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Theorem ismfs 32693
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 (𝑇𝑊 → (𝑇 ∈ mFS ↔ (((𝐶𝑉) = ∅ ∧ 𝑌:𝑉𝐾) ∧ (𝐴𝑆 ∧ ∀𝑣𝐹 ¬ (𝑌 “ {𝑣}) ∈ 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 6663 . . . . . . 7 (𝑡 = 𝑇 → (mCN‘𝑡) = (mCN‘𝑇))
2 ismfs.c . . . . . . 7 𝐶 = (mCN‘𝑇)
31, 2syl6eqr 2871 . . . . . 6 (𝑡 = 𝑇 → (mCN‘𝑡) = 𝐶)
4 fveq2 6663 . . . . . . 7 (𝑡 = 𝑇 → (mVR‘𝑡) = (mVR‘𝑇))
5 ismfs.v . . . . . . 7 𝑉 = (mVR‘𝑇)
64, 5syl6eqr 2871 . . . . . 6 (𝑡 = 𝑇 → (mVR‘𝑡) = 𝑉)
73, 6ineq12d 4187 . . . . 5 (𝑡 = 𝑇 → ((mCN‘𝑡) ∩ (mVR‘𝑡)) = (𝐶𝑉))
87eqeq1d 2820 . . . 4 (𝑡 = 𝑇 → (((mCN‘𝑡) ∩ (mVR‘𝑡)) = ∅ ↔ (𝐶𝑉) = ∅))
9 fveq2 6663 . . . . . 6 (𝑡 = 𝑇 → (mType‘𝑡) = (mType‘𝑇))
10 ismfs.y . . . . . 6 𝑌 = (mType‘𝑇)
119, 10syl6eqr 2871 . . . . 5 (𝑡 = 𝑇 → (mType‘𝑡) = 𝑌)
12 fveq2 6663 . . . . . 6 (𝑡 = 𝑇 → (mTC‘𝑡) = (mTC‘𝑇))
13 ismfs.k . . . . . 6 𝐾 = (mTC‘𝑇)
1412, 13syl6eqr 2871 . . . . 5 (𝑡 = 𝑇 → (mTC‘𝑡) = 𝐾)
1511, 6, 14feq123d 6496 . . . 4 (𝑡 = 𝑇 → ((mType‘𝑡):(mVR‘𝑡)⟶(mTC‘𝑡) ↔ 𝑌:𝑉𝐾))
168, 15anbi12d 630 . . 3 (𝑡 = 𝑇 → ((((mCN‘𝑡) ∩ (mVR‘𝑡)) = ∅ ∧ (mType‘𝑡):(mVR‘𝑡)⟶(mTC‘𝑡)) ↔ ((𝐶𝑉) = ∅ ∧ 𝑌:𝑉𝐾)))
17 fveq2 6663 . . . . . 6 (𝑡 = 𝑇 → (mAx‘𝑡) = (mAx‘𝑇))
18 ismfs.a . . . . . 6 𝐴 = (mAx‘𝑇)
1917, 18syl6eqr 2871 . . . . 5 (𝑡 = 𝑇 → (mAx‘𝑡) = 𝐴)
20 fveq2 6663 . . . . . 6 (𝑡 = 𝑇 → (mStat‘𝑡) = (mStat‘𝑇))
21 ismfs.s . . . . . 6 𝑆 = (mStat‘𝑇)
2220, 21syl6eqr 2871 . . . . 5 (𝑡 = 𝑇 → (mStat‘𝑡) = 𝑆)
2319, 22sseq12d 3997 . . . 4 (𝑡 = 𝑇 → ((mAx‘𝑡) ⊆ (mStat‘𝑡) ↔ 𝐴𝑆))
24 fveq2 6663 . . . . . 6 (𝑡 = 𝑇 → (mVT‘𝑡) = (mVT‘𝑇))
25 ismfs.f . . . . . 6 𝐹 = (mVT‘𝑇)
2624, 25syl6eqr 2871 . . . . 5 (𝑡 = 𝑇 → (mVT‘𝑡) = 𝐹)
2711cnveqd 5739 . . . . . . . 8 (𝑡 = 𝑇(mType‘𝑡) = 𝑌)
2827imaeq1d 5921 . . . . . . 7 (𝑡 = 𝑇 → ((mType‘𝑡) “ {𝑣}) = (𝑌 “ {𝑣}))
2928eleq1d 2894 . . . . . 6 (𝑡 = 𝑇 → (((mType‘𝑡) “ {𝑣}) ∈ Fin ↔ (𝑌 “ {𝑣}) ∈ Fin))
3029notbid 319 . . . . 5 (𝑡 = 𝑇 → (¬ ((mType‘𝑡) “ {𝑣}) ∈ Fin ↔ ¬ (𝑌 “ {𝑣}) ∈ Fin))
3126, 30raleqbidv 3399 . . . 4 (𝑡 = 𝑇 → (∀𝑣 ∈ (mVT‘𝑡) ¬ ((mType‘𝑡) “ {𝑣}) ∈ Fin ↔ ∀𝑣𝐹 ¬ (𝑌 “ {𝑣}) ∈ Fin))
3223, 31anbi12d 630 . . 3 (𝑡 = 𝑇 → (((mAx‘𝑡) ⊆ (mStat‘𝑡) ∧ ∀𝑣 ∈ (mVT‘𝑡) ¬ ((mType‘𝑡) “ {𝑣}) ∈ Fin) ↔ (𝐴𝑆 ∧ ∀𝑣𝐹 ¬ (𝑌 “ {𝑣}) ∈ Fin)))
3316, 32anbi12d 630 . 2 (𝑡 = 𝑇 → (((((mCN‘𝑡) ∩ (mVR‘𝑡)) = ∅ ∧ (mType‘𝑡):(mVR‘𝑡)⟶(mTC‘𝑡)) ∧ ((mAx‘𝑡) ⊆ (mStat‘𝑡) ∧ ∀𝑣 ∈ (mVT‘𝑡) ¬ ((mType‘𝑡) “ {𝑣}) ∈ Fin)) ↔ (((𝐶𝑉) = ∅ ∧ 𝑌:𝑉𝐾) ∧ (𝐴𝑆 ∧ ∀𝑣𝐹 ¬ (𝑌 “ {𝑣}) ∈ Fin))))
34 df-mfs 32640 . 2 mFS = {𝑡 ∣ ((((mCN‘𝑡) ∩ (mVR‘𝑡)) = ∅ ∧ (mType‘𝑡):(mVR‘𝑡)⟶(mTC‘𝑡)) ∧ ((mAx‘𝑡) ⊆ (mStat‘𝑡) ∧ ∀𝑣 ∈ (mVT‘𝑡) ¬ ((mType‘𝑡) “ {𝑣}) ∈ Fin))}
3533, 34elab2g 3665 1 (𝑇𝑊 → (𝑇 ∈ mFS ↔ (((𝐶𝑉) = ∅ ∧ 𝑌:𝑉𝐾) ∧ (𝐴𝑆 ∧ ∀𝑣𝐹 ¬ (𝑌 “ {𝑣}) ∈ Fin))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wral 3135  cin 3932  wss 3933  c0 4288  {csn 4557  ccnv 5547  cima 5551  wf 6344  cfv 6348  Fincfn 8497  mCNcmcn 32604  mVRcmvar 32605  mTypecmty 32606  mVTcmvt 32607  mTCcmtc 32608  mAxcmax 32609  mStatcmsta 32619  mFScmfs 32620
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-mfs 32640
This theorem is referenced by:  mfsdisj  32694  mtyf2  32695  maxsta  32698  mvtinf  32699
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