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Theorem cantnflem1c 7643
Description: Lemma for cantnf 7649. (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 732 . 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 7640 . . . . . . 7  |-  ( ph  ->  ( X  e.  B  /\  ( F `  X
)  e.  ( G `
 X )  /\  A. w  e.  B  ( X  e.  w  -> 
( F `  w
)  =  ( G `
 w ) ) ) )
1110simp3d 971 . . . . . 6  |-  ( ph  ->  A. w  e.  B  ( X  e.  w  ->  ( F `  w
)  =  ( G `
 w ) ) )
1211ad3antrrr 711 . . . . 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 7642 . . . . . . 7  |-  ( (
ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X ) 
C_  u ) )  ->  X  C_  ( O `  u )
)
1514ad2antrr 707 . . . . . 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 734 . . . . . 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 969 . . . . . . . . 9  |-  ( ph  ->  X  e.  B )
18 onelon 4606 . . . . . . . . 9  |-  ( ( B  e.  On  /\  X  e.  B )  ->  X  e.  On )
194, 17, 18syl2anc 643 . . . . . . . 8  |-  ( ph  ->  X  e.  On )
2019ad3antrrr 711 . . . . . . 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 4771 . . . . . . . . . . 11  |-  ( B  e.  On  ->  B  C_  On )
224, 21syl 16 . . . . . . . . . 10  |-  ( ph  ->  B  C_  On )
2322sselda 3348 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  B )  ->  x  e.  On )
2423adantlr 696 . . . . . . . 8  |-  ( ( ( ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X )  C_  u
) )  /\  x  e.  B )  ->  x  e.  On )
2524adantr 452 . . . . . . 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 4628 . . . . . . 7  |-  ( ( X  e.  On  /\  x  e.  On )  ->  ( ( X  C_  ( O `  u )  /\  ( O `  u )  e.  x
)  ->  X  e.  x ) )
2720, 25, 26syl2anc 643 . . . . . 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 661 . . . . 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 2497 . . . . . . 7  |-  ( w  =  x  ->  ( X  e.  w  <->  X  e.  x ) )
30 fveq2 5728 . . . . . . . 8  |-  ( w  =  x  ->  ( F `  w )  =  ( F `  x ) )
31 fveq2 5728 . . . . . . . 8  |-  ( w  =  x  ->  ( G `  w )  =  ( G `  x ) )
3230, 31eqeq12d 2450 . . . . . . 7  |-  ( w  =  x  ->  (
( F `  w
)  =  ( G `
 w )  <->  ( F `  x )  =  ( G `  x ) ) )
3329, 32imbi12d 312 . . . . . 6  |-  ( w  =  x  ->  (
( X  e.  w  ->  ( F `  w
)  =  ( G `
 w ) )  <-> 
( X  e.  x  ->  ( F `  x
)  =  ( G `
 x ) ) ) )
3433rspcv 3048 . . . . 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 59 . . . 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 733 . . . 4  |-  ( ( ( ( ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X )  C_  u
) )  /\  x  e.  B )  /\  (
( F `  x
)  =/=  (/)  /\  ( O `  u )  e.  x ) )  -> 
( F `  x
)  =/=  (/) )
3735, 36eqnetrrd 2621 . . 3  |-  ( ( ( ( ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X )  C_  u
) )  /\  x  e.  B )  /\  (
( F `  x
)  =/=  (/)  /\  ( O `  u )  e.  x ) )  -> 
( G `  x
)  =/=  (/) )
38 fvex 5742 . . . 4  |-  ( G `
 x )  e. 
_V
39 dif1o 6744 . . . 4  |-  ( ( G `  x )  e.  ( _V  \  1o )  <->  ( ( G `
 x )  e. 
_V  /\  ( G `  x )  =/=  (/) ) )
4038, 39mpbiran 885 . . 3  |-  ( ( G `  x )  e.  ( _V  \  1o )  <->  ( G `  x )  =/=  (/) )
4137, 40sylibr 204 . 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 7621 . . . . . . 7  |-  ( ph  ->  ( G  e.  S  <->  ( G : B --> A  /\  ( `' G " ( _V 
\  1o ) )  e.  Fin ) ) )
437, 42mpbid 202 . . . . . 6  |-  ( ph  ->  ( G : B --> A  /\  ( `' G " ( _V  \  1o ) )  e.  Fin ) )
4443simpld 446 . . . . 5  |-  ( ph  ->  G : B --> A )
45 ffn 5591 . . . . 5  |-  ( G : B --> A  ->  G  Fn  B )
4644, 45syl 16 . . . 4  |-  ( ph  ->  G  Fn  B )
4746ad3antrrr 711 . . 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 5850 . . 3  |-  ( G  Fn  B  ->  (
x  e.  ( `' G " ( _V 
\  1o ) )  <-> 
( x  e.  B  /\  ( G `  x
)  e.  ( _V 
\  1o ) ) ) )
4947, 48syl 16 . 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 889 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 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599   A.wral 2705   E.wrex 2706   {crab 2709   _Vcvv 2956    \ cdif 3317    C_ wss 3320   (/)c0 3628   U.cuni 4015   class class class wbr 4212   {copab 4265    _E cep 4492   Oncon0 4581   suc csuc 4583   `'ccnv 4877   dom cdm 4878   "cima 4881    Fn wfn 5449   -->wf 5450   ` cfv 5454  (class class class)co 6081   1oc1o 6717   Fincfn 7109  OrdIsocoi 7478   CNF ccnf 7616
This theorem is referenced by:  cantnflem1  7645
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-riota 6549  df-recs 6633  df-rdg 6668  df-seqom 6705  df-1o 6724  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-oi 7479  df-cnf 7617
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