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Theorem ismfs 35159
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 𝐢 = (mCNβ€˜π‘‡)
ismfs.v 𝑉 = (mVRβ€˜π‘‡)
ismfs.y π‘Œ = (mTypeβ€˜π‘‡)
ismfs.f 𝐹 = (mVTβ€˜π‘‡)
ismfs.k 𝐾 = (mTCβ€˜π‘‡)
ismfs.a 𝐴 = (mAxβ€˜π‘‡)
ismfs.s 𝑆 = (mStatβ€˜π‘‡)
Assertion
Ref Expression
ismfs (𝑇 ∈ π‘Š β†’ (𝑇 ∈ mFS ↔ (((𝐢 ∩ 𝑉) = βˆ… ∧ π‘Œ:π‘‰βŸΆπΎ) ∧ (𝐴 βŠ† 𝑆 ∧ βˆ€π‘£ ∈ 𝐹 Β¬ (β—‘π‘Œ β€œ {𝑣}) ∈ Fin))))
Distinct variable groups:   𝑣,𝐹   𝑣,𝑇
Allowed substitution hints:   𝐴(𝑣)   𝐢(𝑣)   𝑆(𝑣)   𝐾(𝑣)   𝑉(𝑣)   π‘Š(𝑣)   π‘Œ(𝑣)

Proof of Theorem ismfs
Dummy variable 𝑑 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6897 . . . . . . 7 (𝑑 = 𝑇 β†’ (mCNβ€˜π‘‘) = (mCNβ€˜π‘‡))
2 ismfs.c . . . . . . 7 𝐢 = (mCNβ€˜π‘‡)
31, 2eqtr4di 2786 . . . . . 6 (𝑑 = 𝑇 β†’ (mCNβ€˜π‘‘) = 𝐢)
4 fveq2 6897 . . . . . . 7 (𝑑 = 𝑇 β†’ (mVRβ€˜π‘‘) = (mVRβ€˜π‘‡))
5 ismfs.v . . . . . . 7 𝑉 = (mVRβ€˜π‘‡)
64, 5eqtr4di 2786 . . . . . 6 (𝑑 = 𝑇 β†’ (mVRβ€˜π‘‘) = 𝑉)
73, 6ineq12d 4213 . . . . 5 (𝑑 = 𝑇 β†’ ((mCNβ€˜π‘‘) ∩ (mVRβ€˜π‘‘)) = (𝐢 ∩ 𝑉))
87eqeq1d 2730 . . . 4 (𝑑 = 𝑇 β†’ (((mCNβ€˜π‘‘) ∩ (mVRβ€˜π‘‘)) = βˆ… ↔ (𝐢 ∩ 𝑉) = βˆ…))
9 fveq2 6897 . . . . . 6 (𝑑 = 𝑇 β†’ (mTypeβ€˜π‘‘) = (mTypeβ€˜π‘‡))
10 ismfs.y . . . . . 6 π‘Œ = (mTypeβ€˜π‘‡)
119, 10eqtr4di 2786 . . . . 5 (𝑑 = 𝑇 β†’ (mTypeβ€˜π‘‘) = π‘Œ)
12 fveq2 6897 . . . . . 6 (𝑑 = 𝑇 β†’ (mTCβ€˜π‘‘) = (mTCβ€˜π‘‡))
13 ismfs.k . . . . . 6 𝐾 = (mTCβ€˜π‘‡)
1412, 13eqtr4di 2786 . . . . 5 (𝑑 = 𝑇 β†’ (mTCβ€˜π‘‘) = 𝐾)
1511, 6, 14feq123d 6711 . . . 4 (𝑑 = 𝑇 β†’ ((mTypeβ€˜π‘‘):(mVRβ€˜π‘‘)⟢(mTCβ€˜π‘‘) ↔ π‘Œ:π‘‰βŸΆπΎ))
168, 15anbi12d 631 . . 3 (𝑑 = 𝑇 β†’ ((((mCNβ€˜π‘‘) ∩ (mVRβ€˜π‘‘)) = βˆ… ∧ (mTypeβ€˜π‘‘):(mVRβ€˜π‘‘)⟢(mTCβ€˜π‘‘)) ↔ ((𝐢 ∩ 𝑉) = βˆ… ∧ π‘Œ:π‘‰βŸΆπΎ)))
17 fveq2 6897 . . . . . 6 (𝑑 = 𝑇 β†’ (mAxβ€˜π‘‘) = (mAxβ€˜π‘‡))
18 ismfs.a . . . . . 6 𝐴 = (mAxβ€˜π‘‡)
1917, 18eqtr4di 2786 . . . . 5 (𝑑 = 𝑇 β†’ (mAxβ€˜π‘‘) = 𝐴)
20 fveq2 6897 . . . . . 6 (𝑑 = 𝑇 β†’ (mStatβ€˜π‘‘) = (mStatβ€˜π‘‡))
21 ismfs.s . . . . . 6 𝑆 = (mStatβ€˜π‘‡)
2220, 21eqtr4di 2786 . . . . 5 (𝑑 = 𝑇 β†’ (mStatβ€˜π‘‘) = 𝑆)
2319, 22sseq12d 4013 . . . 4 (𝑑 = 𝑇 β†’ ((mAxβ€˜π‘‘) βŠ† (mStatβ€˜π‘‘) ↔ 𝐴 βŠ† 𝑆))
24 fveq2 6897 . . . . . 6 (𝑑 = 𝑇 β†’ (mVTβ€˜π‘‘) = (mVTβ€˜π‘‡))
25 ismfs.f . . . . . 6 𝐹 = (mVTβ€˜π‘‡)
2624, 25eqtr4di 2786 . . . . 5 (𝑑 = 𝑇 β†’ (mVTβ€˜π‘‘) = 𝐹)
2711cnveqd 5878 . . . . . . . 8 (𝑑 = 𝑇 β†’ β—‘(mTypeβ€˜π‘‘) = β—‘π‘Œ)
2827imaeq1d 6062 . . . . . . 7 (𝑑 = 𝑇 β†’ (β—‘(mTypeβ€˜π‘‘) β€œ {𝑣}) = (β—‘π‘Œ β€œ {𝑣}))
2928eleq1d 2814 . . . . . 6 (𝑑 = 𝑇 β†’ ((β—‘(mTypeβ€˜π‘‘) β€œ {𝑣}) ∈ Fin ↔ (β—‘π‘Œ β€œ {𝑣}) ∈ Fin))
3029notbid 318 . . . . 5 (𝑑 = 𝑇 β†’ (Β¬ (β—‘(mTypeβ€˜π‘‘) β€œ {𝑣}) ∈ Fin ↔ Β¬ (β—‘π‘Œ β€œ {𝑣}) ∈ Fin))
3126, 30raleqbidv 3339 . . . 4 (𝑑 = 𝑇 β†’ (βˆ€π‘£ ∈ (mVTβ€˜π‘‘) Β¬ (β—‘(mTypeβ€˜π‘‘) β€œ {𝑣}) ∈ Fin ↔ βˆ€π‘£ ∈ 𝐹 Β¬ (β—‘π‘Œ β€œ {𝑣}) ∈ Fin))
3223, 31anbi12d 631 . . 3 (𝑑 = 𝑇 β†’ (((mAxβ€˜π‘‘) βŠ† (mStatβ€˜π‘‘) ∧ βˆ€π‘£ ∈ (mVTβ€˜π‘‘) Β¬ (β—‘(mTypeβ€˜π‘‘) β€œ {𝑣}) ∈ Fin) ↔ (𝐴 βŠ† 𝑆 ∧ βˆ€π‘£ ∈ 𝐹 Β¬ (β—‘π‘Œ β€œ {𝑣}) ∈ Fin)))
3316, 32anbi12d 631 . 2 (𝑑 = 𝑇 β†’ (((((mCNβ€˜π‘‘) ∩ (mVRβ€˜π‘‘)) = βˆ… ∧ (mTypeβ€˜π‘‘):(mVRβ€˜π‘‘)⟢(mTCβ€˜π‘‘)) ∧ ((mAxβ€˜π‘‘) βŠ† (mStatβ€˜π‘‘) ∧ βˆ€π‘£ ∈ (mVTβ€˜π‘‘) Β¬ (β—‘(mTypeβ€˜π‘‘) β€œ {𝑣}) ∈ Fin)) ↔ (((𝐢 ∩ 𝑉) = βˆ… ∧ π‘Œ:π‘‰βŸΆπΎ) ∧ (𝐴 βŠ† 𝑆 ∧ βˆ€π‘£ ∈ 𝐹 Β¬ (β—‘π‘Œ β€œ {𝑣}) ∈ Fin))))
34 df-mfs 35106 . 2 mFS = {𝑑 ∣ ((((mCNβ€˜π‘‘) ∩ (mVRβ€˜π‘‘)) = βˆ… ∧ (mTypeβ€˜π‘‘):(mVRβ€˜π‘‘)⟢(mTCβ€˜π‘‘)) ∧ ((mAxβ€˜π‘‘) βŠ† (mStatβ€˜π‘‘) ∧ βˆ€π‘£ ∈ (mVTβ€˜π‘‘) Β¬ (β—‘(mTypeβ€˜π‘‘) β€œ {𝑣}) ∈ Fin))}
3533, 34elab2g 3669 1 (𝑇 ∈ π‘Š β†’ (𝑇 ∈ mFS ↔ (((𝐢 ∩ 𝑉) = βˆ… ∧ π‘Œ:π‘‰βŸΆπΎ) ∧ (𝐴 βŠ† 𝑆 ∧ βˆ€π‘£ ∈ 𝐹 Β¬ (β—‘π‘Œ β€œ {𝑣}) ∈ Fin))))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099  βˆ€wral 3058   ∩ cin 3946   βŠ† wss 3947  βˆ…c0 4323  {csn 4629  β—‘ccnv 5677   β€œ cima 5681  βŸΆwf 6544  β€˜cfv 6548  Fincfn 8963  mCNcmcn 35070  mVRcmvar 35071  mTypecmty 35072  mVTcmvt 35073  mTCcmtc 35074  mAxcmax 35075  mStatcmsta 35085  mFScmfs 35086
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-fv 6556  df-mfs 35106
This theorem is referenced by:  mfsdisj  35160  mtyf2  35161  maxsta  35164  mvtinf  35165
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