lim wlim 4810   suc csuc 4811   `'ccnv 4929   dom cdm 4930   "cima 4933    Fn wfn 5508   -->wf 5509   ` cfv 5513  (class class class)co 6218   omcom 6621   1oc1o 7063    ~~ cen 7454   Fincfn 7457  OrdIsocoi 7871   CNF ccnf 8013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-rep 4495  ax-sep 4505  ax-nul 4513  ax-pow 4560  ax-pr 4618  ax-un 6513
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-fal 1405  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2236  df-mo 2237  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-ral 2751  df-rex 2752  df-reu 2753  df-rmo 2754  df-rab 2755  df-v 3053  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3729  df-if 3875  df-pw 3946  df-sn 3962  df-pr 3964  df-tp 3966  df-op 3968  df-uni 4181  df-int 4217  df-iun 4262  df-br 4385  df-opab 4443  df-mpt 4444  df-tr 4478  df-eprel 4722  df-id 4726  df-po 4731  df-so 4732  df-fr 4769  df-se 4770  df-we 4771  df-ord 4812  df-on 4813  df-lim 4814  df-suc 4815  df-xp 4936  df-rel 4937  df-cnv 4938  df-co 4939  df-dm 4940  df-rn 4941  df-res 4942  df-ima 4943  df-iota 5477  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-isom 5522  df-riota 6180  df-ov 6221  df-oprab 6222  df-mpt2 6223  df-om 6622  df-1st 6721  df-2nd 6722  df-supp 6840  df-recs 6982  df-rdg 7016  df-seqom 7053  df-1o 7070  df-oadd 7074  df-er 7251  df-map 7362  df-en 7458  df-dom 7459  df-sdom 7460  df-fin 7461  df-fsupp 7767  df-oi 7872  df-cnf 8014
This theorem is referenced by:  cantnfp1lem3OLD  8060
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