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Theorem cantnflem1b 8122
Description: Lemma for cantnf 8129. (Contributed by Mario Carneiro, 4-Jun-2015.) (Revised by AV, 2-Jul-2019.)
Hypotheses
Ref Expression
cantnfs.s  |-  S  =  dom  ( A CNF  B
)
cantnfs.a  |-  ( ph  ->  A  e.  On )
cantnfs.b  |-  ( ph  ->  B  e.  On )
oemapval.t  |-  T  =  { <. x ,  y
>.  |  E. z  e.  B  ( (
x `  z )  e.  ( y `  z
)  /\  A. w  e.  B  ( z  e.  w  ->  ( x `
 w )  =  ( y `  w
) ) ) }
oemapval.f  |-  ( ph  ->  F  e.  S )
oemapval.g  |-  ( ph  ->  G  e.  S )
oemapvali.r  |-  ( ph  ->  F T G )
oemapvali.x  |-  X  = 
U. { c  e.  B  |  ( F `
 c )  e.  ( G `  c
) }
cantnflem1.o  |-  O  = OrdIso
(  _E  ,  ( G supp  (/) ) )
Assertion
Ref Expression
cantnflem1b  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  X  C_  ( O `  u )
)
Distinct variable groups:    u, c, w, x, y, z, B    A, c, u, w, x, y, z    T, c, u    u, F, w, x, y, z    S, c, u, x, y, z    G, c, u, w, x, y, z    u, O, w, x, y, z    ph, u, x, y, z   
u, X, w, x, y, z    F, c    ph, c
Allowed substitution hints:    ph( w)    S( w)    T( x, y, z, w)    O( c)    X( c)

Proof of Theorem cantnflem1b
StepHypRef Expression
1 simprr 757 . . . 4  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( `' O `  X )  C_  u
)
2 cantnflem1.o . . . . . . . 8  |-  O  = OrdIso
(  _E  ,  ( G supp  (/) ) )
32oicl 7972 . . . . . . 7  |-  Ord  dom  O
4 cantnfs.b . . . . . . . . . . . 12  |-  ( ph  ->  B  e.  On )
5 suppssdm 6930 . . . . . . . . . . . . 13  |-  ( G supp  (/) )  C_  dom  G
6 oemapval.g . . . . . . . . . . . . . . . 16  |-  ( ph  ->  G  e.  S )
7 cantnfs.s . . . . . . . . . . . . . . . . 17  |-  S  =  dom  ( A CNF  B
)
8 cantnfs.a . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  A  e.  On )
97, 8, 4cantnfs 8102 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( G  e.  S  <->  ( G : B --> A  /\  G finSupp 
(/) ) ) )
106, 9mpbid 210 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( G : B --> A  /\  G finSupp  (/) ) )
1110simpld 459 . . . . . . . . . . . . . 14  |-  ( ph  ->  G : B --> A )
12 fdm 5741 . . . . . . . . . . . . . 14  |-  ( G : B --> A  ->  dom  G  =  B )
1311, 12syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  dom  G  =  B )
145, 13syl5sseq 3547 . . . . . . . . . . . 12  |-  ( ph  ->  ( G supp  (/) )  C_  B )
154, 14ssexd 4603 . . . . . . . . . . 11  |-  ( ph  ->  ( G supp  (/) )  e. 
_V )
167, 8, 4, 2, 6cantnfcl 8103 . . . . . . . . . . . 12  |-  ( ph  ->  (  _E  We  ( G supp 
(/) )  /\  dom  O  e.  om ) )
1716simpld 459 . . . . . . . . . . 11  |-  ( ph  ->  _E  We  ( G supp  (/) ) )
182oiiso 7980 . . . . . . . . . . 11  |-  ( ( ( G supp  (/) )  e. 
_V  /\  _E  We  ( G supp  (/) ) )  ->  O  Isom  _E  ,  _E  ( dom  O , 
( G supp  (/) ) ) )
1915, 17, 18syl2anc 661 . . . . . . . . . 10  |-  ( ph  ->  O  Isom  _E  ,  _E  ( dom  O ,  ( G supp  (/) ) ) )
20 isof1o 6222 . . . . . . . . . 10  |-  ( O 
Isom  _E  ,  _E  ( dom  O ,  ( G supp  (/) ) )  ->  O : dom  O -1-1-onto-> ( G supp  (/) ) )
2119, 20syl 16 . . . . . . . . 9  |-  ( ph  ->  O : dom  O -1-1-onto-> ( G supp 
(/) ) )
22 f1ocnv 5834 . . . . . . . . 9  |-  ( O : dom  O -1-1-onto-> ( G supp  (/) )  ->  `' O : ( G supp  (/) ) -1-1-onto-> dom  O
)
23 f1of 5822 . . . . . . . . 9  |-  ( `' O : ( G supp  (/) ) -1-1-onto-> dom  O  ->  `' O : ( G supp  (/) ) --> dom 
O )
2421, 22, 233syl 20 . . . . . . . 8  |-  ( ph  ->  `' O : ( G supp  (/) ) --> dom  O )
25 oemapval.t . . . . . . . . 9  |-  T  =  { <. x ,  y
>.  |  E. z  e.  B  ( (
x `  z )  e.  ( y `  z
)  /\  A. w  e.  B  ( z  e.  w  ->  ( x `
 w )  =  ( y `  w
) ) ) }
26 oemapval.f . . . . . . . . 9  |-  ( ph  ->  F  e.  S )
27 oemapvali.r . . . . . . . . 9  |-  ( ph  ->  F T G )
28 oemapvali.x . . . . . . . . 9  |-  X  = 
U. { c  e.  B  |  ( F `
 c )  e.  ( G `  c
) }
297, 8, 4, 25, 26, 6, 27, 28cantnflem1a 8121 . . . . . . . 8  |-  ( ph  ->  X  e.  ( G supp  (/) ) )
3024, 29ffvelrnd 6033 . . . . . . 7  |-  ( ph  ->  ( `' O `  X )  e.  dom  O )
31 ordelon 4911 . . . . . . 7  |-  ( ( Ord  dom  O  /\  ( `' O `  X )  e.  dom  O )  ->  ( `' O `  X )  e.  On )
323, 30, 31sylancr 663 . . . . . 6  |-  ( ph  ->  ( `' O `  X )  e.  On )
3332adantr 465 . . . . 5  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( `' O `  X )  e.  On )
343a1i 11 . . . . . . . 8  |-  ( ph  ->  Ord  dom  O )
35 ordelon 4911 . . . . . . . 8  |-  ( ( Ord  dom  O  /\  suc  u  e.  dom  O
)  ->  suc  u  e.  On )
3634, 35sylan 471 . . . . . . 7  |-  ( (
ph  /\  suc  u  e. 
dom  O )  ->  suc  u  e.  On )
37 sucelon 6651 . . . . . . 7  |-  ( u  e.  On  <->  suc  u  e.  On )
3836, 37sylibr 212 . . . . . 6  |-  ( (
ph  /\  suc  u  e. 
dom  O )  ->  u  e.  On )
3938adantrr 716 . . . . 5  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  u  e.  On )
40 ontri1 4921 . . . . 5  |-  ( ( ( `' O `  X )  e.  On  /\  u  e.  On )  ->  ( ( `' O `  X ) 
C_  u  <->  -.  u  e.  ( `' O `  X ) ) )
4133, 39, 40syl2anc 661 . . . 4  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( ( `' O `  X ) 
C_  u  <->  -.  u  e.  ( `' O `  X ) ) )
421, 41mpbid 210 . . 3  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  -.  u  e.  ( `' O `  X ) )
4319adantr 465 . . . . . 6  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  O  Isom  _E  ,  _E  ( dom  O , 
( G supp  (/) ) ) )
44 ordtr 4901 . . . . . . . 8  |-  ( Ord 
dom  O  ->  Tr  dom  O )
453, 44mp1i 12 . . . . . . 7  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  Tr  dom  O )
46 simprl 756 . . . . . . 7  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  suc  u  e.  dom  O )
47 trsuc 4971 . . . . . . 7  |-  ( ( Tr  dom  O  /\  suc  u  e.  dom  O
)  ->  u  e.  dom  O )
4845, 46, 47syl2anc 661 . . . . . 6  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  u  e.  dom  O )
4930adantr 465 . . . . . 6  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( `' O `  X )  e.  dom  O )
50 isorel 6223 . . . . . 6  |-  ( ( O  Isom  _E  ,  _E  ( dom  O ,  ( G supp  (/) ) )  /\  ( u  e.  dom  O  /\  ( `' O `  X )  e.  dom  O ) )  ->  (
u  _E  ( `' O `  X )  <-> 
( O `  u
)  _E  ( O `
 ( `' O `  X ) ) ) )
5143, 48, 49, 50syl12anc 1226 . . . . 5  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( u  _E  ( `' O `  X )  <->  ( O `  u )  _E  ( O `  ( `' O `  X )
) ) )
52 fvex 5882 . . . . . 6  |-  ( `' O `  X )  e.  _V
5352epelc 4802 . . . . 5  |-  ( u  _E  ( `' O `  X )  <->  u  e.  ( `' O `  X ) )
54 fvex 5882 . . . . . 6  |-  ( O `
 ( `' O `  X ) )  e. 
_V
5554epelc 4802 . . . . 5  |-  ( ( O `  u )  _E  ( O `  ( `' O `  X ) )  <->  ( O `  u )  e.  ( O `  ( `' O `  X ) ) )
5651, 53, 553bitr3g 287 . . . 4  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( u  e.  ( `' O `  X )  <->  ( O `  u )  e.  ( O `  ( `' O `  X ) ) ) )
57 f1ocnvfv2 6184 . . . . . . 7  |-  ( ( O : dom  O -1-1-onto-> ( G supp 
(/) )  /\  X  e.  ( G supp  (/) ) )  ->  ( O `  ( `' O `  X ) )  =  X )
5821, 29, 57syl2anc 661 . . . . . 6  |-  ( ph  ->  ( O `  ( `' O `  X ) )  =  X )
5958adantr 465 . . . . 5  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( O `  ( `' O `  X ) )  =  X )
6059eleq2d 2527 . . . 4  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( ( O `
 u )  e.  ( O `  ( `' O `  X ) )  <->  ( O `  u )  e.  X
) )
6156, 60bitrd 253 . . 3  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( u  e.  ( `' O `  X )  <->  ( O `  u )  e.  X
) )
6242, 61mtbid 300 . 2  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  -.  ( O `  u )  e.  X
)
637, 8, 4, 25, 26, 6, 27, 28oemapvali 8120 . . . . . 6  |-  ( ph  ->  ( X  e.  B  /\  ( F `  X
)  e.  ( G `
 X )  /\  A. w  e.  B  ( X  e.  w  -> 
( F `  w
)  =  ( G `
 w ) ) ) )
6463simp1d 1008 . . . . 5  |-  ( ph  ->  X  e.  B )
65 onelon 4912 . . . . 5  |-  ( ( B  e.  On  /\  X  e.  B )  ->  X  e.  On )
664, 64, 65syl2anc 661 . . . 4  |-  ( ph  ->  X  e.  On )
6766adantr 465 . . 3  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  X  e.  On )
684adantr 465 . . . 4  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  B  e.  On )
6914adantr 465 . . . . 5  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( G supp  (/) )  C_  B )
702oif 7973 . . . . . . 7  |-  O : dom  O --> ( G supp  (/) )
7170ffvelrni 6031 . . . . . 6  |-  ( u  e.  dom  O  -> 
( O `  u
)  e.  ( G supp  (/) ) )
7248, 71syl 16 . . . . 5  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( O `  u )  e.  ( G supp  (/) ) )
7369, 72sseldd 3500 . . . 4  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( O `  u )  e.  B
)
74 onelon 4912 . . . 4  |-  ( ( B  e.  On  /\  ( O `  u )  e.  B )  -> 
( O `  u
)  e.  On )
7568, 73, 74syl2anc 661 . . 3  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( O `  u )  e.  On )
76 ontri1 4921 . . 3  |-  ( ( X  e.  On  /\  ( O `  u )  e.  On )  -> 
( X  C_  ( O `  u )  <->  -.  ( O `  u
)  e.  X ) )
7767, 75, 76syl2anc 661 . 2  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( X  C_  ( O `  u )  <->  -.  ( O `  u
)  e.  X ) )
7862, 77mpbird 232 1  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  X  C_  ( O `  u )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   A.wral 2807   E.wrex 2808   {crab 2811   _Vcvv 3109    C_ wss 3471   (/)c0 3793   U.cuni 4251   class class class wbr 4456   {copab 4514   Tr wtr 4550    _E cep 4798    We wwe 4846   Ord word 4886   Oncon0 4887   suc csuc 4889   `'ccnv 5007   dom cdm 5008   -->wf 5590   -1-1-onto->wf1o 5593   ` cfv 5594    Isom wiso 5595  (class class class)co 6296   omcom 6699   supp csupp 6917   finSupp cfsupp 7847  OrdIsocoi 7952   CNF ccnf 8095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-supp 6918  df-recs 7060  df-rdg 7094  df-seqom 7131  df-1o 7148  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-oi 7953  df-cnf 8096
This theorem is referenced by:  cantnflem1c  8123
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