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Theorem ismfs 31778
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 6332 . . . . . . 7 (𝑡 = 𝑇 → (mCN‘𝑡) = (mCN‘𝑇))
2 ismfs.c . . . . . . 7 𝐶 = (mCN‘𝑇)
31, 2syl6eqr 2822 . . . . . 6 (𝑡 = 𝑇 → (mCN‘𝑡) = 𝐶)
4 fveq2 6332 . . . . . . 7 (𝑡 = 𝑇 → (mVR‘𝑡) = (mVR‘𝑇))
5 ismfs.v . . . . . . 7 𝑉 = (mVR‘𝑇)
64, 5syl6eqr 2822 . . . . . 6 (𝑡 = 𝑇 → (mVR‘𝑡) = 𝑉)
73, 6ineq12d 3964 . . . . 5 (𝑡 = 𝑇 → ((mCN‘𝑡) ∩ (mVR‘𝑡)) = (𝐶𝑉))
87eqeq1d 2772 . . . 4 (𝑡 = 𝑇 → (((mCN‘𝑡) ∩ (mVR‘𝑡)) = ∅ ↔ (𝐶𝑉) = ∅))
9 fveq2 6332 . . . . . 6 (𝑡 = 𝑇 → (mType‘𝑡) = (mType‘𝑇))
10 ismfs.y . . . . . 6 𝑌 = (mType‘𝑇)
119, 10syl6eqr 2822 . . . . 5 (𝑡 = 𝑇 → (mType‘𝑡) = 𝑌)
12 fveq2 6332 . . . . . 6 (𝑡 = 𝑇 → (mTC‘𝑡) = (mTC‘𝑇))
13 ismfs.k . . . . . 6 𝐾 = (mTC‘𝑇)
1412, 13syl6eqr 2822 . . . . 5 (𝑡 = 𝑇 → (mTC‘𝑡) = 𝐾)
1511, 6, 14feq123d 6174 . . . 4 (𝑡 = 𝑇 → ((mType‘𝑡):(mVR‘𝑡)⟶(mTC‘𝑡) ↔ 𝑌:𝑉𝐾))
168, 15anbi12d 608 . . 3 (𝑡 = 𝑇 → ((((mCN‘𝑡) ∩ (mVR‘𝑡)) = ∅ ∧ (mType‘𝑡):(mVR‘𝑡)⟶(mTC‘𝑡)) ↔ ((𝐶𝑉) = ∅ ∧ 𝑌:𝑉𝐾)))
17 fveq2 6332 . . . . . 6 (𝑡 = 𝑇 → (mAx‘𝑡) = (mAx‘𝑇))
18 ismfs.a . . . . . 6 𝐴 = (mAx‘𝑇)
1917, 18syl6eqr 2822 . . . . 5 (𝑡 = 𝑇 → (mAx‘𝑡) = 𝐴)
20 fveq2 6332 . . . . . 6 (𝑡 = 𝑇 → (mStat‘𝑡) = (mStat‘𝑇))
21 ismfs.s . . . . . 6 𝑆 = (mStat‘𝑇)
2220, 21syl6eqr 2822 . . . . 5 (𝑡 = 𝑇 → (mStat‘𝑡) = 𝑆)
2319, 22sseq12d 3781 . . . 4 (𝑡 = 𝑇 → ((mAx‘𝑡) ⊆ (mStat‘𝑡) ↔ 𝐴𝑆))
24 fveq2 6332 . . . . . 6 (𝑡 = 𝑇 → (mVT‘𝑡) = (mVT‘𝑇))
25 ismfs.f . . . . . 6 𝐹 = (mVT‘𝑇)
2624, 25syl6eqr 2822 . . . . 5 (𝑡 = 𝑇 → (mVT‘𝑡) = 𝐹)
2711cnveqd 5436 . . . . . . . 8 (𝑡 = 𝑇(mType‘𝑡) = 𝑌)
2827imaeq1d 5606 . . . . . . 7 (𝑡 = 𝑇 → ((mType‘𝑡) “ {𝑣}) = (𝑌 “ {𝑣}))
2928eleq1d 2834 . . . . . 6 (𝑡 = 𝑇 → (((mType‘𝑡) “ {𝑣}) ∈ Fin ↔ (𝑌 “ {𝑣}) ∈ Fin))
3029notbid 307 . . . . 5 (𝑡 = 𝑇 → (¬ ((mType‘𝑡) “ {𝑣}) ∈ Fin ↔ ¬ (𝑌 “ {𝑣}) ∈ Fin))
3126, 30raleqbidv 3300 . . . 4 (𝑡 = 𝑇 → (∀𝑣 ∈ (mVT‘𝑡) ¬ ((mType‘𝑡) “ {𝑣}) ∈ Fin ↔ ∀𝑣𝐹 ¬ (𝑌 “ {𝑣}) ∈ Fin))
3223, 31anbi12d 608 . . 3 (𝑡 = 𝑇 → (((mAx‘𝑡) ⊆ (mStat‘𝑡) ∧ ∀𝑣 ∈ (mVT‘𝑡) ¬ ((mType‘𝑡) “ {𝑣}) ∈ Fin) ↔ (𝐴𝑆 ∧ ∀𝑣𝐹 ¬ (𝑌 “ {𝑣}) ∈ Fin)))
3316, 32anbi12d 608 . 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 31725 . 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 3502 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 196  wa 382   = wceq 1630  wcel 2144  wral 3060  cin 3720  wss 3721  c0 4061  {csn 4314  ccnv 5248  cima 5252  wf 6027  cfv 6031  Fincfn 8108  mCNcmcn 31689  mVRcmvar 31690  mTypecmty 31691  mVTcmvt 31692  mTCcmtc 31693  mAxcmax 31694  mStatcmsta 31704  mFScmfs 31705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039  df-mfs 31725
This theorem is referenced by:  mfsdisj  31779  mtyf2  31780  maxsta  31783  mvtinf  31784
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