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Theorem cantnfp1lem1 8088
Description: Lemma for cantnfp1 8091. (Contributed by Mario Carneiro, 20-Jun-2015.) (Revised by AV, 30-Jun-2019.)
Hypotheses
Ref Expression
cantnfs.s  |-  S  =  dom  ( A CNF  B
)
cantnfs.a  |-  ( ph  ->  A  e.  On )
cantnfs.b  |-  ( ph  ->  B  e.  On )
cantnfp1.g  |-  ( ph  ->  G  e.  S )
cantnfp1.x  |-  ( ph  ->  X  e.  B )
cantnfp1.y  |-  ( ph  ->  Y  e.  A )
cantnfp1.s  |-  ( ph  ->  ( G supp  (/) )  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.y . . . . 5  |-  ( ph  ->  Y  e.  A )
21adantr 465 . . . 4  |-  ( (
ph  /\  t  e.  B )  ->  Y  e.  A )
3 cantnfp1.g . . . . . . 7  |-  ( ph  ->  G  e.  S )
4 cantnfs.s . . . . . . . 8  |-  S  =  dom  ( A CNF  B
)
5 cantnfs.a . . . . . . . 8  |-  ( ph  ->  A  e.  On )
6 cantnfs.b . . . . . . . 8  |-  ( ph  ->  B  e.  On )
74, 5, 6cantnfs 8076 . . . . . . 7  |-  ( ph  ->  ( G  e.  S  <->  ( G : B --> A  /\  G finSupp 
(/) ) ) )
83, 7mpbid 210 . . . . . 6  |-  ( ph  ->  ( G : B --> A  /\  G finSupp  (/) ) )
98simpld 459 . . . . 5  |-  ( ph  ->  G : B --> A )
109ffvelrnda 6014 . . . 4  |-  ( (
ph  /\  t  e.  B )  ->  ( G `  t )  e.  A )
112, 10ifcld 3977 . . 3  |-  ( (
ph  /\  t  e.  B )  ->  if ( t  =  X ,  Y ,  ( G `  t ) )  e.  A )
12 cantnfp1.f . . 3  |-  F  =  ( t  e.  B  |->  if ( t  =  X ,  Y , 
( G `  t
) ) )
1311, 12fmptd 6038 . 2  |-  ( ph  ->  F : B --> A )
148simprd 463 . . . . . 6  |-  ( ph  ->  G finSupp  (/) )
1514fsuppimpd 7827 . . . . 5  |-  ( ph  ->  ( G supp  (/) )  e. 
Fin )
16 snfi 7588 . . . . 5  |-  { X }  e.  Fin
17 unfi 7778 . . . . 5  |-  ( ( ( G supp  (/) )  e. 
Fin  /\  { X }  e.  Fin )  ->  ( ( G supp  (/) )  u. 
{ X } )  e.  Fin )
1815, 16, 17sylancl 662 . . . 4  |-  ( ph  ->  ( ( G supp  (/) )  u. 
{ X } )  e.  Fin )
19 eldifi 3621 . . . . . . . 8  |-  ( k  e.  ( B  \ 
( ( G supp  (/) )  u. 
{ X } ) )  ->  k  e.  B )
2019adantl 466 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( B  \  (
( G supp  (/) )  u. 
{ X } ) ) )  ->  k  e.  B )
211adantr 465 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( B  \  (
( G supp  (/) )  u. 
{ X } ) ) )  ->  Y  e.  A )
22 fvex 5869 . . . . . . . 8  |-  ( G `
 k )  e. 
_V
23 ifexg 4004 . . . . . . . 8  |-  ( ( Y  e.  A  /\  ( G `  k )  e.  _V )  ->  if ( k  =  X ,  Y ,  ( G `  k ) )  e.  _V )
2421, 22, 23sylancl 662 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( B  \  (
( G supp  (/) )  u. 
{ X } ) ) )  ->  if ( k  =  X ,  Y ,  ( G `  k ) )  e.  _V )
25 eqeq1 2466 . . . . . . . . 9  |-  ( t  =  k  ->  (
t  =  X  <->  k  =  X ) )
26 fveq2 5859 . . . . . . . . 9  |-  ( t  =  k  ->  ( G `  t )  =  ( G `  k ) )
2725, 26ifbieq2d 3959 . . . . . . . 8  |-  ( t  =  k  ->  if ( t  =  X ,  Y ,  ( G `  t ) )  =  if ( k  =  X ,  Y ,  ( G `  k ) ) )
2827, 12fvmptg 5941 . . . . . . 7  |-  ( ( k  e.  B  /\  if ( k  =  X ,  Y ,  ( G `  k ) )  e.  _V )  ->  ( F `  k
)  =  if ( k  =  X ,  Y ,  ( G `  k ) ) )
2920, 24, 28syl2anc 661 . . . . . 6  |-  ( (
ph  /\  k  e.  ( B  \  (
( G supp  (/) )  u. 
{ X } ) ) )  ->  ( F `  k )  =  if ( k  =  X ,  Y , 
( G `  k
) ) )
30 eldifn 3622 . . . . . . . . 9  |-  ( k  e.  ( B  \ 
( ( G supp  (/) )  u. 
{ X } ) )  ->  -.  k  e.  ( ( G supp  (/) )  u. 
{ X } ) )
3130adantl 466 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( B  \  (
( G supp  (/) )  u. 
{ X } ) ) )  ->  -.  k  e.  ( ( G supp 
(/) )  u.  { X } ) )
32 elsn 4036 . . . . . . . . 9  |-  ( k  e.  { X }  <->  k  =  X )
33 elun2 3667 . . . . . . . . 9  |-  ( k  e.  { X }  ->  k  e.  ( ( G supp  (/) )  u.  { X } ) )
3432, 33sylbir 213 . . . . . . . 8  |-  ( k  =  X  ->  k  e.  ( ( G supp  (/) )  u. 
{ X } ) )
3531, 34nsyl 121 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( B  \  (
( G supp  (/) )  u. 
{ X } ) ) )  ->  -.  k  =  X )
36 iffalse 3943 . . . . . . 7  |-  ( -.  k  =  X  ->  if ( k  =  X ,  Y ,  ( G `  k ) )  =  ( G `
 k ) )
3735, 36syl 16 . . . . . 6  |-  ( (
ph  /\  k  e.  ( B  \  (
( G supp  (/) )  u. 
{ X } ) ) )  ->  if ( k  =  X ,  Y ,  ( G `  k ) )  =  ( G `
 k ) )
38 ssun1 3662 . . . . . . . . 9  |-  ( G supp  (/) )  C_  ( ( G supp  (/) )  u.  { X } )
39 sscon 3633 . . . . . . . . 9  |-  ( ( G supp  (/) )  C_  (
( G supp  (/) )  u. 
{ X } )  ->  ( B  \ 
( ( G supp  (/) )  u. 
{ X } ) )  C_  ( B  \  ( G supp  (/) ) ) )
4038, 39ax-mp 5 . . . . . . . 8  |-  ( B 
\  ( ( G supp  (/) )  u.  { X } ) )  C_  ( B  \  ( G supp 
(/) ) )
4140sseli 3495 . . . . . . 7  |-  ( k  e.  ( B  \ 
( ( G supp  (/) )  u. 
{ X } ) )  ->  k  e.  ( B  \  ( G supp 
(/) ) ) )
42 eqid 2462 . . . . . . . . 9  |-  ( G supp  (/) )  =  ( G supp 
(/) )
43 eqimss2 3552 . . . . . . . . 9  |-  ( ( G supp  (/) )  =  ( G supp  (/) )  ->  ( G supp 
(/) )  C_  ( G supp 
(/) ) )
4442, 43mp1i 12 . . . . . . . 8  |-  ( ph  ->  ( G supp  (/) )  C_  ( G supp  (/) ) )
45 0ex 4572 . . . . . . . . 9  |-  (/)  e.  _V
4645a1i 11 . . . . . . . 8  |-  ( ph  -> 
(/)  e.  _V )
479, 44, 6, 46suppssr 6923 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( B  \  ( G supp 
(/) ) ) )  ->  ( G `  k )  =  (/) )
4841, 47sylan2 474 . . . . . 6  |-  ( (
ph  /\  k  e.  ( B  \  (
( G supp  (/) )  u. 
{ X } ) ) )  ->  ( G `  k )  =  (/) )
4929, 37, 483eqtrd 2507 . . . . 5  |-  ( (
ph  /\  k  e.  ( B  \  (
( G supp  (/) )  u. 
{ X } ) ) )  ->  ( F `  k )  =  (/) )
5013, 49suppss 6922 . . . 4  |-  ( ph  ->  ( F supp  (/) )  C_  ( ( G supp  (/) )  u. 
{ X } ) )
51 ssfi 7732 . . . 4  |-  ( ( ( ( G supp  (/) )  u. 
{ X } )  e.  Fin  /\  ( F supp 
(/) )  C_  (
( G supp  (/) )  u. 
{ X } ) )  ->  ( F supp  (/) )  e.  Fin )
5218, 50, 51syl2anc 661 . . 3  |-  ( ph  ->  ( F supp  (/) )  e. 
Fin )
5312funmpt2 5618 . . . . 5  |-  Fun  F
5453a1i 11 . . . 4  |-  ( ph  ->  Fun  F )
55 mptexg 6123 . . . . . 6  |-  ( B  e.  On  ->  (
t  e.  B  |->  if ( t  =  X ,  Y ,  ( G `  t ) ) )  e.  _V )
5612, 55syl5eqel 2554 . . . . 5  |-  ( B  e.  On  ->  F  e.  _V )
576, 56syl 16 . . . 4  |-  ( ph  ->  F  e.  _V )
58 funisfsupp 7825 . . . 4  |-  ( ( Fun  F  /\  F  e.  _V  /\  (/)  e.  _V )  ->  ( F finSupp  (/)  <->  ( F supp  (/) )  e.  Fin )
)
5954, 57, 46, 58syl3anc 1223 . . 3  |-  ( ph  ->  ( F finSupp  (/)  <->  ( F supp  (/) )  e.  Fin )
)
6052, 59mpbird 232 . 2  |-  ( ph  ->  F finSupp  (/) )
614, 5, 6cantnfs 8076 . 2  |-  ( ph  ->  ( F  e.  S  <->  ( F : B --> A  /\  F finSupp 
(/) ) ) )
6213, 60, 61mpbir2and 915 1  |-  ( ph  ->  F  e.  S )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1374    e. wcel 1762   _Vcvv 3108    \ cdif 3468    u. cun 3469    C_ wss 3471   (/)c0 3780   ifcif 3934   {csn 4022   class class class wbr 4442    |-> cmpt 4500   Oncon0 4873   dom cdm 4994   Fun wfun 5575   -->wf 5577   ` cfv 5581  (class class class)co 6277   supp csupp 6893   Fincfn 7508   finSupp cfsupp 7820   CNF ccnf 8069
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 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-supp 6894  df-recs 7034  df-rdg 7068  df-seqom 7105  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-en 7509  df-fin 7512  df-fsupp 7821  df-cnf 8070
This theorem is referenced by:  cantnfp1lem2  8089  cantnfp1lem3  8090  cantnfp1  8091
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