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Theorem ismfs 34143
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 6842 . . . . . . 7 (𝑡 = 𝑇 → (mCN‘𝑡) = (mCN‘𝑇))
2 ismfs.c . . . . . . 7 𝐶 = (mCN‘𝑇)
31, 2eqtr4di 2794 . . . . . 6 (𝑡 = 𝑇 → (mCN‘𝑡) = 𝐶)
4 fveq2 6842 . . . . . . 7 (𝑡 = 𝑇 → (mVR‘𝑡) = (mVR‘𝑇))
5 ismfs.v . . . . . . 7 𝑉 = (mVR‘𝑇)
64, 5eqtr4di 2794 . . . . . 6 (𝑡 = 𝑇 → (mVR‘𝑡) = 𝑉)
73, 6ineq12d 4173 . . . . 5 (𝑡 = 𝑇 → ((mCN‘𝑡) ∩ (mVR‘𝑡)) = (𝐶𝑉))
87eqeq1d 2738 . . . 4 (𝑡 = 𝑇 → (((mCN‘𝑡) ∩ (mVR‘𝑡)) = ∅ ↔ (𝐶𝑉) = ∅))
9 fveq2 6842 . . . . . 6 (𝑡 = 𝑇 → (mType‘𝑡) = (mType‘𝑇))
10 ismfs.y . . . . . 6 𝑌 = (mType‘𝑇)
119, 10eqtr4di 2794 . . . . 5 (𝑡 = 𝑇 → (mType‘𝑡) = 𝑌)
12 fveq2 6842 . . . . . 6 (𝑡 = 𝑇 → (mTC‘𝑡) = (mTC‘𝑇))
13 ismfs.k . . . . . 6 𝐾 = (mTC‘𝑇)
1412, 13eqtr4di 2794 . . . . 5 (𝑡 = 𝑇 → (mTC‘𝑡) = 𝐾)
1511, 6, 14feq123d 6657 . . . 4 (𝑡 = 𝑇 → ((mType‘𝑡):(mVR‘𝑡)⟶(mTC‘𝑡) ↔ 𝑌:𝑉𝐾))
168, 15anbi12d 631 . . 3 (𝑡 = 𝑇 → ((((mCN‘𝑡) ∩ (mVR‘𝑡)) = ∅ ∧ (mType‘𝑡):(mVR‘𝑡)⟶(mTC‘𝑡)) ↔ ((𝐶𝑉) = ∅ ∧ 𝑌:𝑉𝐾)))
17 fveq2 6842 . . . . . 6 (𝑡 = 𝑇 → (mAx‘𝑡) = (mAx‘𝑇))
18 ismfs.a . . . . . 6 𝐴 = (mAx‘𝑇)
1917, 18eqtr4di 2794 . . . . 5 (𝑡 = 𝑇 → (mAx‘𝑡) = 𝐴)
20 fveq2 6842 . . . . . 6 (𝑡 = 𝑇 → (mStat‘𝑡) = (mStat‘𝑇))
21 ismfs.s . . . . . 6 𝑆 = (mStat‘𝑇)
2220, 21eqtr4di 2794 . . . . 5 (𝑡 = 𝑇 → (mStat‘𝑡) = 𝑆)
2319, 22sseq12d 3977 . . . 4 (𝑡 = 𝑇 → ((mAx‘𝑡) ⊆ (mStat‘𝑡) ↔ 𝐴𝑆))
24 fveq2 6842 . . . . . 6 (𝑡 = 𝑇 → (mVT‘𝑡) = (mVT‘𝑇))
25 ismfs.f . . . . . 6 𝐹 = (mVT‘𝑇)
2624, 25eqtr4di 2794 . . . . 5 (𝑡 = 𝑇 → (mVT‘𝑡) = 𝐹)
2711cnveqd 5831 . . . . . . . 8 (𝑡 = 𝑇(mType‘𝑡) = 𝑌)
2827imaeq1d 6012 . . . . . . 7 (𝑡 = 𝑇 → ((mType‘𝑡) “ {𝑣}) = (𝑌 “ {𝑣}))
2928eleq1d 2822 . . . . . 6 (𝑡 = 𝑇 → (((mType‘𝑡) “ {𝑣}) ∈ Fin ↔ (𝑌 “ {𝑣}) ∈ Fin))
3029notbid 317 . . . . 5 (𝑡 = 𝑇 → (¬ ((mType‘𝑡) “ {𝑣}) ∈ Fin ↔ ¬ (𝑌 “ {𝑣}) ∈ Fin))
3126, 30raleqbidv 3319 . . . 4 (𝑡 = 𝑇 → (∀𝑣 ∈ (mVT‘𝑡) ¬ ((mType‘𝑡) “ {𝑣}) ∈ Fin ↔ ∀𝑣𝐹 ¬ (𝑌 “ {𝑣}) ∈ Fin))
3223, 31anbi12d 631 . . 3 (𝑡 = 𝑇 → (((mAx‘𝑡) ⊆ (mStat‘𝑡) ∧ ∀𝑣 ∈ (mVT‘𝑡) ¬ ((mType‘𝑡) “ {𝑣}) ∈ Fin) ↔ (𝐴𝑆 ∧ ∀𝑣𝐹 ¬ (𝑌 “ {𝑣}) ∈ Fin)))
3316, 32anbi12d 631 . 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 34090 . 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 3632 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 205  wa 396   = wceq 1541  wcel 2106  wral 3064  cin 3909  wss 3910  c0 4282  {csn 4586  ccnv 5632  cima 5636  wf 6492  cfv 6496  Fincfn 8883  mCNcmcn 34054  mVRcmvar 34055  mTypecmty 34056  mVTcmvt 34057  mTCcmtc 34058  mAxcmax 34059  mStatcmsta 34069  mFScmfs 34070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3065  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fv 6504  df-mfs 34090
This theorem is referenced by:  mfsdisj  34144  mtyf2  34145  maxsta  34148  mvtinf  34149
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