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Theorem cantnfp1lem2 7627
Description: Lemma for cantnfp1 7629. (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 3737 . . . . . . . . . . 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 5796 . . . . . . . . . 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 4598 . . . . . . . . . . . 12  |-  ( ( A  e.  On  /\  Y  e.  A )  ->  Y  e.  On )
108, 2, 9syl2anc 643 . . . . . . . . . . 11  |-  ( ph  ->  Y  e.  On )
11 on0eln0 4628 . . . . . . . . . . 11  |-  ( Y  e.  On  ->  ( (/) 
e.  Y  <->  Y  =/=  (/) ) )
1210, 11syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( (/)  e.  Y  <->  Y  =/=  (/) ) )
137, 12mpbid 202 . . . . . . . . 9  |-  ( ph  ->  Y  =/=  (/) )
146, 13eqnetrd 2616 . . . . . . . 8  |-  ( ph  ->  ( F `  X
)  =/=  (/) )
15 fvex 5734 . . . . . . . . 9  |-  ( F `
 X )  e. 
_V
16 dif1o 6736 . . . . . . . . 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 7613 . . . . . . . . . . . . 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 5862 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  B )  ->  ( G `  t )  e.  A )
27 ifcl 3767 . . . . . . . . . 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 5885 . . . . . . . 8  |-  ( ph  ->  F : B --> A )
30 ffn 5583 . . . . . . . 8  |-  ( F : B --> A  ->  F  Fn  B )
31 elpreima 5842 . . . . . . . 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 3625 . . . . . 6  |-  ( X  e.  ( `' F " ( _V  \  1o ) )  ->  -.  ( `' F " ( _V 
\  1o ) )  =  (/) )
3533, 34syl 16 . . . . 5  |-  ( ph  ->  -.  ( `' F " ( _V  \  1o ) )  =  (/) )
36 cnvimass 5216 . . . . . . . . 9  |-  ( `' F " ( _V 
\  1o ) ) 
C_  dom  F
37 fdm 5587 . . . . . . . . . 10  |-  ( F : B --> A  ->  dom  F  =  B )
3829, 37syl 16 . . . . . . . . 9  |-  ( ph  ->  dom  F  =  B )
3936, 38syl5sseq 3388 . . . . . . . 8  |-  ( ph  ->  ( `' F "
( _V  \  1o ) )  C_  B
)
4022, 39ssexd 4342 . . . . . . 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 7626 . . . . . . . . 9  |-  ( ph  ->  F  e.  S )
4421, 8, 22, 41, 43cantnfcl 7614 . . . . . . . 8  |-  ( ph  ->  (  _E  We  ( `' F " ( _V 
\  1o ) )  /\  dom  O  e. 
om ) )
4544simpld 446 . . . . . . 7  |-  ( ph  ->  _E  We  ( `' F " ( _V 
\  1o ) ) )
4641oien 7499 . . . . . . 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 4207 . . . . . . 7  |-  ( dom 
O  =  (/)  ->  ( dom  O  ~~  ( `' F " ( _V 
\  1o ) )  <->  (/)  ~~  ( `' F "
( _V  \  1o ) ) ) )
49 ensymb 7147 . . . . . . . 8  |-  ( (/)  ~~  ( `' F "
( _V  \  1o ) )  <->  ( `' F " ( _V  \  1o ) )  ~~  (/) )
50 en0 7162 . . . . . . . 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 4850 . . . . 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 4845 . . . 4  |-  ( dom 
O  e.  om  ->  Ord 
dom  O )
61 unizlim 4690 . . . 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 4802 . . . 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 1652    e. wcel 1725    =/= wne 2598   _Vcvv 2948    \ cdif 3309    C_ wss 3312   (/)c0 3620   ifcif 3731   U.cuni 4007   class class class wbr 4204    e. cmpt 4258    _E cep 4484    We wwe 4532   Ord word 4572   Oncon0 4573   Lim wlim 4574   suc csuc 4575   omcom 4837   `'ccnv 4869   dom cdm 4870   "cima 4873    Fn wfn 5441   -->wf 5442   ` cfv 5446  (class class class)co 6073   1oc1o 6709    ~~ cen 7098   Fincfn 7101  OrdIsocoi 7470   CNF ccnf 7608
This theorem is referenced by:  cantnfp1lem3  7628
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541  df-recs 6625  df-rdg 6660  df-seqom 6697  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-oi 7471  df-cnf 7609
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