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Theorem ismfs 30183
Description: A formal system is a tuple  <.mCN , mVR , mType , mVT , mTC , mAx >. such that: mCN and mVR are disjoint; mType is a function from mVR to mVT; mVT is a subset of mTC; mAx is a set of statements; and for each variable typecode, there are infinitely many variables of that type. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
ismfs.c  |-  C  =  (mCN `  T )
ismfs.v  |-  V  =  (mVR `  T )
ismfs.y  |-  Y  =  (mType `  T )
ismfs.f  |-  F  =  (mVT `  T )
ismfs.k  |-  K  =  (mTC `  T )
ismfs.a  |-  A  =  (mAx `  T )
ismfs.s  |-  S  =  (mStat `  T )
Assertion
Ref Expression
ismfs  |-  ( T  e.  W  ->  ( T  e. mFS  <->  ( ( ( C  i^i  V )  =  (/)  /\  Y : V
--> K )  /\  ( A  C_  S  /\  A. v  e.  F  -.  ( `' Y " { v } )  e.  Fin ) ) ) )
Distinct variable groups:    v, F    v, T
Allowed substitution hints:    A( v)    C( v)    S( v)    K( v)    V( v)    W( v)    Y( v)

Proof of Theorem ismfs
Dummy variable  t is distinct from all other variables.
StepHypRef Expression
1 fveq2 5878 . . . . . . 7  |-  ( t  =  T  ->  (mCN `  t )  =  (mCN
`  T ) )
2 ismfs.c . . . . . . 7  |-  C  =  (mCN `  T )
31, 2syl6eqr 2481 . . . . . 6  |-  ( t  =  T  ->  (mCN `  t )  =  C )
4 fveq2 5878 . . . . . . 7  |-  ( t  =  T  ->  (mVR `  t )  =  (mVR
`  T ) )
5 ismfs.v . . . . . . 7  |-  V  =  (mVR `  T )
64, 5syl6eqr 2481 . . . . . 6  |-  ( t  =  T  ->  (mVR `  t )  =  V )
73, 6ineq12d 3665 . . . . 5  |-  ( t  =  T  ->  (
(mCN `  t )  i^i  (mVR `  t )
)  =  ( C  i^i  V ) )
87eqeq1d 2424 . . . 4  |-  ( t  =  T  ->  (
( (mCN `  t
)  i^i  (mVR `  t
) )  =  (/)  <->  ( C  i^i  V )  =  (/) ) )
9 fveq2 5878 . . . . . 6  |-  ( t  =  T  ->  (mType `  t )  =  (mType `  T ) )
10 ismfs.y . . . . . 6  |-  Y  =  (mType `  T )
119, 10syl6eqr 2481 . . . . 5  |-  ( t  =  T  ->  (mType `  t )  =  Y )
12 fveq2 5878 . . . . . 6  |-  ( t  =  T  ->  (mTC `  t )  =  (mTC
`  T ) )
13 ismfs.k . . . . . 6  |-  K  =  (mTC `  T )
1412, 13syl6eqr 2481 . . . . 5  |-  ( t  =  T  ->  (mTC `  t )  =  K )
1511, 6, 14feq123d 5733 . . . 4  |-  ( t  =  T  ->  (
(mType `  t ) : (mVR `  t ) --> (mTC `  t )  <->  Y : V
--> K ) )
168, 15anbi12d 715 . . 3  |-  ( t  =  T  ->  (
( ( (mCN `  t )  i^i  (mVR `  t ) )  =  (/)  /\  (mType `  t
) : (mVR `  t ) --> (mTC `  t ) )  <->  ( ( C  i^i  V )  =  (/)  /\  Y : V --> K ) ) )
17 fveq2 5878 . . . . . 6  |-  ( t  =  T  ->  (mAx `  t )  =  (mAx
`  T ) )
18 ismfs.a . . . . . 6  |-  A  =  (mAx `  T )
1917, 18syl6eqr 2481 . . . . 5  |-  ( t  =  T  ->  (mAx `  t )  =  A )
20 fveq2 5878 . . . . . 6  |-  ( t  =  T  ->  (mStat `  t )  =  (mStat `  T ) )
21 ismfs.s . . . . . 6  |-  S  =  (mStat `  T )
2220, 21syl6eqr 2481 . . . . 5  |-  ( t  =  T  ->  (mStat `  t )  =  S )
2319, 22sseq12d 3493 . . . 4  |-  ( t  =  T  ->  (
(mAx `  t )  C_  (mStat `  t )  <->  A 
C_  S ) )
24 fveq2 5878 . . . . . 6  |-  ( t  =  T  ->  (mVT `  t )  =  (mVT
`  T ) )
25 ismfs.f . . . . . 6  |-  F  =  (mVT `  T )
2624, 25syl6eqr 2481 . . . . 5  |-  ( t  =  T  ->  (mVT `  t )  =  F )
2711cnveqd 5026 . . . . . . . 8  |-  ( t  =  T  ->  `' (mType `  t )  =  `' Y )
2827imaeq1d 5183 . . . . . . 7  |-  ( t  =  T  ->  ( `' (mType `  t ) " { v } )  =  ( `' Y " { v } ) )
2928eleq1d 2491 . . . . . 6  |-  ( t  =  T  ->  (
( `' (mType `  t ) " {
v } )  e. 
Fin 
<->  ( `' Y " { v } )  e.  Fin ) )
3029notbid 295 . . . . 5  |-  ( t  =  T  ->  ( -.  ( `' (mType `  t ) " {
v } )  e. 
Fin 
<->  -.  ( `' Y " { v } )  e.  Fin ) )
3126, 30raleqbidv 3039 . . . 4  |-  ( t  =  T  ->  ( A. v  e.  (mVT `  t )  -.  ( `' (mType `  t ) " { v } )  e.  Fin  <->  A. v  e.  F  -.  ( `' Y " { v } )  e.  Fin ) )
3223, 31anbi12d 715 . . 3  |-  ( t  =  T  ->  (
( (mAx `  t
)  C_  (mStat `  t
)  /\  A. v  e.  (mVT `  t )  -.  ( `' (mType `  t ) " {
v } )  e. 
Fin )  <->  ( A  C_  S  /\  A. v  e.  F  -.  ( `' Y " { v } )  e.  Fin ) ) )
3316, 32anbi12d 715 . 2  |-  ( t  =  T  ->  (
( ( ( (mCN
`  t )  i^i  (mVR `  t )
)  =  (/)  /\  (mType `  t ) : (mVR
`  t ) --> (mTC
`  t ) )  /\  ( (mAx `  t )  C_  (mStat `  t )  /\  A. v  e.  (mVT `  t
)  -.  ( `' (mType `  t ) " { v } )  e.  Fin ) )  <-> 
( ( ( C  i^i  V )  =  (/)  /\  Y : V --> K )  /\  ( A  C_  S  /\  A. v  e.  F  -.  ( `' Y " { v } )  e.  Fin ) ) ) )
34 df-mfs 30130 . 2  |- mFS  =  {
t  |  ( ( ( (mCN `  t
)  i^i  (mVR `  t
) )  =  (/)  /\  (mType `  t ) : (mVR `  t ) --> (mTC `  t ) )  /\  ( (mAx `  t )  C_  (mStat `  t )  /\  A. v  e.  (mVT `  t
)  -.  ( `' (mType `  t ) " { v } )  e.  Fin ) ) }
3533, 34elab2g 3220 1  |-  ( T  e.  W  ->  ( T  e. mFS  <->  ( ( ( C  i^i  V )  =  (/)  /\  Y : V
--> K )  /\  ( A  C_  S  /\  A. v  e.  F  -.  ( `' Y " { v } )  e.  Fin ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1868   A.wral 2775    i^i cin 3435    C_ wss 3436   (/)c0 3761   {csn 3996   `'ccnv 4849   "cima 4853   -->wf 5594   ` cfv 5598   Fincfn 7574  mCNcmcn 30094  mVRcmvar 30095  mTypecmty 30096  mVTcmvt 30097  mTCcmtc 30098  mAxcmax 30099  mStatcmsta 30109  mFScmfs 30110
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-br 4421  df-opab 4480  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-iota 5562  df-fun 5600  df-fn 5601  df-f 5602  df-fv 5606  df-mfs 30130
This theorem is referenced by:  mfsdisj  30184  mtyf2  30185  maxsta  30188  mvtinf  30189
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