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Theorem cantnfp1lem1OLD 7908
Description: Lemma for cantnfp1OLD 7911. (Contributed by Mario Carneiro, 20-Jun-2015.) Obsolete version of cantnfp1lem1 7882 as of 30-Jun-2019. (New usage 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 462 . . . 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 7900 . . . . . . 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 456 . . . . 5  |-  ( ph  ->  G : B --> A )
109ffvelrnda 5840 . . . 4  |-  ( (
ph  /\  t  e.  B )  ->  ( G `  t )  e.  A )
11 ifcl 3828 . . . 4  |-  ( ( Y  e.  A  /\  ( G `  t )  e.  A )  ->  if ( t  =  X ,  Y ,  ( G `  t ) )  e.  A )
122, 10, 11syl2anc 656 . . 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 5864 . 2  |-  ( ph  ->  F : B --> A )
158simprd 460 . . . 4  |-  ( ph  ->  ( `' G "
( _V  \  1o ) )  e.  Fin )
16 snfi 7386 . . . 4  |-  { X }  e.  Fin
17 unfi 7575 . . . 4  |-  ( ( ( `' G "
( _V  \  1o ) )  e.  Fin  /\ 
{ X }  e.  Fin )  ->  ( ( `' G " ( _V 
\  1o ) )  u.  { X }
)  e.  Fin )
1815, 16, 17sylancl 657 . . 3  |-  ( ph  ->  ( ( `' G " ( _V  \  1o ) )  u.  { X } )  e.  Fin )
19 df1o2 6928 . . . . . 6  |-  1o  =  { (/) }
2019difeq2i 3468 . . . . 5  |-  ( _V 
\  1o )  =  ( _V  \  { (/)
} )
2120imaeq2i 5164 . . . 4  |-  ( `' F " ( _V 
\  1o ) )  =  ( `' F " ( _V  \  { (/)
} ) )
22 eldifi 3475 . . . . . . . 8  |-  ( k  e.  ( B  \ 
( ( `' G " ( _V  \  1o ) )  u.  { X } ) )  -> 
k  e.  B )
2322adantl 463 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( B  \  (
( `' G "
( _V  \  1o ) )  u.  { X } ) ) )  ->  k  e.  B
)
241adantr 462 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( B  \  (
( `' G "
( _V  \  1o ) )  u.  { X } ) ) )  ->  Y  e.  A
)
25 fvex 5698 . . . . . . . 8  |-  ( G `
 k )  e. 
_V
26 ifexg 3856 . . . . . . . 8  |-  ( ( Y  e.  A  /\  ( G `  k )  e.  _V )  ->  if ( k  =  X ,  Y ,  ( G `  k ) )  e.  _V )
2724, 25, 26sylancl 657 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( B  \  (
( `' G "
( _V  \  1o ) )  u.  { X } ) ) )  ->  if ( k  =  X ,  Y ,  ( G `  k ) )  e. 
_V )
28 eqeq1 2447 . . . . . . . . 9  |-  ( t  =  k  ->  (
t  =  X  <->  k  =  X ) )
29 fveq2 5688 . . . . . . . . 9  |-  ( t  =  k  ->  ( G `  t )  =  ( G `  k ) )
3028, 29ifbieq2d 3811 . . . . . . . 8  |-  ( t  =  k  ->  if ( t  =  X ,  Y ,  ( G `  t ) )  =  if ( k  =  X ,  Y ,  ( G `  k ) ) )
3130, 13fvmptg 5769 . . . . . . 7  |-  ( ( k  e.  B  /\  if ( k  =  X ,  Y ,  ( G `  k ) )  e.  _V )  ->  ( F `  k
)  =  if ( k  =  X ,  Y ,  ( G `  k ) ) )
3223, 27, 31syl2anc 656 . . . . . 6  |-  ( (
ph  /\  k  e.  ( B  \  (
( `' G "
( _V  \  1o ) )  u.  { X } ) ) )  ->  ( F `  k )  =  if ( k  =  X ,  Y ,  ( G `  k ) ) )
33 eldifn 3476 . . . . . . . . 9  |-  ( k  e.  ( B  \ 
( ( `' G " ( _V  \  1o ) )  u.  { X } ) )  ->  -.  k  e.  (
( `' G "
( _V  \  1o ) )  u.  { X } ) )
3433adantl 463 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( B  \  (
( `' G "
( _V  \  1o ) )  u.  { X } ) ) )  ->  -.  k  e.  ( ( `' G " ( _V  \  1o ) )  u.  { X } ) )
35 elsn 3888 . . . . . . . . 9  |-  ( k  e.  { X }  <->  k  =  X )
36 elun2 3521 . . . . . . . . 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 3796 . . . . . . 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 3516 . . . . . . . . 9  |-  ( `' G " ( _V 
\  1o ) ) 
C_  ( ( `' G " ( _V 
\  1o ) )  u.  { X }
)
42 sscon 3487 . . . . . . . . 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 3349 . . . . . . 7  |-  ( k  e.  ( B  \ 
( ( `' G " ( _V  \  1o ) )  u.  { X } ) )  -> 
k  e.  ( B 
\  ( `' G " ( _V  \  1o ) ) ) )
4520imaeq2i 5164 . . . . . . . . 9  |-  ( `' G " ( _V 
\  1o ) )  =  ( `' G " ( _V  \  { (/)
} ) )
46 eqimss2 3406 . . . . . . . . 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 5834 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( B  \  ( `' G " ( _V 
\  1o ) ) ) )  ->  ( G `  k )  =  (/) )
4944, 48sylan2 471 . . . . . 6  |-  ( (
ph  /\  k  e.  ( B  \  (
( `' G "
( _V  \  1o ) )  u.  { X } ) ) )  ->  ( G `  k )  =  (/) )
5032, 40, 493eqtrd 2477 . . . . 5  |-  ( (
ph  /\  k  e.  ( B  \  (
( `' G "
( _V  \  1o ) )  u.  { X } ) ) )  ->  ( F `  k )  =  (/) )
5114, 50suppssOLD 5833 . . . 4  |-  ( ph  ->  ( `' F "
( _V  \  { (/)
} ) )  C_  ( ( `' G " ( _V  \  1o ) )  u.  { X } ) )
5221, 51syl5eqss 3397 . . 3  |-  ( ph  ->  ( `' F "
( _V  \  1o ) )  C_  (
( `' G "
( _V  \  1o ) )  u.  { X } ) )
53 ssfi 7529 . . 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 656 . 2  |-  ( ph  ->  ( `' F "
( _V  \  1o ) )  e.  Fin )
554, 5, 6cantnfsOLD 7900 . 2  |-  ( ph  ->  ( F  e.  S  <->  ( F : B --> A  /\  ( `' F " ( _V 
\  1o ) )  e.  Fin ) ) )
5614, 54, 55mpbir2and 908 1  |-  ( ph  ->  F  e.  S )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1761   _Vcvv 2970    \ cdif 3322    u. cun 3323    C_ wss 3325   (/)c0 3634   ifcif 3788   {csn 3874    e. cmpt 4347   Oncon0 4715   `'ccnv 4835   dom cdm 4836   "cima 4839   -->wf 5411   ` cfv 5415  (class class class)co 6090   1oc1o 6909   Fincfn 7306   CNF ccnf 7863
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-recs 6828  df-rdg 6862  df-seqom 6899  df-1o 6916  df-oadd 6920  df-er 7097  df-map 7212  df-en 7307  df-fin 7310  df-fsupp 7617  df-oi 7720  df-cnf 7864
This theorem is referenced by:  cantnfp1lem2OLD  7909  cantnfp1lem3OLD  7910  cantnfp1OLD  7911
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