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Theorem ismfs 31431
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 6189 . . . . . . 7 (𝑡 = 𝑇 → (mCN‘𝑡) = (mCN‘𝑇))
2 ismfs.c . . . . . . 7 𝐶 = (mCN‘𝑇)
31, 2syl6eqr 2673 . . . . . 6 (𝑡 = 𝑇 → (mCN‘𝑡) = 𝐶)
4 fveq2 6189 . . . . . . 7 (𝑡 = 𝑇 → (mVR‘𝑡) = (mVR‘𝑇))
5 ismfs.v . . . . . . 7 𝑉 = (mVR‘𝑇)
64, 5syl6eqr 2673 . . . . . 6 (𝑡 = 𝑇 → (mVR‘𝑡) = 𝑉)
73, 6ineq12d 3813 . . . . 5 (𝑡 = 𝑇 → ((mCN‘𝑡) ∩ (mVR‘𝑡)) = (𝐶𝑉))
87eqeq1d 2623 . . . 4 (𝑡 = 𝑇 → (((mCN‘𝑡) ∩ (mVR‘𝑡)) = ∅ ↔ (𝐶𝑉) = ∅))
9 fveq2 6189 . . . . . 6 (𝑡 = 𝑇 → (mType‘𝑡) = (mType‘𝑇))
10 ismfs.y . . . . . 6 𝑌 = (mType‘𝑇)
119, 10syl6eqr 2673 . . . . 5 (𝑡 = 𝑇 → (mType‘𝑡) = 𝑌)
12 fveq2 6189 . . . . . 6 (𝑡 = 𝑇 → (mTC‘𝑡) = (mTC‘𝑇))
13 ismfs.k . . . . . 6 𝐾 = (mTC‘𝑇)
1412, 13syl6eqr 2673 . . . . 5 (𝑡 = 𝑇 → (mTC‘𝑡) = 𝐾)
1511, 6, 14feq123d 6032 . . . 4 (𝑡 = 𝑇 → ((mType‘𝑡):(mVR‘𝑡)⟶(mTC‘𝑡) ↔ 𝑌:𝑉𝐾))
168, 15anbi12d 747 . . 3 (𝑡 = 𝑇 → ((((mCN‘𝑡) ∩ (mVR‘𝑡)) = ∅ ∧ (mType‘𝑡):(mVR‘𝑡)⟶(mTC‘𝑡)) ↔ ((𝐶𝑉) = ∅ ∧ 𝑌:𝑉𝐾)))
17 fveq2 6189 . . . . . 6 (𝑡 = 𝑇 → (mAx‘𝑡) = (mAx‘𝑇))
18 ismfs.a . . . . . 6 𝐴 = (mAx‘𝑇)
1917, 18syl6eqr 2673 . . . . 5 (𝑡 = 𝑇 → (mAx‘𝑡) = 𝐴)
20 fveq2 6189 . . . . . 6 (𝑡 = 𝑇 → (mStat‘𝑡) = (mStat‘𝑇))
21 ismfs.s . . . . . 6 𝑆 = (mStat‘𝑇)
2220, 21syl6eqr 2673 . . . . 5 (𝑡 = 𝑇 → (mStat‘𝑡) = 𝑆)
2319, 22sseq12d 3632 . . . 4 (𝑡 = 𝑇 → ((mAx‘𝑡) ⊆ (mStat‘𝑡) ↔ 𝐴𝑆))
24 fveq2 6189 . . . . . 6 (𝑡 = 𝑇 → (mVT‘𝑡) = (mVT‘𝑇))
25 ismfs.f . . . . . 6 𝐹 = (mVT‘𝑇)
2624, 25syl6eqr 2673 . . . . 5 (𝑡 = 𝑇 → (mVT‘𝑡) = 𝐹)
2711cnveqd 5296 . . . . . . . 8 (𝑡 = 𝑇(mType‘𝑡) = 𝑌)
2827imaeq1d 5463 . . . . . . 7 (𝑡 = 𝑇 → ((mType‘𝑡) “ {𝑣}) = (𝑌 “ {𝑣}))
2928eleq1d 2685 . . . . . 6 (𝑡 = 𝑇 → (((mType‘𝑡) “ {𝑣}) ∈ Fin ↔ (𝑌 “ {𝑣}) ∈ Fin))
3029notbid 308 . . . . 5 (𝑡 = 𝑇 → (¬ ((mType‘𝑡) “ {𝑣}) ∈ Fin ↔ ¬ (𝑌 “ {𝑣}) ∈ Fin))
3126, 30raleqbidv 3150 . . . 4 (𝑡 = 𝑇 → (∀𝑣 ∈ (mVT‘𝑡) ¬ ((mType‘𝑡) “ {𝑣}) ∈ Fin ↔ ∀𝑣𝐹 ¬ (𝑌 “ {𝑣}) ∈ Fin))
3223, 31anbi12d 747 . . 3 (𝑡 = 𝑇 → (((mAx‘𝑡) ⊆ (mStat‘𝑡) ∧ ∀𝑣 ∈ (mVT‘𝑡) ¬ ((mType‘𝑡) “ {𝑣}) ∈ Fin) ↔ (𝐴𝑆 ∧ ∀𝑣𝐹 ¬ (𝑌 “ {𝑣}) ∈ Fin)))
3316, 32anbi12d 747 . 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 31378 . 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 3351 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 384   = wceq 1482  wcel 1989  wral 2911  cin 3571  wss 3572  c0 3913  {csn 4175  ccnv 5111  cima 5115  wf 5882  cfv 5886  Fincfn 7952  mCNcmcn 31342  mVRcmvar 31343  mTypecmty 31344  mVTcmvt 31345  mTCcmtc 31346  mAxcmax 31347  mStatcmsta 31357  mFScmfs 31358
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-fv 5894  df-mfs 31378
This theorem is referenced by:  mfsdisj  31432  mtyf2  31433  maxsta  31436  mvtinf  31437
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