Mathbox for Mario Carneiro < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ismfs Structured version   Visualization version   GIF version

Theorem ismfs 30700
 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 6103 . . . . . . 7 (𝑡 = 𝑇 → (mCN‘𝑡) = (mCN‘𝑇))
2 ismfs.c . . . . . . 7 𝐶 = (mCN‘𝑇)
31, 2syl6eqr 2662 . . . . . 6 (𝑡 = 𝑇 → (mCN‘𝑡) = 𝐶)
4 fveq2 6103 . . . . . . 7 (𝑡 = 𝑇 → (mVR‘𝑡) = (mVR‘𝑇))
5 ismfs.v . . . . . . 7 𝑉 = (mVR‘𝑇)
64, 5syl6eqr 2662 . . . . . 6 (𝑡 = 𝑇 → (mVR‘𝑡) = 𝑉)
73, 6ineq12d 3777 . . . . 5 (𝑡 = 𝑇 → ((mCN‘𝑡) ∩ (mVR‘𝑡)) = (𝐶𝑉))
87eqeq1d 2612 . . . 4 (𝑡 = 𝑇 → (((mCN‘𝑡) ∩ (mVR‘𝑡)) = ∅ ↔ (𝐶𝑉) = ∅))
9 fveq2 6103 . . . . . 6 (𝑡 = 𝑇 → (mType‘𝑡) = (mType‘𝑇))
10 ismfs.y . . . . . 6 𝑌 = (mType‘𝑇)
119, 10syl6eqr 2662 . . . . 5 (𝑡 = 𝑇 → (mType‘𝑡) = 𝑌)
12 fveq2 6103 . . . . . 6 (𝑡 = 𝑇 → (mTC‘𝑡) = (mTC‘𝑇))
13 ismfs.k . . . . . 6 𝐾 = (mTC‘𝑇)
1412, 13syl6eqr 2662 . . . . 5 (𝑡 = 𝑇 → (mTC‘𝑡) = 𝐾)
1511, 6, 14feq123d 5947 . . . 4 (𝑡 = 𝑇 → ((mType‘𝑡):(mVR‘𝑡)⟶(mTC‘𝑡) ↔ 𝑌:𝑉𝐾))
168, 15anbi12d 743 . . 3 (𝑡 = 𝑇 → ((((mCN‘𝑡) ∩ (mVR‘𝑡)) = ∅ ∧ (mType‘𝑡):(mVR‘𝑡)⟶(mTC‘𝑡)) ↔ ((𝐶𝑉) = ∅ ∧ 𝑌:𝑉𝐾)))
17 fveq2 6103 . . . . . 6 (𝑡 = 𝑇 → (mAx‘𝑡) = (mAx‘𝑇))
18 ismfs.a . . . . . 6 𝐴 = (mAx‘𝑇)
1917, 18syl6eqr 2662 . . . . 5 (𝑡 = 𝑇 → (mAx‘𝑡) = 𝐴)
20 fveq2 6103 . . . . . 6 (𝑡 = 𝑇 → (mStat‘𝑡) = (mStat‘𝑇))
21 ismfs.s . . . . . 6 𝑆 = (mStat‘𝑇)
2220, 21syl6eqr 2662 . . . . 5 (𝑡 = 𝑇 → (mStat‘𝑡) = 𝑆)
2319, 22sseq12d 3597 . . . 4 (𝑡 = 𝑇 → ((mAx‘𝑡) ⊆ (mStat‘𝑡) ↔ 𝐴𝑆))
24 fveq2 6103 . . . . . 6 (𝑡 = 𝑇 → (mVT‘𝑡) = (mVT‘𝑇))
25 ismfs.f . . . . . 6 𝐹 = (mVT‘𝑇)
2624, 25syl6eqr 2662 . . . . 5 (𝑡 = 𝑇 → (mVT‘𝑡) = 𝐹)
2711cnveqd 5220 . . . . . . . 8 (𝑡 = 𝑇(mType‘𝑡) = 𝑌)
2827imaeq1d 5384 . . . . . . 7 (𝑡 = 𝑇 → ((mType‘𝑡) “ {𝑣}) = (𝑌 “ {𝑣}))
2928eleq1d 2672 . . . . . 6 (𝑡 = 𝑇 → (((mType‘𝑡) “ {𝑣}) ∈ Fin ↔ (𝑌 “ {𝑣}) ∈ Fin))
3029notbid 307 . . . . 5 (𝑡 = 𝑇 → (¬ ((mType‘𝑡) “ {𝑣}) ∈ Fin ↔ ¬ (𝑌 “ {𝑣}) ∈ Fin))
3126, 30raleqbidv 3129 . . . 4 (𝑡 = 𝑇 → (∀𝑣 ∈ (mVT‘𝑡) ¬ ((mType‘𝑡) “ {𝑣}) ∈ Fin ↔ ∀𝑣𝐹 ¬ (𝑌 “ {𝑣}) ∈ Fin))
3223, 31anbi12d 743 . . 3 (𝑡 = 𝑇 → (((mAx‘𝑡) ⊆ (mStat‘𝑡) ∧ ∀𝑣 ∈ (mVT‘𝑡) ¬ ((mType‘𝑡) “ {𝑣}) ∈ Fin) ↔ (𝐴𝑆 ∧ ∀𝑣𝐹 ¬ (𝑌 “ {𝑣}) ∈ Fin)))
3316, 32anbi12d 743 . 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 30647 . 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 3322 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 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  {csn 4125  ◡ccnv 5037   “ cima 5041  ⟶wf 5800  ‘cfv 5804  Fincfn 7841  mCNcmcn 30611  mVRcmvar 30612  mTypecmty 30613  mVTcmvt 30614  mTCcmtc 30615  mAxcmax 30616  mStatcmsta 30626  mFScmfs 30627 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-mfs 30647 This theorem is referenced by:  mfsdisj  30701  mtyf2  30702  maxsta  30705  mvtinf  30706
 Copyright terms: Public domain W3C validator