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Theorem cantnfp1lem1 8128
Description: Lemma for cantnfp1 8131. (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 463 . . . 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 8116 . . . . . . 7  |-  ( ph  ->  ( G  e.  S  <->  ( G : B --> A  /\  G finSupp 
(/) ) ) )
83, 7mpbid 210 . . . . . 6  |-  ( ph  ->  ( G : B --> A  /\  G finSupp  (/) ) )
98simpld 457 . . . . 5  |-  ( ph  ->  G : B --> A )
109ffvelrnda 6008 . . . 4  |-  ( (
ph  /\  t  e.  B )  ->  ( G `  t )  e.  A )
112, 10ifcld 3927 . . 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 6032 . 2  |-  ( ph  ->  F : B --> A )
148simprd 461 . . . . . 6  |-  ( ph  ->  G finSupp  (/) )
1514fsuppimpd 7869 . . . . 5  |-  ( ph  ->  ( G supp  (/) )  e. 
Fin )
16 snfi 7633 . . . . 5  |-  { X }  e.  Fin
17 unfi 7820 . . . . 5  |-  ( ( ( G supp  (/) )  e. 
Fin  /\  { X }  e.  Fin )  ->  ( ( G supp  (/) )  u. 
{ X } )  e.  Fin )
1815, 16, 17sylancl 660 . . . 4  |-  ( ph  ->  ( ( G supp  (/) )  u. 
{ X } )  e.  Fin )
19 eldifi 3564 . . . . . . . 8  |-  ( k  e.  ( B  \ 
( ( G supp  (/) )  u. 
{ X } ) )  ->  k  e.  B )
2019adantl 464 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( B  \  (
( G supp  (/) )  u. 
{ X } ) ) )  ->  k  e.  B )
211adantr 463 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( B  \  (
( G supp  (/) )  u. 
{ X } ) ) )  ->  Y  e.  A )
22 fvex 5858 . . . . . . . 8  |-  ( G `
 k )  e. 
_V
23 ifexg 3953 . . . . . . . 8  |-  ( ( Y  e.  A  /\  ( G `  k )  e.  _V )  ->  if ( k  =  X ,  Y ,  ( G `  k ) )  e.  _V )
2421, 22, 23sylancl 660 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( B  \  (
( G supp  (/) )  u. 
{ X } ) ) )  ->  if ( k  =  X ,  Y ,  ( G `  k ) )  e.  _V )
25 eqeq1 2406 . . . . . . . . 9  |-  ( t  =  k  ->  (
t  =  X  <->  k  =  X ) )
26 fveq2 5848 . . . . . . . . 9  |-  ( t  =  k  ->  ( G `  t )  =  ( G `  k ) )
2725, 26ifbieq2d 3909 . . . . . . . 8  |-  ( t  =  k  ->  if ( t  =  X ,  Y ,  ( G `  t ) )  =  if ( k  =  X ,  Y ,  ( G `  k ) ) )
2827, 12fvmptg 5929 . . . . . . 7  |-  ( ( k  e.  B  /\  if ( k  =  X ,  Y ,  ( G `  k ) )  e.  _V )  ->  ( F `  k
)  =  if ( k  =  X ,  Y ,  ( G `  k ) ) )
2920, 24, 28syl2anc 659 . . . . . 6  |-  ( (
ph  /\  k  e.  ( B  \  (
( G supp  (/) )  u. 
{ X } ) ) )  ->  ( F `  k )  =  if ( k  =  X ,  Y , 
( G `  k
) ) )
30 eldifn 3565 . . . . . . . . 9  |-  ( k  e.  ( B  \ 
( ( G supp  (/) )  u. 
{ X } ) )  ->  -.  k  e.  ( ( G supp  (/) )  u. 
{ X } ) )
3130adantl 464 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( B  \  (
( G supp  (/) )  u. 
{ X } ) ) )  ->  -.  k  e.  ( ( G supp 
(/) )  u.  { X } ) )
32 elsn 3985 . . . . . . . . 9  |-  ( k  e.  { X }  <->  k  =  X )
33 elun2 3610 . . . . . . . . 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 )
3635iffalsed 3895 . . . . . 6  |-  ( (
ph  /\  k  e.  ( B  \  (
( G supp  (/) )  u. 
{ X } ) ) )  ->  if ( k  =  X ,  Y ,  ( G `  k ) )  =  ( G `
 k ) )
37 ssun1 3605 . . . . . . . . 9  |-  ( G supp  (/) )  C_  ( ( G supp  (/) )  u.  { X } )
38 sscon 3576 . . . . . . . . 9  |-  ( ( G supp  (/) )  C_  (
( G supp  (/) )  u. 
{ X } )  ->  ( B  \ 
( ( G supp  (/) )  u. 
{ X } ) )  C_  ( B  \  ( G supp  (/) ) ) )
3937, 38ax-mp 5 . . . . . . . 8  |-  ( B 
\  ( ( G supp  (/) )  u.  { X } ) )  C_  ( B  \  ( G supp 
(/) ) )
4039sseli 3437 . . . . . . 7  |-  ( k  e.  ( B  \ 
( ( G supp  (/) )  u. 
{ X } ) )  ->  k  e.  ( B  \  ( G supp 
(/) ) ) )
41 eqid 2402 . . . . . . . . 9  |-  ( G supp  (/) )  =  ( G supp 
(/) )
42 eqimss2 3494 . . . . . . . . 9  |-  ( ( G supp  (/) )  =  ( G supp  (/) )  ->  ( G supp 
(/) )  C_  ( G supp 
(/) ) )
4341, 42mp1i 13 . . . . . . . 8  |-  ( ph  ->  ( G supp  (/) )  C_  ( G supp  (/) ) )
44 0ex 4525 . . . . . . . . 9  |-  (/)  e.  _V
4544a1i 11 . . . . . . . 8  |-  ( ph  -> 
(/)  e.  _V )
469, 43, 6, 45suppssr 6933 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( B  \  ( G supp 
(/) ) ) )  ->  ( G `  k )  =  (/) )
4740, 46sylan2 472 . . . . . 6  |-  ( (
ph  /\  k  e.  ( B  \  (
( G supp  (/) )  u. 
{ X } ) ) )  ->  ( G `  k )  =  (/) )
4829, 36, 473eqtrd 2447 . . . . 5  |-  ( (
ph  /\  k  e.  ( B  \  (
( G supp  (/) )  u. 
{ X } ) ) )  ->  ( F `  k )  =  (/) )
4913, 48suppss 6932 . . . 4  |-  ( ph  ->  ( F supp  (/) )  C_  ( ( G supp  (/) )  u. 
{ X } ) )
50 ssfi 7774 . . . 4  |-  ( ( ( ( G supp  (/) )  u. 
{ X } )  e.  Fin  /\  ( F supp 
(/) )  C_  (
( G supp  (/) )  u. 
{ X } ) )  ->  ( F supp  (/) )  e.  Fin )
5118, 49, 50syl2anc 659 . . 3  |-  ( ph  ->  ( F supp  (/) )  e. 
Fin )
5212funmpt2 5605 . . . . 5  |-  Fun  F
5352a1i 11 . . . 4  |-  ( ph  ->  Fun  F )
54 mptexg 6122 . . . . . 6  |-  ( B  e.  On  ->  (
t  e.  B  |->  if ( t  =  X ,  Y ,  ( G `  t ) ) )  e.  _V )
5512, 54syl5eqel 2494 . . . . 5  |-  ( B  e.  On  ->  F  e.  _V )
566, 55syl 17 . . . 4  |-  ( ph  ->  F  e.  _V )
57 funisfsupp 7867 . . . 4  |-  ( ( Fun  F  /\  F  e.  _V  /\  (/)  e.  _V )  ->  ( F finSupp  (/)  <->  ( F supp  (/) )  e.  Fin )
)
5853, 56, 45, 57syl3anc 1230 . . 3  |-  ( ph  ->  ( F finSupp  (/)  <->  ( F supp  (/) )  e.  Fin )
)
5951, 58mpbird 232 . 2  |-  ( ph  ->  F finSupp  (/) )
604, 5, 6cantnfs 8116 . 2  |-  ( ph  ->  ( F  e.  S  <->  ( F : B --> A  /\  F finSupp 
(/) ) ) )
6113, 59, 60mpbir2and 923 1  |-  ( ph  ->  F  e.  S )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842   _Vcvv 3058    \ cdif 3410    u. cun 3411    C_ wss 3413   (/)c0 3737   ifcif 3884   {csn 3971   class class class wbr 4394    |-> cmpt 4452   dom cdm 4822   Oncon0 5409   Fun wfun 5562   -->wf 5564   ` cfv 5568  (class class class)co 6277   supp csupp 6901   Fincfn 7553   finSupp cfsupp 7862   CNF ccnf 8109
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-supp 6902  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-seqom 7149  df-1o 7166  df-oadd 7170  df-er 7347  df-map 7458  df-en 7554  df-fin 7557  df-fsupp 7863  df-cnf 8110
This theorem is referenced by:  cantnfp1lem2  8129  cantnfp1lem3  8130  cantnfp1  8131
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