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Theorem cantnflem1c 7389
Description: Lemma for cantnf 7395. (Contributed by Mario Carneiro, 4-Jun-2015.)
Hypotheses
Ref Expression
cantnfs.1  |-  S  =  dom  ( A CNF  B
)
cantnfs.2  |-  ( ph  ->  A  e.  On )
cantnfs.3  |-  ( 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.3  |-  ( ph  ->  F  e.  S )
oemapval.4  |-  ( ph  ->  G  e.  S )
oemapvali.5  |-  ( ph  ->  F T G )
oemapvali.6  |-  X  = 
U. { c  e.  B  |  ( F `
 c )  e.  ( G `  c
) }
cantnflem1.o  |-  O  = OrdIso
(  _E  ,  ( `' G " ( _V 
\  1o ) ) )
Assertion
Ref Expression
cantnflem1c  |-  ( ( ( ( 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 cantnflem1c
StepHypRef Expression
1 simplr 731 . 2  |-  ( ( ( ( ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X )  C_  u
) )  /\  x  e.  B )  /\  (
( F `  x
)  =/=  (/)  /\  ( O `  u )  e.  x ) )  ->  x  e.  B )
2 cantnfs.1 . . . . . . . 8  |-  S  =  dom  ( A CNF  B
)
3 cantnfs.2 . . . . . . . 8  |-  ( ph  ->  A  e.  On )
4 cantnfs.3 . . . . . . . 8  |-  ( ph  ->  B  e.  On )
5 oemapval.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 oemapval.3 . . . . . . . 8  |-  ( ph  ->  F  e.  S )
7 oemapval.4 . . . . . . . 8  |-  ( ph  ->  G  e.  S )
8 oemapvali.5 . . . . . . . 8  |-  ( ph  ->  F T G )
9 oemapvali.6 . . . . . . . 8  |-  X  = 
U. { c  e.  B  |  ( F `
 c )  e.  ( G `  c
) }
102, 3, 4, 5, 6, 7, 8, 9oemapvali 7386 . . . . . . 7  |-  ( ph  ->  ( X  e.  B  /\  ( F `  X
)  e.  ( G `
 X )  /\  A. w  e.  B  ( X  e.  w  -> 
( F `  w
)  =  ( G `
 w ) ) ) )
1110simp3d 969 . . . . . 6  |-  ( ph  ->  A. w  e.  B  ( X  e.  w  ->  ( F `  w
)  =  ( G `
 w ) ) )
1211ad3antrrr 710 . . . . 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 cantnflem1.o . . . . . . . 8  |-  O  = OrdIso
(  _E  ,  ( `' G " ( _V 
\  1o ) ) )
142, 3, 4, 5, 6, 7, 8, 9, 13cantnflem1b 7388 . . . . . . 7  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  X  C_  ( O `  u )
)
1514ad2antrr 706 . . . . . 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 733 . . . . . 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 967 . . . . . . . . 9  |-  ( ph  ->  X  e.  B )
18 onelon 4417 . . . . . . . . 9  |-  ( ( B  e.  On  /\  X  e.  B )  ->  X  e.  On )
194, 17, 18syl2anc 642 . . . . . . . 8  |-  ( ph  ->  X  e.  On )
2019ad3antrrr 710 . . . . . . 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 4582 . . . . . . . . . . 11  |-  ( B  e.  On  ->  B  C_  On )
224, 21syl 15 . . . . . . . . . 10  |-  ( ph  ->  B  C_  On )
2322sselda 3180 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  B )  ->  x  e.  On )
2423adantlr 695 . . . . . . . 8  |-  ( ( ( ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X )  C_  u
) )  /\  x  e.  B )  ->  x  e.  On )
2524adantr 451 . . . . . . 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 4439 . . . . . . 7  |-  ( ( X  e.  On  /\  x  e.  On )  ->  ( ( X  C_  ( O `  u )  /\  ( O `  u )  e.  x
)  ->  X  e.  x ) )
2720, 25, 26syl2anc 642 . . . . . 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 660 . . . . 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 2344 . . . . . . 7  |-  ( w  =  x  ->  ( X  e.  w  <->  X  e.  x ) )
30 fveq2 5525 . . . . . . . 8  |-  ( w  =  x  ->  ( F `  w )  =  ( F `  x ) )
31 fveq2 5525 . . . . . . . 8  |-  ( w  =  x  ->  ( G `  w )  =  ( G `  x ) )
3230, 31eqeq12d 2297 . . . . . . 7  |-  ( w  =  x  ->  (
( F `  w
)  =  ( G `
 w )  <->  ( F `  x )  =  ( G `  x ) ) )
3329, 32imbi12d 311 . . . . . 6  |-  ( w  =  x  ->  (
( X  e.  w  ->  ( F `  w
)  =  ( G `
 w ) )  <-> 
( X  e.  x  ->  ( F `  x
)  =  ( G `
 x ) ) ) )
3433rspcv 2880 . . . . 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 57 . . . 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 732 . . . 4  |-  ( ( ( ( ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X )  C_  u
) )  /\  x  e.  B )  /\  (
( F `  x
)  =/=  (/)  /\  ( O `  u )  e.  x ) )  -> 
( F `  x
)  =/=  (/) )
3735, 36eqnetrrd 2466 . . 3  |-  ( ( ( ( ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X )  C_  u
) )  /\  x  e.  B )  /\  (
( F `  x
)  =/=  (/)  /\  ( O `  u )  e.  x ) )  -> 
( G `  x
)  =/=  (/) )
38 fvex 5539 . . . 4  |-  ( G `
 x )  e. 
_V
39 dif1o 6499 . . . 4  |-  ( ( G `  x )  e.  ( _V  \  1o )  <->  ( ( G `
 x )  e. 
_V  /\  ( G `  x )  =/=  (/) ) )
4038, 39mpbiran 884 . . 3  |-  ( ( G `  x )  e.  ( _V  \  1o )  <->  ( G `  x )  =/=  (/) )
4137, 40sylibr 203 . 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, 4cantnfs 7367 . . . . . . 7  |-  ( ph  ->  ( G  e.  S  <->  ( G : B --> A  /\  ( `' G " ( _V 
\  1o ) )  e.  Fin ) ) )
437, 42mpbid 201 . . . . . 6  |-  ( ph  ->  ( G : B --> A  /\  ( `' G " ( _V  \  1o ) )  e.  Fin ) )
4443simpld 445 . . . . 5  |-  ( ph  ->  G : B --> A )
45 ffn 5389 . . . . 5  |-  ( G : B --> A  ->  G  Fn  B )
4644, 45syl 15 . . . 4  |-  ( ph  ->  G  Fn  B )
4746ad3antrrr 710 . . 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 5645 . . 3  |-  ( G  Fn  B  ->  (
x  e.  ( `' G " ( _V 
\  1o ) )  <-> 
( x  e.  B  /\  ( G `  x
)  e.  ( _V 
\  1o ) ) ) )
4947, 48syl 15 . 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 888 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 set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544   {crab 2547   _Vcvv 2788    \ cdif 3149    C_ wss 3152   (/)c0 3455   U.cuni 3827   class class class wbr 4023   {copab 4076    _E cep 4303   Oncon0 4392   suc csuc 4394   `'ccnv 4688   dom cdm 4689   "cima 4692    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   1oc1o 6472   Fincfn 6863  OrdIsocoi 7224   CNF ccnf 7362
This theorem is referenced by:  cantnflem1  7391
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-recs 6388  df-rdg 6423  df-seqom 6460  df-1o 6479  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-oi 7225  df-cnf 7363
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