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Theorem cantnfp1lem2 7569
Description: Lemma for cantnfp1 7571. (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 3689 . . . . . . . . . . 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 5744 . . . . . . . . . 10  |-  ( ( X  e.  B  /\  Y  e.  A )  ->  ( F `  X
)  =  Y )
61, 2, 5syl2anc 643 . . . . . . . . 9  |-  ( ph  ->  ( F `  X
)  =  Y )
7 cantnfp1.8 . . . . . . . . . 10  |-  ( ph  -> 
(/)  e.  Y )
8 cantnfs.2 . . . . . . . . . . . 12  |-  ( ph  ->  A  e.  On )
9 onelon 4548 . . . . . . . . . . . 12  |-  ( ( A  e.  On  /\  Y  e.  A )  ->  Y  e.  On )
108, 2, 9syl2anc 643 . . . . . . . . . . 11  |-  ( ph  ->  Y  e.  On )
11 on0eln0 4578 . . . . . . . . . . 11  |-  ( Y  e.  On  ->  ( (/) 
e.  Y  <->  Y  =/=  (/) ) )
1210, 11syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( (/)  e.  Y  <->  Y  =/=  (/) ) )
137, 12mpbid 202 . . . . . . . . 9  |-  ( ph  ->  Y  =/=  (/) )
146, 13eqnetrd 2569 . . . . . . . 8  |-  ( ph  ->  ( F `  X
)  =/=  (/) )
15 fvex 5683 . . . . . . . . 9  |-  ( F `
 X )  e. 
_V
16 dif1o 6681 . . . . . . . . 9  |-  ( ( F `  X )  e.  ( _V  \  1o )  <->  ( ( F `
 X )  e. 
_V  /\  ( F `  X )  =/=  (/) ) )
1715, 16mpbiran 885 . . . . . . . 8  |-  ( ( F `  X )  e.  ( _V  \  1o )  <->  ( F `  X )  =/=  (/) )
1814, 17sylibr 204 . . . . . . 7  |-  ( ph  ->  ( F `  X
)  e.  ( _V 
\  1o ) )
192adantr 452 . . . . . . . . . 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 7555 . . . . . . . . . . . . 13  |-  ( ph  ->  ( G  e.  S  <->  ( G : B --> A  /\  ( `' G " ( _V 
\  1o ) )  e.  Fin ) ) )
2420, 23mpbid 202 . . . . . . . . . . . 12  |-  ( ph  ->  ( G : B --> A  /\  ( `' G " ( _V  \  1o ) )  e.  Fin ) )
2524simpld 446 . . . . . . . . . . 11  |-  ( ph  ->  G : B --> A )
2625ffvelrnda 5810 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  B )  ->  ( G `  t )  e.  A )
27 ifcl 3719 . . . . . . . . . 10  |-  ( ( Y  e.  A  /\  ( G `  t )  e.  A )  ->  if ( t  =  X ,  Y ,  ( G `  t ) )  e.  A )
2819, 26, 27syl2anc 643 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  B )  ->  if ( t  =  X ,  Y ,  ( G `  t ) )  e.  A )
2928, 4fmptd 5833 . . . . . . . 8  |-  ( ph  ->  F : B --> A )
30 ffn 5532 . . . . . . . 8  |-  ( F : B --> A  ->  F  Fn  B )
31 elpreima 5790 . . . . . . . 8  |-  ( F  Fn  B  ->  ( X  e.  ( `' F " ( _V  \  1o ) )  <->  ( X  e.  B  /\  ( F `  X )  e.  ( _V  \  1o ) ) ) )
3229, 30, 313syl 19 . . . . . . 7  |-  ( ph  ->  ( X  e.  ( `' F " ( _V 
\  1o ) )  <-> 
( X  e.  B  /\  ( F `  X
)  e.  ( _V 
\  1o ) ) ) )
331, 18, 32mpbir2and 889 . . . . . 6  |-  ( ph  ->  X  e.  ( `' F " ( _V 
\  1o ) ) )
34 n0i 3577 . . . . . 6  |-  ( X  e.  ( `' F " ( _V  \  1o ) )  ->  -.  ( `' F " ( _V 
\  1o ) )  =  (/) )
3533, 34syl 16 . . . . 5  |-  ( ph  ->  -.  ( `' F " ( _V  \  1o ) )  =  (/) )
36 cnvimass 5165 . . . . . . . . 9  |-  ( `' F " ( _V 
\  1o ) ) 
C_  dom  F
37 fdm 5536 . . . . . . . . . 10  |-  ( F : B --> A  ->  dom  F  =  B )
3829, 37syl 16 . . . . . . . . 9  |-  ( ph  ->  dom  F  =  B )
3936, 38syl5sseq 3340 . . . . . . . 8  |-  ( ph  ->  ( `' F "
( _V  \  1o ) )  C_  B
)
4022, 39ssexd 4292 . . . . . . 7  |-  ( ph  ->  ( `' F "
( _V  \  1o ) )  e.  _V )
41 cantnfp1.o . . . . . . . . 9  |-  O  = OrdIso
(  _E  ,  ( `' F " ( _V 
\  1o ) ) )
42 cantnfp1.7 . . . . . . . . . 10  |-  ( ph  ->  ( `' G "
( _V  \  1o ) )  C_  X
)
4321, 8, 22, 20, 1, 2, 42, 4cantnfp1lem1 7568 . . . . . . . . 9  |-  ( ph  ->  F  e.  S )
4421, 8, 22, 41, 43cantnfcl 7556 . . . . . . . 8  |-  ( ph  ->  (  _E  We  ( `' F " ( _V 
\  1o ) )  /\  dom  O  e. 
om ) )
4544simpld 446 . . . . . . 7  |-  ( ph  ->  _E  We  ( `' F " ( _V 
\  1o ) ) )
4641oien 7441 . . . . . . 7  |-  ( ( ( `' F "
( _V  \  1o ) )  e.  _V  /\  _E  We  ( `' F " ( _V 
\  1o ) ) )  ->  dom  O  ~~  ( `' F " ( _V 
\  1o ) ) )
4740, 45, 46syl2anc 643 . . . . . 6  |-  ( ph  ->  dom  O  ~~  ( `' F " ( _V 
\  1o ) ) )
48 breq1 4157 . . . . . . 7  |-  ( dom 
O  =  (/)  ->  ( dom  O  ~~  ( `' F " ( _V 
\  1o ) )  <->  (/)  ~~  ( `' F "
( _V  \  1o ) ) ) )
49 ensymb 7092 . . . . . . . 8  |-  ( (/)  ~~  ( `' F "
( _V  \  1o ) )  <->  ( `' F " ( _V  \  1o ) )  ~~  (/) )
50 en0 7107 . . . . . . . 8  |-  ( ( `' F " ( _V 
\  1o ) ) 
~~  (/)  <->  ( `' F " ( _V  \  1o ) )  =  (/) )
5149, 50bitri 241 . . . . . . 7  |-  ( (/)  ~~  ( `' F "
( _V  \  1o ) )  <->  ( `' F " ( _V  \  1o ) )  =  (/) )
5248, 51syl6bb 253 . . . . . 6  |-  ( dom 
O  =  (/)  ->  ( dom  O  ~~  ( `' F " ( _V 
\  1o ) )  <-> 
( `' F "
( _V  \  1o ) )  =  (/) ) )
5347, 52syl5ibcom 212 . . . . 5  |-  ( ph  ->  ( dom  O  =  (/)  ->  ( `' F " ( _V  \  1o ) )  =  (/) ) )
5435, 53mtod 170 . . . 4  |-  ( ph  ->  -.  dom  O  =  (/) )
5544simprd 450 . . . . 5  |-  ( ph  ->  dom  O  e.  om )
56 nnlim 4799 . . . . 5  |-  ( dom 
O  e.  om  ->  -. 
Lim  dom  O )
5755, 56syl 16 . . . 4  |-  ( ph  ->  -.  Lim  dom  O
)
58 ioran 477 . . . 4  |-  ( -.  ( dom  O  =  (/)  \/  Lim  dom  O
)  <->  ( -.  dom  O  =  (/)  /\  -.  Lim  dom 
O ) )
5954, 57, 58sylanbrc 646 . . 3  |-  ( ph  ->  -.  ( dom  O  =  (/)  \/  Lim  dom  O ) )
60 nnord 4794 . . . 4  |-  ( dom 
O  e.  om  ->  Ord 
dom  O )
61 unizlim 4639 . . . 4  |-  ( Ord 
dom  O  ->  ( dom 
O  =  U. dom  O  <-> 
( dom  O  =  (/) 
\/  Lim  dom  O ) ) )
6255, 60, 613syl 19 . . 3  |-  ( ph  ->  ( dom  O  = 
U. dom  O  <->  ( dom  O  =  (/)  \/  Lim  dom 
O ) ) )
6359, 62mtbird 293 . 2  |-  ( ph  ->  -.  dom  O  = 
U. dom  O )
64 orduniorsuc 4751 . . . 4  |-  ( Ord 
dom  O  ->  ( dom 
O  =  U. dom  O  \/  dom  O  =  suc  U. dom  O
) )
6555, 60, 643syl 19 . . 3  |-  ( ph  ->  ( dom  O  = 
U. dom  O  \/  dom  O  =  suc  U. dom  O ) )
6665ord 367 . 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 set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2551   _Vcvv 2900    \ cdif 3261    C_ wss 3264   (/)c0 3572   ifcif 3683   U.cuni 3958   class class class wbr 4154    e. cmpt 4208    _E cep 4434    We wwe 4482   Ord word 4522   Oncon0 4523   Lim wlim 4524   suc csuc 4525   omcom 4786   `'ccnv 4818   dom cdm 4819   "cima 4822    Fn wfn 5390   -->wf 5391   ` cfv 5395  (class class class)co 6021   1oc1o 6654    ~~ cen 7043   Fincfn 7046  OrdIsocoi 7412   CNF ccnf 7550
This theorem is referenced by:  cantnfp1lem3  7570
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-se 4484  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-isom 5404  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-riota 6486  df-recs 6570  df-rdg 6605  df-seqom 6642  df-1o 6661  df-oadd 6665  df-er 6842  df-map 6957  df-en 7047  df-dom 7048  df-sdom 7049  df-fin 7050  df-oi 7413  df-cnf 7551
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