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Theorem cantnfp1lem1OLD 8121
Description: Lemma for cantnfp1OLD 8124. (Contributed by Mario Carneiro, 20-Jun-2015.) Obsolete version of cantnfp1lem1 8095 as of 30-Jun-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 )
cantnfp1OLD.4  |-  ( ph  ->  G  e.  S )
cantnfp1OLD.5  |-  ( ph  ->  X  e.  B )
cantnfp1OLD.6  |-  ( ph  ->  Y  e.  A )
cantnfp1OLD.7  |-  ( ph  ->  ( `' G "
( _V  \  1o ) )  C_  X
)
cantnfp1OLD.f  |-  F  =  ( t  e.  B  |->  if ( t  =  X ,  Y , 
( G `  t
) ) )
Assertion
Ref Expression
cantnfp1lem1OLD  |-  ( ph  ->  F  e.  S )
Distinct variable groups:    t, B    t, A    t, S    t, G    ph, t    t, Y   
t, X
Allowed substitution hint:    F( t)

Proof of Theorem cantnfp1lem1OLD
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 cantnfp1OLD.6 . . . . 5  |-  ( ph  ->  Y  e.  A )
21adantr 465 . . . 4  |-  ( (
ph  /\  t  e.  B )  ->  Y  e.  A )
3 cantnfp1OLD.4 . . . . . . 7  |-  ( ph  ->  G  e.  S )
4 cantnfsOLD.1 . . . . . . . 8  |-  S  =  dom  ( A CNF  B
)
5 cantnfsOLD.2 . . . . . . . 8  |-  ( ph  ->  A  e.  On )
6 cantnfsOLD.3 . . . . . . . 8  |-  ( ph  ->  B  e.  On )
74, 5, 6cantnfsOLD 8113 . . . . . . 7  |-  ( ph  ->  ( G  e.  S  <->  ( G : B --> A  /\  ( `' G " ( _V 
\  1o ) )  e.  Fin ) ) )
83, 7mpbid 210 . . . . . 6  |-  ( ph  ->  ( G : B --> A  /\  ( `' G " ( _V  \  1o ) )  e.  Fin ) )
98simpld 459 . . . . 5  |-  ( ph  ->  G : B --> A )
109ffvelrnda 6012 . . . 4  |-  ( (
ph  /\  t  e.  B )  ->  ( G `  t )  e.  A )
11 ifcl 3964 . . . 4  |-  ( ( Y  e.  A  /\  ( G `  t )  e.  A )  ->  if ( t  =  X ,  Y ,  ( G `  t ) )  e.  A )
122, 10, 11syl2anc 661 . . 3  |-  ( (
ph  /\  t  e.  B )  ->  if ( t  =  X ,  Y ,  ( G `  t ) )  e.  A )
13 cantnfp1OLD.f . . 3  |-  F  =  ( t  e.  B  |->  if ( t  =  X ,  Y , 
( G `  t
) ) )
1412, 13fmptd 6036 . 2  |-  ( ph  ->  F : B --> A )
158simprd 463 . . . 4  |-  ( ph  ->  ( `' G "
( _V  \  1o ) )  e.  Fin )
16 snfi 7594 . . . 4  |-  { X }  e.  Fin
17 unfi 7785 . . . 4  |-  ( ( ( `' G "
( _V  \  1o ) )  e.  Fin  /\ 
{ X }  e.  Fin )  ->  ( ( `' G " ( _V 
\  1o ) )  u.  { X }
)  e.  Fin )
1815, 16, 17sylancl 662 . . 3  |-  ( ph  ->  ( ( `' G " ( _V  \  1o ) )  u.  { X } )  e.  Fin )
19 df1o2 7140 . . . . . 6  |-  1o  =  { (/) }
2019difeq2i 3601 . . . . 5  |-  ( _V 
\  1o )  =  ( _V  \  { (/)
} )
2120imaeq2i 5321 . . . 4  |-  ( `' F " ( _V 
\  1o ) )  =  ( `' F " ( _V  \  { (/)
} ) )
22 eldifi 3608 . . . . . . . 8  |-  ( k  e.  ( B  \ 
( ( `' G " ( _V  \  1o ) )  u.  { X } ) )  -> 
k  e.  B )
2322adantl 466 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( B  \  (
( `' G "
( _V  \  1o ) )  u.  { X } ) ) )  ->  k  e.  B
)
241adantr 465 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( B  \  (
( `' G "
( _V  \  1o ) )  u.  { X } ) ) )  ->  Y  e.  A
)
25 fvex 5862 . . . . . . . 8  |-  ( G `
 k )  e. 
_V
26 ifexg 3992 . . . . . . . 8  |-  ( ( Y  e.  A  /\  ( G `  k )  e.  _V )  ->  if ( k  =  X ,  Y ,  ( G `  k ) )  e.  _V )
2724, 25, 26sylancl 662 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( B  \  (
( `' G "
( _V  \  1o ) )  u.  { X } ) ) )  ->  if ( k  =  X ,  Y ,  ( G `  k ) )  e. 
_V )
28 eqeq1 2445 . . . . . . . . 9  |-  ( t  =  k  ->  (
t  =  X  <->  k  =  X ) )
29 fveq2 5852 . . . . . . . . 9  |-  ( t  =  k  ->  ( G `  t )  =  ( G `  k ) )
3028, 29ifbieq2d 3947 . . . . . . . 8  |-  ( t  =  k  ->  if ( t  =  X ,  Y ,  ( G `  t ) )  =  if ( k  =  X ,  Y ,  ( G `  k ) ) )
3130, 13fvmptg 5935 . . . . . . 7  |-  ( ( k  e.  B  /\  if ( k  =  X ,  Y ,  ( G `  k ) )  e.  _V )  ->  ( F `  k
)  =  if ( k  =  X ,  Y ,  ( G `  k ) ) )
3223, 27, 31syl2anc 661 . . . . . 6  |-  ( (
ph  /\  k  e.  ( B  \  (
( `' G "
( _V  \  1o ) )  u.  { X } ) ) )  ->  ( F `  k )  =  if ( k  =  X ,  Y ,  ( G `  k ) ) )
33 eldifn 3609 . . . . . . . . 9  |-  ( k  e.  ( B  \ 
( ( `' G " ( _V  \  1o ) )  u.  { X } ) )  ->  -.  k  e.  (
( `' G "
( _V  \  1o ) )  u.  { X } ) )
3433adantl 466 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( B  \  (
( `' G "
( _V  \  1o ) )  u.  { X } ) ) )  ->  -.  k  e.  ( ( `' G " ( _V  \  1o ) )  u.  { X } ) )
35 elsn 4024 . . . . . . . . 9  |-  ( k  e.  { X }  <->  k  =  X )
36 elun2 3654 . . . . . . . . 9  |-  ( k  e.  { X }  ->  k  e.  ( ( `' G " ( _V 
\  1o ) )  u.  { X }
) )
3735, 36sylbir 213 . . . . . . . 8  |-  ( k  =  X  ->  k  e.  ( ( `' G " ( _V  \  1o ) )  u.  { X } ) )
3834, 37nsyl 121 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( B  \  (
( `' G "
( _V  \  1o ) )  u.  { X } ) ) )  ->  -.  k  =  X )
39 iffalse 3931 . . . . . . 7  |-  ( -.  k  =  X  ->  if ( k  =  X ,  Y ,  ( G `  k ) )  =  ( G `
 k ) )
4038, 39syl 16 . . . . . 6  |-  ( (
ph  /\  k  e.  ( B  \  (
( `' G "
( _V  \  1o ) )  u.  { X } ) ) )  ->  if ( k  =  X ,  Y ,  ( G `  k ) )  =  ( G `  k
) )
41 ssun1 3649 . . . . . . . . 9  |-  ( `' G " ( _V 
\  1o ) ) 
C_  ( ( `' G " ( _V 
\  1o ) )  u.  { X }
)
42 sscon 3620 . . . . . . . . 9  |-  ( ( `' G " ( _V 
\  1o ) ) 
C_  ( ( `' G " ( _V 
\  1o ) )  u.  { X }
)  ->  ( B  \  ( ( `' G " ( _V  \  1o ) )  u.  { X } ) )  C_  ( B  \  ( `' G " ( _V 
\  1o ) ) ) )
4341, 42ax-mp 5 . . . . . . . 8  |-  ( B 
\  ( ( `' G " ( _V 
\  1o ) )  u.  { X }
) )  C_  ( B  \  ( `' G " ( _V  \  1o ) ) )
4443sseli 3482 . . . . . . 7  |-  ( k  e.  ( B  \ 
( ( `' G " ( _V  \  1o ) )  u.  { X } ) )  -> 
k  e.  ( B 
\  ( `' G " ( _V  \  1o ) ) ) )
4520imaeq2i 5321 . . . . . . . . 9  |-  ( `' G " ( _V 
\  1o ) )  =  ( `' G " ( _V  \  { (/)
} ) )
46 eqimss2 3539 . . . . . . . . 9  |-  ( ( `' G " ( _V 
\  1o ) )  =  ( `' G " ( _V  \  { (/)
} ) )  -> 
( `' G "
( _V  \  { (/)
} ) )  C_  ( `' G " ( _V 
\  1o ) ) )
4745, 46mp1i 12 . . . . . . . 8  |-  ( ph  ->  ( `' G "
( _V  \  { (/)
} ) )  C_  ( `' G " ( _V 
\  1o ) ) )
489, 47suppssrOLD 6002 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( B  \  ( `' G " ( _V 
\  1o ) ) ) )  ->  ( G `  k )  =  (/) )
4944, 48sylan2 474 . . . . . 6  |-  ( (
ph  /\  k  e.  ( B  \  (
( `' G "
( _V  \  1o ) )  u.  { X } ) ) )  ->  ( G `  k )  =  (/) )
5032, 40, 493eqtrd 2486 . . . . 5  |-  ( (
ph  /\  k  e.  ( B  \  (
( `' G "
( _V  \  1o ) )  u.  { X } ) ) )  ->  ( F `  k )  =  (/) )
5114, 50suppssOLD 6001 . . . 4  |-  ( ph  ->  ( `' F "
( _V  \  { (/)
} ) )  C_  ( ( `' G " ( _V  \  1o ) )  u.  { X } ) )
5221, 51syl5eqss 3530 . . 3  |-  ( ph  ->  ( `' F "
( _V  \  1o ) )  C_  (
( `' G "
( _V  \  1o ) )  u.  { X } ) )
53 ssfi 7738 . . 3  |-  ( ( ( ( `' G " ( _V  \  1o ) )  u.  { X } )  e.  Fin  /\  ( `' F "
( _V  \  1o ) )  C_  (
( `' G "
( _V  \  1o ) )  u.  { X } ) )  -> 
( `' F "
( _V  \  1o ) )  e.  Fin )
5418, 52, 53syl2anc 661 . 2  |-  ( ph  ->  ( `' F "
( _V  \  1o ) )  e.  Fin )
554, 5, 6cantnfsOLD 8113 . 2  |-  ( ph  ->  ( F  e.  S  <->  ( F : B --> A  /\  ( `' F " ( _V 
\  1o ) )  e.  Fin ) ) )
5614, 54, 55mpbir2and 920 1  |-  ( ph  ->  F  e.  S )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1381    e. wcel 1802   _Vcvv 3093    \ cdif 3455    u. cun 3456    C_ wss 3458   (/)c0 3767   ifcif 3922   {csn 4010    |-> cmpt 4491   Oncon0 4864   `'ccnv 4984   dom cdm 4985   "cima 4988   -->wf 5570   ` cfv 5574  (class class class)co 6277   1oc1o 7121   Fincfn 7514   CNF ccnf 8076
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-fal 1387  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-se 4825  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-supp 6900  df-recs 7040  df-rdg 7074  df-seqom 7111  df-1o 7128  df-oadd 7132  df-er 7309  df-map 7420  df-en 7515  df-fin 7518  df-fsupp 7828  df-oi 7933  df-cnf 8077
This theorem is referenced by:  cantnfp1lem2OLD  8122  cantnfp1lem3OLD  8123  cantnfp1OLD  8124
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