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Theorem cantnflem1b 8188
Description: Lemma for cantnf 8195. (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 765 . . . 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 8041 . . . . . . 7  |-  Ord  dom  O
4 cantnfs.b . . . . . . . . . . . 12  |-  ( ph  ->  B  e.  On )
5 suppssdm 6924 . . . . . . . . . . . . 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 8168 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( G  e.  S  <->  ( G : B --> A  /\  G finSupp 
(/) ) ) )
106, 9mpbid 214 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( G : B --> A  /\  G finSupp  (/) ) )
1110simpld 461 . . . . . . . . . . . . . 14  |-  ( ph  ->  G : B --> A )
12 fdm 5731 . . . . . . . . . . . . . 14  |-  ( G : B --> A  ->  dom  G  =  B )
1311, 12syl 17 . . . . . . . . . . . . 13  |-  ( ph  ->  dom  G  =  B )
145, 13syl5sseq 3479 . . . . . . . . . . . 12  |-  ( ph  ->  ( G supp  (/) )  C_  B )
154, 14ssexd 4549 . . . . . . . . . . 11  |-  ( ph  ->  ( G supp  (/) )  e. 
_V )
167, 8, 4, 2, 6cantnfcl 8169 . . . . . . . . . . . 12  |-  ( ph  ->  (  _E  We  ( G supp 
(/) )  /\  dom  O  e.  om ) )
1716simpld 461 . . . . . . . . . . 11  |-  ( ph  ->  _E  We  ( G supp  (/) ) )
182oiiso 8049 . . . . . . . . . . 11  |-  ( ( ( G supp  (/) )  e. 
_V  /\  _E  We  ( G supp  (/) ) )  ->  O  Isom  _E  ,  _E  ( dom  O , 
( G supp  (/) ) ) )
1915, 17, 18syl2anc 666 . . . . . . . . . 10  |-  ( ph  ->  O  Isom  _E  ,  _E  ( dom  O ,  ( G supp  (/) ) ) )
20 isof1o 6214 . . . . . . . . . 10  |-  ( O 
Isom  _E  ,  _E  ( dom  O ,  ( G supp  (/) ) )  ->  O : dom  O -1-1-onto-> ( G supp  (/) ) )
2119, 20syl 17 . . . . . . . . 9  |-  ( ph  ->  O : dom  O -1-1-onto-> ( G supp 
(/) ) )
22 f1ocnv 5824 . . . . . . . . 9  |-  ( O : dom  O -1-1-onto-> ( G supp  (/) )  ->  `' O : ( G supp  (/) ) -1-1-onto-> dom  O
)
23 f1of 5812 . . . . . . . . 9  |-  ( `' O : ( G supp  (/) ) -1-1-onto-> dom  O  ->  `' O : ( G supp  (/) ) --> dom 
O )
2421, 22, 233syl 18 . . . . . . . 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 8187 . . . . . . . 8  |-  ( ph  ->  X  e.  ( G supp  (/) ) )
3024, 29ffvelrnd 6021 . . . . . . 7  |-  ( ph  ->  ( `' O `  X )  e.  dom  O )
31 ordelon 5446 . . . . . . 7  |-  ( ( Ord  dom  O  /\  ( `' O `  X )  e.  dom  O )  ->  ( `' O `  X )  e.  On )
323, 30, 31sylancr 668 . . . . . 6  |-  ( ph  ->  ( `' O `  X )  e.  On )
3332adantr 467 . . . . 5  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( `' O `  X )  e.  On )
343a1i 11 . . . . . . . 8  |-  ( ph  ->  Ord  dom  O )
35 ordelon 5446 . . . . . . . 8  |-  ( ( Ord  dom  O  /\  suc  u  e.  dom  O
)  ->  suc  u  e.  On )
3634, 35sylan 474 . . . . . . 7  |-  ( (
ph  /\  suc  u  e. 
dom  O )  ->  suc  u  e.  On )
37 sucelon 6641 . . . . . . 7  |-  ( u  e.  On  <->  suc  u  e.  On )
3836, 37sylibr 216 . . . . . 6  |-  ( (
ph  /\  suc  u  e. 
dom  O )  ->  u  e.  On )
3938adantrr 722 . . . . 5  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  u  e.  On )
40 ontri1 5456 . . . . 5  |-  ( ( ( `' O `  X )  e.  On  /\  u  e.  On )  ->  ( ( `' O `  X ) 
C_  u  <->  -.  u  e.  ( `' O `  X ) ) )
4133, 39, 40syl2anc 666 . . . 4  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( ( `' O `  X ) 
C_  u  <->  -.  u  e.  ( `' O `  X ) ) )
421, 41mpbid 214 . . 3  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  -.  u  e.  ( `' O `  X ) )
4319adantr 467 . . . . . 6  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  O  Isom  _E  ,  _E  ( dom  O , 
( G supp  (/) ) ) )
44 ordtr 5436 . . . . . . . 8  |-  ( Ord 
dom  O  ->  Tr  dom  O )
453, 44mp1i 13 . . . . . . 7  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  Tr  dom  O )
46 simprl 763 . . . . . . 7  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  suc  u  e.  dom  O )
47 trsuc 5506 . . . . . . 7  |-  ( ( Tr  dom  O  /\  suc  u  e.  dom  O
)  ->  u  e.  dom  O )
4845, 46, 47syl2anc 666 . . . . . 6  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  u  e.  dom  O )
4930adantr 467 . . . . . 6  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( `' O `  X )  e.  dom  O )
50 isorel 6215 . . . . . 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 1265 . . . . 5  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( u  _E  ( `' O `  X )  <->  ( O `  u )  _E  ( O `  ( `' O `  X )
) ) )
52 fvex 5873 . . . . . 6  |-  ( `' O `  X )  e.  _V
5352epelc 4746 . . . . 5  |-  ( u  _E  ( `' O `  X )  <->  u  e.  ( `' O `  X ) )
54 fvex 5873 . . . . . 6  |-  ( O `
 ( `' O `  X ) )  e. 
_V
5554epelc 4746 . . . . 5  |-  ( ( O `  u )  _E  ( O `  ( `' O `  X ) )  <->  ( O `  u )  e.  ( O `  ( `' O `  X ) ) )
5651, 53, 553bitr3g 291 . . . 4  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( u  e.  ( `' O `  X )  <->  ( O `  u )  e.  ( O `  ( `' O `  X ) ) ) )
57 f1ocnvfv2 6174 . . . . . . 7  |-  ( ( O : dom  O -1-1-onto-> ( G supp 
(/) )  /\  X  e.  ( G supp  (/) ) )  ->  ( O `  ( `' O `  X ) )  =  X )
5821, 29, 57syl2anc 666 . . . . . 6  |-  ( ph  ->  ( O `  ( `' O `  X ) )  =  X )
5958adantr 467 . . . . 5  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( O `  ( `' O `  X ) )  =  X )
6059eleq2d 2513 . . . 4  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( ( O `
 u )  e.  ( O `  ( `' O `  X ) )  <->  ( O `  u )  e.  X
) )
6156, 60bitrd 257 . . 3  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( u  e.  ( `' O `  X )  <->  ( O `  u )  e.  X
) )
6242, 61mtbid 302 . 2  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  -.  ( O `  u )  e.  X
)
637, 8, 4, 25, 26, 6, 27, 28oemapvali 8186 . . . . . 6  |-  ( ph  ->  ( X  e.  B  /\  ( F `  X
)  e.  ( G `
 X )  /\  A. w  e.  B  ( X  e.  w  -> 
( F `  w
)  =  ( G `
 w ) ) ) )
6463simp1d 1019 . . . . 5  |-  ( ph  ->  X  e.  B )
65 onelon 5447 . . . . 5  |-  ( ( B  e.  On  /\  X  e.  B )  ->  X  e.  On )
664, 64, 65syl2anc 666 . . . 4  |-  ( ph  ->  X  e.  On )
6766adantr 467 . . 3  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  X  e.  On )
684adantr 467 . . . 4  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  B  e.  On )
6914adantr 467 . . . . 5  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( G supp  (/) )  C_  B )
702oif 8042 . . . . . . 7  |-  O : dom  O --> ( G supp  (/) )
7170ffvelrni 6019 . . . . . 6  |-  ( u  e.  dom  O  -> 
( O `  u
)  e.  ( G supp  (/) ) )
7248, 71syl 17 . . . . 5  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( O `  u )  e.  ( G supp  (/) ) )
7369, 72sseldd 3432 . . . 4  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( O `  u )  e.  B
)
74 onelon 5447 . . . 4  |-  ( ( B  e.  On  /\  ( O `  u )  e.  B )  -> 
( O `  u
)  e.  On )
7568, 73, 74syl2anc 666 . . 3  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( O `  u )  e.  On )
76 ontri1 5456 . . 3  |-  ( ( X  e.  On  /\  ( O `  u )  e.  On )  -> 
( X  C_  ( O `  u )  <->  -.  ( O `  u
)  e.  X ) )
7767, 75, 76syl2anc 666 . 2  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  ( X  C_  ( O `  u )  <->  -.  ( O `  u
)  e.  X ) )
7862, 77mpbird 236 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 188    /\ wa 371    = wceq 1443    e. wcel 1886   A.wral 2736   E.wrex 2737   {crab 2740   _Vcvv 3044    C_ wss 3403   (/)c0 3730   U.cuni 4197   class class class wbr 4401   {copab 4459   Tr wtr 4496    _E cep 4742    We wwe 4791   `'ccnv 4832   dom cdm 4833   Ord word 5421   Oncon0 5422   suc csuc 5424   -->wf 5577   -1-1-onto->wf1o 5580   ` cfv 5581    Isom wiso 5582  (class class class)co 6288   omcom 6689   supp csupp 6911   finSupp cfsupp 7880  OrdIsocoi 8021   CNF ccnf 8163
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-fal 1449  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-se 4793  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-om 6690  df-supp 6912  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-seqom 7162  df-1o 7179  df-er 7360  df-map 7471  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-fsupp 7881  df-oi 8022  df-cnf 8164
This theorem is referenced by:  cantnflem1c  8189
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