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Theorem cantnfp1lem1 7636
Description: Lemma for cantnfp1 7639. (Contributed by Mario Carneiro, 20-Jun-2015.)
Hypotheses
Ref Expression
cantnfs.1  |-  S  =  dom  ( A CNF  B
)
cantnfs.2  |-  ( ph  ->  A  e.  On )
cantnfs.3  |-  ( ph  ->  B  e.  On )
cantnfp1.4  |-  ( ph  ->  G  e.  S )
cantnfp1.5  |-  ( ph  ->  X  e.  B )
cantnfp1.6  |-  ( ph  ->  Y  e.  A )
cantnfp1.7  |-  ( ph  ->  ( `' G "
( _V  \  1o ) )  C_  X
)
cantnfp1.f  |-  F  =  ( t  e.  B  |->  if ( t  =  X ,  Y , 
( G `  t
) ) )
Assertion
Ref Expression
cantnfp1lem1  |-  ( 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 cantnfp1lem1
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 cantnfp1.6 . . . . 5  |-  ( ph  ->  Y  e.  A )
21adantr 453 . . . 4  |-  ( (
ph  /\  t  e.  B )  ->  Y  e.  A )
3 cantnfp1.4 . . . . . . 7  |-  ( ph  ->  G  e.  S )
4 cantnfs.1 . . . . . . . 8  |-  S  =  dom  ( A CNF  B
)
5 cantnfs.2 . . . . . . . 8  |-  ( ph  ->  A  e.  On )
6 cantnfs.3 . . . . . . . 8  |-  ( ph  ->  B  e.  On )
74, 5, 6cantnfs 7623 . . . . . . 7  |-  ( ph  ->  ( G  e.  S  <->  ( G : B --> A  /\  ( `' G " ( _V 
\  1o ) )  e.  Fin ) ) )
83, 7mpbid 203 . . . . . 6  |-  ( ph  ->  ( G : B --> A  /\  ( `' G " ( _V  \  1o ) )  e.  Fin ) )
98simpld 447 . . . . 5  |-  ( ph  ->  G : B --> A )
109ffvelrnda 5872 . . . 4  |-  ( (
ph  /\  t  e.  B )  ->  ( G `  t )  e.  A )
11 ifcl 3777 . . . 4  |-  ( ( Y  e.  A  /\  ( G `  t )  e.  A )  ->  if ( t  =  X ,  Y ,  ( G `  t ) )  e.  A )
122, 10, 11syl2anc 644 . . 3  |-  ( (
ph  /\  t  e.  B )  ->  if ( t  =  X ,  Y ,  ( G `  t ) )  e.  A )
13 cantnfp1.f . . 3  |-  F  =  ( t  e.  B  |->  if ( t  =  X ,  Y , 
( G `  t
) ) )
1412, 13fmptd 5895 . 2  |-  ( ph  ->  F : B --> A )
158simprd 451 . . . 4  |-  ( ph  ->  ( `' G "
( _V  \  1o ) )  e.  Fin )
16 snfi 7189 . . . 4  |-  { X }  e.  Fin
17 unfi 7376 . . . 4  |-  ( ( ( `' G "
( _V  \  1o ) )  e.  Fin  /\ 
{ X }  e.  Fin )  ->  ( ( `' G " ( _V 
\  1o ) )  u.  { X }
)  e.  Fin )
1815, 16, 17sylancl 645 . . 3  |-  ( ph  ->  ( ( `' G " ( _V  \  1o ) )  u.  { X } )  e.  Fin )
19 df1o2 6738 . . . . . 6  |-  1o  =  { (/) }
2019difeq2i 3464 . . . . 5  |-  ( _V 
\  1o )  =  ( _V  \  { (/)
} )
2120imaeq2i 5203 . . . 4  |-  ( `' F " ( _V 
\  1o ) )  =  ( `' F " ( _V  \  { (/)
} ) )
22 eldifi 3471 . . . . . . . 8  |-  ( k  e.  ( B  \ 
( ( `' G " ( _V  \  1o ) )  u.  { X } ) )  -> 
k  e.  B )
2322adantl 454 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( B  \  (
( `' G "
( _V  \  1o ) )  u.  { X } ) ) )  ->  k  e.  B
)
241adantr 453 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( B  \  (
( `' G "
( _V  \  1o ) )  u.  { X } ) ) )  ->  Y  e.  A
)
25 fvex 5744 . . . . . . . 8  |-  ( G `
 k )  e. 
_V
26 ifexg 3800 . . . . . . . 8  |-  ( ( Y  e.  A  /\  ( G `  k )  e.  _V )  ->  if ( k  =  X ,  Y ,  ( G `  k ) )  e.  _V )
2724, 25, 26sylancl 645 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( B  \  (
( `' G "
( _V  \  1o ) )  u.  { X } ) ) )  ->  if ( k  =  X ,  Y ,  ( G `  k ) )  e. 
_V )
28 eqeq1 2444 . . . . . . . . 9  |-  ( t  =  k  ->  (
t  =  X  <->  k  =  X ) )
29 fveq2 5730 . . . . . . . . 9  |-  ( t  =  k  ->  ( G `  t )  =  ( G `  k ) )
3028, 29ifbieq2d 3761 . . . . . . . 8  |-  ( t  =  k  ->  if ( t  =  X ,  Y ,  ( G `  t ) )  =  if ( k  =  X ,  Y ,  ( G `  k ) ) )
3130, 13fvmptg 5806 . . . . . . 7  |-  ( ( k  e.  B  /\  if ( k  =  X ,  Y ,  ( G `  k ) )  e.  _V )  ->  ( F `  k
)  =  if ( k  =  X ,  Y ,  ( G `  k ) ) )
3223, 27, 31syl2anc 644 . . . . . 6  |-  ( (
ph  /\  k  e.  ( B  \  (
( `' G "
( _V  \  1o ) )  u.  { X } ) ) )  ->  ( F `  k )  =  if ( k  =  X ,  Y ,  ( G `  k ) ) )
33 eldifn 3472 . . . . . . . . 9  |-  ( k  e.  ( B  \ 
( ( `' G " ( _V  \  1o ) )  u.  { X } ) )  ->  -.  k  e.  (
( `' G "
( _V  \  1o ) )  u.  { X } ) )
3433adantl 454 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( B  \  (
( `' G "
( _V  \  1o ) )  u.  { X } ) ) )  ->  -.  k  e.  ( ( `' G " ( _V  \  1o ) )  u.  { X } ) )
35 elsn 3831 . . . . . . . . 9  |-  ( k  e.  { X }  <->  k  =  X )
36 elun2 3517 . . . . . . . . 9  |-  ( k  e.  { X }  ->  k  e.  ( ( `' G " ( _V 
\  1o ) )  u.  { X }
) )
3735, 36sylbir 206 . . . . . . . 8  |-  ( k  =  X  ->  k  e.  ( ( `' G " ( _V  \  1o ) )  u.  { X } ) )
3834, 37nsyl 116 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( B  \  (
( `' G "
( _V  \  1o ) )  u.  { X } ) ) )  ->  -.  k  =  X )
39 iffalse 3748 . . . . . . 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 3512 . . . . . . . . 9  |-  ( `' G " ( _V 
\  1o ) ) 
C_  ( ( `' G " ( _V 
\  1o ) )  u.  { X }
)
42 sscon 3483 . . . . . . . . 9  |-  ( ( `' G " ( _V 
\  1o ) ) 
C_  ( ( `' G " ( _V 
\  1o ) )  u.  { X }
)  ->  ( B  \  ( ( `' G " ( _V  \  1o ) )  u.  { X } ) )  C_  ( B  \  ( `' G " ( _V 
\  1o ) ) ) )
4341, 42ax-mp 8 . . . . . . . 8  |-  ( B 
\  ( ( `' G " ( _V 
\  1o ) )  u.  { X }
) )  C_  ( B  \  ( `' G " ( _V  \  1o ) ) )
4443sseli 3346 . . . . . . 7  |-  ( k  e.  ( B  \ 
( ( `' G " ( _V  \  1o ) )  u.  { X } ) )  -> 
k  e.  ( B 
\  ( `' G " ( _V  \  1o ) ) ) )
4520imaeq2i 5203 . . . . . . . . 9  |-  ( `' G " ( _V 
\  1o ) )  =  ( `' G " ( _V  \  { (/)
} ) )
46 eqimss2 3403 . . . . . . . . 9  |-  ( ( `' G " ( _V 
\  1o ) )  =  ( `' G " ( _V  \  { (/)
} ) )  -> 
( `' G "
( _V  \  { (/)
} ) )  C_  ( `' G " ( _V 
\  1o ) ) )
4745, 46mp1i 12 . . . . . . . 8  |-  ( ph  ->  ( `' G "
( _V  \  { (/)
} ) )  C_  ( `' G " ( _V 
\  1o ) ) )
489, 47suppssr 5866 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( B  \  ( `' G " ( _V 
\  1o ) ) ) )  ->  ( G `  k )  =  (/) )
4944, 48sylan2 462 . . . . . 6  |-  ( (
ph  /\  k  e.  ( B  \  (
( `' G "
( _V  \  1o ) )  u.  { X } ) ) )  ->  ( G `  k )  =  (/) )
5032, 40, 493eqtrd 2474 . . . . 5  |-  ( (
ph  /\  k  e.  ( B  \  (
( `' G "
( _V  \  1o ) )  u.  { X } ) ) )  ->  ( F `  k )  =  (/) )
5114, 50suppss 5865 . . . 4  |-  ( ph  ->  ( `' F "
( _V  \  { (/)
} ) )  C_  ( ( `' G " ( _V  \  1o ) )  u.  { X } ) )
5221, 51syl5eqss 3394 . . 3  |-  ( ph  ->  ( `' F "
( _V  \  1o ) )  C_  (
( `' G "
( _V  \  1o ) )  u.  { X } ) )
53 ssfi 7331 . . 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 644 . 2  |-  ( ph  ->  ( `' F "
( _V  \  1o ) )  e.  Fin )
554, 5, 6cantnfs 7623 . 2  |-  ( ph  ->  ( F  e.  S  <->  ( F : B --> A  /\  ( `' F " ( _V 
\  1o ) )  e.  Fin ) ) )
5614, 54, 55mpbir2and 890 1  |-  ( ph  ->  F  e.  S )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2958    \ cdif 3319    u. cun 3320    C_ wss 3322   (/)c0 3630   ifcif 3741   {csn 3816    e. cmpt 4268   Oncon0 4583   `'ccnv 4879   dom cdm 4880   "cima 4883   -->wf 5452   ` cfv 5456  (class class class)co 6083   1oc1o 6719   Fincfn 7111   CNF ccnf 7618
This theorem is referenced by:  cantnfp1lem2  7637  cantnfp1lem3  7638  cantnfp1  7639
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-riota 6551  df-recs 6635  df-rdg 6670  df-seqom 6707  df-1o 6726  df-oadd 6730  df-er 6907  df-map 7022  df-en 7112  df-fin 7115  df-oi 7481  df-cnf 7619
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