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Theorem cantnflem1c 7405
Description: Lemma for cantnf 7411. (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 7402 . . . . . . 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 7404 . . . . . . 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 4433 . . . . . . . . 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 4598 . . . . . . . . . . 11  |-  ( B  e.  On  ->  B  C_  On )
224, 21syl 15 . . . . . . . . . 10  |-  ( ph  ->  B  C_  On )
2322sselda 3193 . . . . . . . . 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 4455 . . . . . . 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 2357 . . . . . . 7  |-  ( w  =  x  ->  ( X  e.  w  <->  X  e.  x ) )
30 fveq2 5541 . . . . . . . 8  |-  ( w  =  x  ->  ( F `  w )  =  ( F `  x ) )
31 fveq2 5541 . . . . . . . 8  |-  ( w  =  x  ->  ( G `  w )  =  ( G `  x ) )
3230, 31eqeq12d 2310 . . . . . . 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 2893 . . . . 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 2479 . . 3  |-  ( ( ( ( ph  /\  ( suc  u  e.  dom  O  /\  ( `' O `  X )  C_  u
) )  /\  x  e.  B )  /\  (
( F `  x
)  =/=  (/)  /\  ( O `  u )  e.  x ) )  -> 
( G `  x
)  =/=  (/) )
38 fvex 5555 . . . 4  |-  ( G `
 x )  e. 
_V
39 dif1o 6515 . . . 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 7383 . . . . . . 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 5405 . . . . 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 5661 . . 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 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557   {crab 2560   _Vcvv 2801    \ cdif 3162    C_ wss 3165   (/)c0 3468   U.cuni 3843   class class class wbr 4039   {copab 4092    _E cep 4319   Oncon0 4408   suc csuc 4410   `'ccnv 4704   dom cdm 4705   "cima 4708    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   1oc1o 6488   Fincfn 6879  OrdIsocoi 7240   CNF ccnf 7378
This theorem is referenced by:  cantnflem1  7407
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-recs 6404  df-rdg 6439  df-seqom 6476  df-1o 6495  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-oi 7241  df-cnf 7379
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