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Theorem cantnfp1lem2 7381
Description: Lemma for cantnfp1 7383. (Contributed by Mario Carneiro, 28-May-2015.)
Hypotheses
Ref Expression
cantnfs.1  |-  S  =  dom  ( A CNF  B
)
cantnfs.2  |-  ( ph  ->  A  e.  On )
cantnfs.3  |-  ( ph  ->  B  e.  On )
cantnfp1.4  |-  ( ph  ->  G  e.  S )
cantnfp1.5  |-  ( ph  ->  X  e.  B )
cantnfp1.6  |-  ( ph  ->  Y  e.  A )
cantnfp1.7  |-  ( ph  ->  ( `' G "
( _V  \  1o ) )  C_  X
)
cantnfp1.f  |-  F  =  ( t  e.  B  |->  if ( t  =  X ,  Y , 
( G `  t
) ) )
cantnfp1.8  |-  ( ph  -> 
(/)  e.  Y )
cantnfp1.o  |-  O  = OrdIso
(  _E  ,  ( `' F " ( _V 
\  1o ) ) )
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 cantnfp1.5 . . . . . . 7  |-  ( ph  ->  X  e.  B )
2 cantnfp1.6 . . . . . . . . . 10  |-  ( ph  ->  Y  e.  A )
3 iftrue 3571 . . . . . . . . . . 11  |-  ( t  =  X  ->  if ( t  =  X ,  Y ,  ( G `  t ) )  =  Y )
4 cantnfp1.f . . . . . . . . . . 11  |-  F  =  ( t  e.  B  |->  if ( t  =  X ,  Y , 
( G `  t
) ) )
53, 4fvmptg 5600 . . . . . . . . . 10  |-  ( ( X  e.  B  /\  Y  e.  A )  ->  ( F `  X
)  =  Y )
61, 2, 5syl2anc 642 . . . . . . . . 9  |-  ( ph  ->  ( F `  X
)  =  Y )
7 cantnfp1.8 . . . . . . . . . 10  |-  ( ph  -> 
(/)  e.  Y )
8 cantnfs.2 . . . . . . . . . . . 12  |-  ( ph  ->  A  e.  On )
9 onelon 4417 . . . . . . . . . . . 12  |-  ( ( A  e.  On  /\  Y  e.  A )  ->  Y  e.  On )
108, 2, 9syl2anc 642 . . . . . . . . . . 11  |-  ( ph  ->  Y  e.  On )
11 on0eln0 4447 . . . . . . . . . . 11  |-  ( Y  e.  On  ->  ( (/) 
e.  Y  <->  Y  =/=  (/) ) )
1210, 11syl 15 . . . . . . . . . 10  |-  ( ph  ->  ( (/)  e.  Y  <->  Y  =/=  (/) ) )
137, 12mpbid 201 . . . . . . . . 9  |-  ( ph  ->  Y  =/=  (/) )
146, 13eqnetrd 2464 . . . . . . . 8  |-  ( ph  ->  ( F `  X
)  =/=  (/) )
15 fvex 5539 . . . . . . . . 9  |-  ( F `
 X )  e. 
_V
16 dif1o 6499 . . . . . . . . 9  |-  ( ( F `  X )  e.  ( _V  \  1o )  <->  ( ( F `
 X )  e. 
_V  /\  ( F `  X )  =/=  (/) ) )
1715, 16mpbiran 884 . . . . . . . 8  |-  ( ( F `  X )  e.  ( _V  \  1o )  <->  ( F `  X )  =/=  (/) )
1814, 17sylibr 203 . . . . . . 7  |-  ( ph  ->  ( F `  X
)  e.  ( _V 
\  1o ) )
192adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  B )  ->  Y  e.  A )
20 cantnfp1.4 . . . . . . . . . . . . 13  |-  ( ph  ->  G  e.  S )
21 cantnfs.1 . . . . . . . . . . . . . 14  |-  S  =  dom  ( A CNF  B
)
22 cantnfs.3 . . . . . . . . . . . . . 14  |-  ( ph  ->  B  e.  On )
2321, 8, 22cantnfs 7367 . . . . . . . . . . . . 13  |-  ( ph  ->  ( G  e.  S  <->  ( G : B --> A  /\  ( `' G " ( _V 
\  1o ) )  e.  Fin ) ) )
2420, 23mpbid 201 . . . . . . . . . . . 12  |-  ( ph  ->  ( G : B --> A  /\  ( `' G " ( _V  \  1o ) )  e.  Fin ) )
2524simpld 445 . . . . . . . . . . 11  |-  ( ph  ->  G : B --> A )
26 ffvelrn 5663 . . . . . . . . . . 11  |-  ( ( G : B --> A  /\  t  e.  B )  ->  ( G `  t
)  e.  A )
2725, 26sylan 457 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  B )  ->  ( G `  t )  e.  A )
28 ifcl 3601 . . . . . . . . . 10  |-  ( ( Y  e.  A  /\  ( G `  t )  e.  A )  ->  if ( t  =  X ,  Y ,  ( G `  t ) )  e.  A )
2919, 27, 28syl2anc 642 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  B )  ->  if ( t  =  X ,  Y ,  ( G `  t ) )  e.  A )
3029, 4fmptd 5684 . . . . . . . 8  |-  ( ph  ->  F : B --> A )
31 ffn 5389 . . . . . . . 8  |-  ( F : B --> A  ->  F  Fn  B )
32 elpreima 5645 . . . . . . . 8  |-  ( F  Fn  B  ->  ( X  e.  ( `' F " ( _V  \  1o ) )  <->  ( X  e.  B  /\  ( F `  X )  e.  ( _V  \  1o ) ) ) )
3330, 31, 323syl 18 . . . . . . 7  |-  ( ph  ->  ( X  e.  ( `' F " ( _V 
\  1o ) )  <-> 
( X  e.  B  /\  ( F `  X
)  e.  ( _V 
\  1o ) ) ) )
341, 18, 33mpbir2and 888 . . . . . 6  |-  ( ph  ->  X  e.  ( `' F " ( _V 
\  1o ) ) )
35 n0i 3460 . . . . . 6  |-  ( X  e.  ( `' F " ( _V  \  1o ) )  ->  -.  ( `' F " ( _V 
\  1o ) )  =  (/) )
3634, 35syl 15 . . . . 5  |-  ( ph  ->  -.  ( `' F " ( _V  \  1o ) )  =  (/) )
37 cnvimass 5033 . . . . . . . . 9  |-  ( `' F " ( _V 
\  1o ) ) 
C_  dom  F
38 fdm 5393 . . . . . . . . . 10  |-  ( F : B --> A  ->  dom  F  =  B )
3930, 38syl 15 . . . . . . . . 9  |-  ( ph  ->  dom  F  =  B )
4037, 39syl5sseq 3226 . . . . . . . 8  |-  ( ph  ->  ( `' F "
( _V  \  1o ) )  C_  B
)
41 ssexg 4160 . . . . . . . 8  |-  ( ( ( `' F "
( _V  \  1o ) )  C_  B  /\  B  e.  On )  ->  ( `' F " ( _V  \  1o ) )  e.  _V )
4240, 22, 41syl2anc 642 . . . . . . 7  |-  ( ph  ->  ( `' F "
( _V  \  1o ) )  e.  _V )
43 cantnfp1.o . . . . . . . . 9  |-  O  = OrdIso
(  _E  ,  ( `' F " ( _V 
\  1o ) ) )
44 cantnfp1.7 . . . . . . . . . 10  |-  ( ph  ->  ( `' G "
( _V  \  1o ) )  C_  X
)
4521, 8, 22, 20, 1, 2, 44, 4cantnfp1lem1 7380 . . . . . . . . 9  |-  ( ph  ->  F  e.  S )
4621, 8, 22, 43, 45cantnfcl 7368 . . . . . . . 8  |-  ( ph  ->  (  _E  We  ( `' F " ( _V 
\  1o ) )  /\  dom  O  e. 
om ) )
4746simpld 445 . . . . . . 7  |-  ( ph  ->  _E  We  ( `' F " ( _V 
\  1o ) ) )
4843oien 7253 . . . . . . 7  |-  ( ( ( `' F "
( _V  \  1o ) )  e.  _V  /\  _E  We  ( `' F " ( _V 
\  1o ) ) )  ->  dom  O  ~~  ( `' F " ( _V 
\  1o ) ) )
4942, 47, 48syl2anc 642 . . . . . 6  |-  ( ph  ->  dom  O  ~~  ( `' F " ( _V 
\  1o ) ) )
50 breq1 4026 . . . . . . 7  |-  ( dom 
O  =  (/)  ->  ( dom  O  ~~  ( `' F " ( _V 
\  1o ) )  <->  (/)  ~~  ( `' F "
( _V  \  1o ) ) ) )
51 ensymb 6909 . . . . . . . 8  |-  ( (/)  ~~  ( `' F "
( _V  \  1o ) )  <->  ( `' F " ( _V  \  1o ) )  ~~  (/) )
52 en0 6924 . . . . . . . 8  |-  ( ( `' F " ( _V 
\  1o ) ) 
~~  (/)  <->  ( `' F " ( _V  \  1o ) )  =  (/) )
5351, 52bitri 240 . . . . . . 7  |-  ( (/)  ~~  ( `' F "
( _V  \  1o ) )  <->  ( `' F " ( _V  \  1o ) )  =  (/) )
5450, 53syl6bb 252 . . . . . 6  |-  ( dom 
O  =  (/)  ->  ( dom  O  ~~  ( `' F " ( _V 
\  1o ) )  <-> 
( `' F "
( _V  \  1o ) )  =  (/) ) )
5549, 54syl5ibcom 211 . . . . 5  |-  ( ph  ->  ( dom  O  =  (/)  ->  ( `' F " ( _V  \  1o ) )  =  (/) ) )
5636, 55mtod 168 . . . 4  |-  ( ph  ->  -.  dom  O  =  (/) )
5746simprd 449 . . . . 5  |-  ( ph  ->  dom  O  e.  om )
58 nnlim 4669 . . . . 5  |-  ( dom 
O  e.  om  ->  -. 
Lim  dom  O )
5957, 58syl 15 . . . 4  |-  ( ph  ->  -.  Lim  dom  O
)
60 ioran 476 . . . 4  |-  ( -.  ( dom  O  =  (/)  \/  Lim  dom  O
)  <->  ( -.  dom  O  =  (/)  /\  -.  Lim  dom 
O ) )
6156, 59, 60sylanbrc 645 . . 3  |-  ( ph  ->  -.  ( dom  O  =  (/)  \/  Lim  dom  O ) )
62 nnord 4664 . . . 4  |-  ( dom 
O  e.  om  ->  Ord 
dom  O )
63 unizlim 4509 . . . 4  |-  ( Ord 
dom  O  ->  ( dom 
O  =  U. dom  O  <-> 
( dom  O  =  (/) 
\/  Lim  dom  O ) ) )
6457, 62, 633syl 18 . . 3  |-  ( ph  ->  ( dom  O  = 
U. dom  O  <->  ( dom  O  =  (/)  \/  Lim  dom 
O ) ) )
6561, 64mtbird 292 . 2  |-  ( ph  ->  -.  dom  O  = 
U. dom  O )
66 orduniorsuc 4621 . . . 4  |-  ( Ord 
dom  O  ->  ( dom 
O  =  U. dom  O  \/  dom  O  =  suc  U. dom  O
) )
6757, 62, 663syl 18 . . 3  |-  ( ph  ->  ( dom  O  = 
U. dom  O  \/  dom  O  =  suc  U. dom  O ) )
6867ord 366 . 2  |-  ( ph  ->  ( -.  dom  O  =  U. dom  O  ->  dom  O  =  suc  U. dom  O ) )
6965, 68mpd 14 1  |-  ( ph  ->  dom  O  =  suc  U.
dom  O )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788    \ cdif 3149    C_ wss 3152   (/)c0 3455   ifcif 3565   U.cuni 3827   class class class wbr 4023    e. cmpt 4077    _E cep 4303    We wwe 4351   Ord word 4391   Oncon0 4392   Lim wlim 4393   suc csuc 4394   omcom 4656   `'ccnv 4688   dom cdm 4689   "cima 4692    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   1oc1o 6472    ~~ cen 6860   Fincfn 6863  OrdIsocoi 7224   CNF ccnf 7362
This theorem is referenced by:  cantnfp1lem3  7382
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-recs 6388  df-rdg 6423  df-seqom 6460  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-oi 7225  df-cnf 7363
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