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Theorem cantnflem1cOLD 8161
Description: Lemma for cantnfOLD 8166. (Contributed by Mario Carneiro, 4-Jun-2015.)

Obsolete version of cantnflem1a 8136 as of 2-Jul-2019. (New usage is discouraged.) (Proof modification is discouraged.)

Hypotheses
Ref Expression
cantnfsOLD.1  |-  S  =  dom  ( A CNF  B
)
cantnfsOLD.2  |-  ( ph  ->  A  e.  On )
cantnfsOLD.3  |-  ( ph  ->  B  e.  On )
oemapvalOLD.t  |-  T  =  { <. x ,  y
>.  |  E. z  e.  B  ( (
x `  z )  e.  ( y `  z
)  /\  A. w  e.  B  ( z  e.  w  ->  ( x `
 w )  =  ( y `  w
) ) ) }
oemapvalOLD.3  |-  ( ph  ->  F  e.  S )
oemapvalOLD.4  |-  ( ph  ->  G  e.  S )
oemapvalOLD.5  |-  ( ph  ->  F T G )
oemapvalOLD.6  |-  X  = 
U. { c  e.  B  |  ( F `
 c )  e.  ( G `  c
) }
cantnflem1OLD.o  |-  O  = OrdIso
(  _E  ,  ( `' G " ( _V 
\  1o ) ) )
Assertion
Ref Expression
cantnflem1cOLD  |-  ( ( ( ( ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X )  C_  u
) )  /\  x  e.  B )  /\  (
( F `  x
)  =/=  (/)  /\  ( O `  u )  e.  x ) )  ->  x  e.  ( `' G " ( _V  \  1o ) ) )
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 cantnflem1cOLD
StepHypRef Expression
1 simplr 754 . 2  |-  ( ( ( ( ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X )  C_  u
) )  /\  x  e.  B )  /\  (
( F `  x
)  =/=  (/)  /\  ( O `  u )  e.  x ) )  ->  x  e.  B )
2 cantnfsOLD.1 . . . . . . . 8  |-  S  =  dom  ( A CNF  B
)
3 cantnfsOLD.2 . . . . . . . 8  |-  ( ph  ->  A  e.  On )
4 cantnfsOLD.3 . . . . . . . 8  |-  ( ph  ->  B  e.  On )
5 oemapvalOLD.t . . . . . . . 8  |-  T  =  { <. x ,  y
>.  |  E. z  e.  B  ( (
x `  z )  e.  ( y `  z
)  /\  A. w  e.  B  ( z  e.  w  ->  ( x `
 w )  =  ( y `  w
) ) ) }
6 oemapvalOLD.3 . . . . . . . 8  |-  ( ph  ->  F  e.  S )
7 oemapvalOLD.4 . . . . . . . 8  |-  ( ph  ->  G  e.  S )
8 oemapvalOLD.5 . . . . . . . 8  |-  ( ph  ->  F T G )
9 oemapvalOLD.6 . . . . . . . 8  |-  X  = 
U. { c  e.  B  |  ( F `
 c )  e.  ( G `  c
) }
102, 3, 4, 5, 6, 7, 8, 9oemapvali 8135 . . . . . . 7  |-  ( ph  ->  ( X  e.  B  /\  ( F `  X
)  e.  ( G `
 X )  /\  A. w  e.  B  ( X  e.  w  -> 
( F `  w
)  =  ( G `
 w ) ) ) )
1110simp3d 1011 . . . . . 6  |-  ( ph  ->  A. w  e.  B  ( X  e.  w  ->  ( F `  w
)  =  ( G `
 w ) ) )
1211ad3antrrr 728 . . . . 5  |-  ( ( ( ( ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X )  C_  u
) )  /\  x  e.  B )  /\  (
( F `  x
)  =/=  (/)  /\  ( O `  u )  e.  x ) )  ->  A. w  e.  B  ( X  e.  w  ->  ( F `  w
)  =  ( G `
 w ) ) )
13 cantnflem1OLD.o . . . . . . . 8  |-  O  = OrdIso
(  _E  ,  ( `' G " ( _V 
\  1o ) ) )
142, 3, 4, 5, 6, 7, 8, 9, 13cantnflem1bOLD 8160 . . . . . . 7  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  X  C_  ( O `  u )
)
1514ad2antrr 724 . . . . . 6  |-  ( ( ( ( ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X )  C_  u
) )  /\  x  e.  B )  /\  (
( F `  x
)  =/=  (/)  /\  ( O `  u )  e.  x ) )  ->  X  C_  ( O `  u ) )
16 simprr 758 . . . . . 6  |-  ( ( ( ( ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X )  C_  u
) )  /\  x  e.  B )  /\  (
( F `  x
)  =/=  (/)  /\  ( O `  u )  e.  x ) )  -> 
( O `  u
)  e.  x )
1710simp1d 1009 . . . . . . . . 9  |-  ( ph  ->  X  e.  B )
18 onelon 5435 . . . . . . . . 9  |-  ( ( B  e.  On  /\  X  e.  B )  ->  X  e.  On )
194, 17, 18syl2anc 659 . . . . . . . 8  |-  ( ph  ->  X  e.  On )
2019ad3antrrr 728 . . . . . . 7  |-  ( ( ( ( ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X )  C_  u
) )  /\  x  e.  B )  /\  (
( F `  x
)  =/=  (/)  /\  ( O `  u )  e.  x ) )  ->  X  e.  On )
21 onss 6608 . . . . . . . . . . 11  |-  ( B  e.  On  ->  B  C_  On )
224, 21syl 17 . . . . . . . . . 10  |-  ( ph  ->  B  C_  On )
2322sselda 3442 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  B )  ->  x  e.  On )
2423adantlr 713 . . . . . . . 8  |-  ( ( ( ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X )  C_  u
) )  /\  x  e.  B )  ->  x  e.  On )
2524adantr 463 . . . . . . 7  |-  ( ( ( ( ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X )  C_  u
) )  /\  x  e.  B )  /\  (
( F `  x
)  =/=  (/)  /\  ( O `  u )  e.  x ) )  ->  x  e.  On )
26 ontr2 5457 . . . . . . 7  |-  ( ( X  e.  On  /\  x  e.  On )  ->  ( ( X  C_  ( O `  u )  /\  ( O `  u )  e.  x
)  ->  X  e.  x ) )
2720, 25, 26syl2anc 659 . . . . . 6  |-  ( ( ( ( ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X )  C_  u
) )  /\  x  e.  B )  /\  (
( F `  x
)  =/=  (/)  /\  ( O `  u )  e.  x ) )  -> 
( ( X  C_  ( O `  u )  /\  ( O `  u )  e.  x
)  ->  X  e.  x ) )
2815, 16, 27mp2and 677 . . . . 5  |-  ( ( ( ( ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X )  C_  u
) )  /\  x  e.  B )  /\  (
( F `  x
)  =/=  (/)  /\  ( O `  u )  e.  x ) )  ->  X  e.  x )
29 eleq2 2475 . . . . . . 7  |-  ( w  =  x  ->  ( X  e.  w  <->  X  e.  x ) )
30 fveq2 5849 . . . . . . . 8  |-  ( w  =  x  ->  ( F `  w )  =  ( F `  x ) )
31 fveq2 5849 . . . . . . . 8  |-  ( w  =  x  ->  ( G `  w )  =  ( G `  x ) )
3230, 31eqeq12d 2424 . . . . . . 7  |-  ( w  =  x  ->  (
( F `  w
)  =  ( G `
 w )  <->  ( F `  x )  =  ( G `  x ) ) )
3329, 32imbi12d 318 . . . . . 6  |-  ( w  =  x  ->  (
( X  e.  w  ->  ( F `  w
)  =  ( G `
 w ) )  <-> 
( X  e.  x  ->  ( F `  x
)  =  ( G `
 x ) ) ) )
3433rspcv 3156 . . . . 5  |-  ( x  e.  B  ->  ( A. w  e.  B  ( X  e.  w  ->  ( F `  w
)  =  ( G `
 w ) )  ->  ( X  e.  x  ->  ( F `  x )  =  ( G `  x ) ) ) )
351, 12, 28, 34syl3c 60 . . . 4  |-  ( ( ( ( ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X )  C_  u
) )  /\  x  e.  B )  /\  (
( F `  x
)  =/=  (/)  /\  ( O `  u )  e.  x ) )  -> 
( F `  x
)  =  ( G `
 x ) )
36 simprl 756 . . . 4  |-  ( ( ( ( ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X )  C_  u
) )  /\  x  e.  B )  /\  (
( F `  x
)  =/=  (/)  /\  ( O `  u )  e.  x ) )  -> 
( F `  x
)  =/=  (/) )
3735, 36eqnetrrd 2697 . . 3  |-  ( ( ( ( ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X )  C_  u
) )  /\  x  e.  B )  /\  (
( F `  x
)  =/=  (/)  /\  ( O `  u )  e.  x ) )  -> 
( G `  x
)  =/=  (/) )
38 fvex 5859 . . . 4  |-  ( G `
 x )  e. 
_V
39 dif1o 7187 . . . 4  |-  ( ( G `  x )  e.  ( _V  \  1o )  <->  ( ( G `
 x )  e. 
_V  /\  ( G `  x )  =/=  (/) ) )
4038, 39mpbiran 919 . . 3  |-  ( ( G `  x )  e.  ( _V  \  1o )  <->  ( G `  x )  =/=  (/) )
4137, 40sylibr 212 . 2  |-  ( ( ( ( ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X )  C_  u
) )  /\  x  e.  B )  /\  (
( F `  x
)  =/=  (/)  /\  ( O `  u )  e.  x ) )  -> 
( G `  x
)  e.  ( _V 
\  1o ) )
422, 3, 4cantnfsOLD 8147 . . . . . . 7  |-  ( ph  ->  ( G  e.  S  <->  ( G : B --> A  /\  ( `' G " ( _V 
\  1o ) )  e.  Fin ) ) )
437, 42mpbid 210 . . . . . 6  |-  ( ph  ->  ( G : B --> A  /\  ( `' G " ( _V  \  1o ) )  e.  Fin ) )
4443simpld 457 . . . . 5  |-  ( ph  ->  G : B --> A )
45 ffn 5714 . . . . 5  |-  ( G : B --> A  ->  G  Fn  B )
4644, 45syl 17 . . . 4  |-  ( ph  ->  G  Fn  B )
4746ad3antrrr 728 . . 3  |-  ( ( ( ( ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X )  C_  u
) )  /\  x  e.  B )  /\  (
( F `  x
)  =/=  (/)  /\  ( O `  u )  e.  x ) )  ->  G  Fn  B )
48 elpreima 5985 . . 3  |-  ( G  Fn  B  ->  (
x  e.  ( `' G " ( _V 
\  1o ) )  <-> 
( x  e.  B  /\  ( G `  x
)  e.  ( _V 
\  1o ) ) ) )
4947, 48syl 17 . 2  |-  ( ( ( ( ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X )  C_  u
) )  /\  x  e.  B )  /\  (
( F `  x
)  =/=  (/)  /\  ( O `  u )  e.  x ) )  -> 
( x  e.  ( `' G " ( _V 
\  1o ) )  <-> 
( x  e.  B  /\  ( G `  x
)  e.  ( _V 
\  1o ) ) ) )
501, 41, 49mpbir2and 923 1  |-  ( ( ( ( ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X )  C_  u
) )  /\  x  e.  B )  /\  (
( F `  x
)  =/=  (/)  /\  ( O `  u )  e.  x ) )  ->  x  e.  ( `' G " ( _V  \  1o ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842    =/= wne 2598   A.wral 2754   E.wrex 2755   {crab 2758   _Vcvv 3059    \ cdif 3411    C_ wss 3414   (/)c0 3738   U.cuni 4191   class class class wbr 4395   {copab 4452    _E cep 4732   `'ccnv 4822   dom cdm 4823   "cima 4826   Oncon0 5410   suc csuc 5412    Fn wfn 5564   -->wf 5565   ` cfv 5569  (class class class)co 6278   1oc1o 7160   Fincfn 7554  OrdIsocoi 7968   CNF ccnf 8110
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-isom 5578  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-supp 6903  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-seqom 7150  df-1o 7167  df-er 7348  df-map 7459  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-fsupp 7864  df-oi 7969  df-cnf 8111
This theorem is referenced by:  cantnflem1OLD  8163
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