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Theorem ismfs 35057
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 6882 . . . . . . 7 (𝑑 = 𝑇 β†’ (mCNβ€˜π‘‘) = (mCNβ€˜π‘‡))
2 ismfs.c . . . . . . 7 𝐢 = (mCNβ€˜π‘‡)
31, 2eqtr4di 2782 . . . . . 6 (𝑑 = 𝑇 β†’ (mCNβ€˜π‘‘) = 𝐢)
4 fveq2 6882 . . . . . . 7 (𝑑 = 𝑇 β†’ (mVRβ€˜π‘‘) = (mVRβ€˜π‘‡))
5 ismfs.v . . . . . . 7 𝑉 = (mVRβ€˜π‘‡)
64, 5eqtr4di 2782 . . . . . 6 (𝑑 = 𝑇 β†’ (mVRβ€˜π‘‘) = 𝑉)
73, 6ineq12d 4206 . . . . 5 (𝑑 = 𝑇 β†’ ((mCNβ€˜π‘‘) ∩ (mVRβ€˜π‘‘)) = (𝐢 ∩ 𝑉))
87eqeq1d 2726 . . . 4 (𝑑 = 𝑇 β†’ (((mCNβ€˜π‘‘) ∩ (mVRβ€˜π‘‘)) = βˆ… ↔ (𝐢 ∩ 𝑉) = βˆ…))
9 fveq2 6882 . . . . . 6 (𝑑 = 𝑇 β†’ (mTypeβ€˜π‘‘) = (mTypeβ€˜π‘‡))
10 ismfs.y . . . . . 6 π‘Œ = (mTypeβ€˜π‘‡)
119, 10eqtr4di 2782 . . . . 5 (𝑑 = 𝑇 β†’ (mTypeβ€˜π‘‘) = π‘Œ)
12 fveq2 6882 . . . . . 6 (𝑑 = 𝑇 β†’ (mTCβ€˜π‘‘) = (mTCβ€˜π‘‡))
13 ismfs.k . . . . . 6 𝐾 = (mTCβ€˜π‘‡)
1412, 13eqtr4di 2782 . . . . 5 (𝑑 = 𝑇 β†’ (mTCβ€˜π‘‘) = 𝐾)
1511, 6, 14feq123d 6697 . . . 4 (𝑑 = 𝑇 β†’ ((mTypeβ€˜π‘‘):(mVRβ€˜π‘‘)⟢(mTCβ€˜π‘‘) ↔ π‘Œ:π‘‰βŸΆπΎ))
168, 15anbi12d 630 . . 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 6882 . . . . . 6 (𝑑 = 𝑇 β†’ (mAxβ€˜π‘‘) = (mAxβ€˜π‘‡))
18 ismfs.a . . . . . 6 𝐴 = (mAxβ€˜π‘‡)
1917, 18eqtr4di 2782 . . . . 5 (𝑑 = 𝑇 β†’ (mAxβ€˜π‘‘) = 𝐴)
20 fveq2 6882 . . . . . 6 (𝑑 = 𝑇 β†’ (mStatβ€˜π‘‘) = (mStatβ€˜π‘‡))
21 ismfs.s . . . . . 6 𝑆 = (mStatβ€˜π‘‡)
2220, 21eqtr4di 2782 . . . . 5 (𝑑 = 𝑇 β†’ (mStatβ€˜π‘‘) = 𝑆)
2319, 22sseq12d 4008 . . . 4 (𝑑 = 𝑇 β†’ ((mAxβ€˜π‘‘) βŠ† (mStatβ€˜π‘‘) ↔ 𝐴 βŠ† 𝑆))
24 fveq2 6882 . . . . . 6 (𝑑 = 𝑇 β†’ (mVTβ€˜π‘‘) = (mVTβ€˜π‘‡))
25 ismfs.f . . . . . 6 𝐹 = (mVTβ€˜π‘‡)
2624, 25eqtr4di 2782 . . . . 5 (𝑑 = 𝑇 β†’ (mVTβ€˜π‘‘) = 𝐹)
2711cnveqd 5866 . . . . . . . 8 (𝑑 = 𝑇 β†’ β—‘(mTypeβ€˜π‘‘) = β—‘π‘Œ)
2827imaeq1d 6049 . . . . . . 7 (𝑑 = 𝑇 β†’ (β—‘(mTypeβ€˜π‘‘) β€œ {𝑣}) = (β—‘π‘Œ β€œ {𝑣}))
2928eleq1d 2810 . . . . . 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 3334 . . . 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 630 . . 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 630 . 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 35004 . 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 3663 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 1533   ∈ wcel 2098  βˆ€wral 3053   ∩ cin 3940   βŠ† wss 3941  βˆ…c0 4315  {csn 4621  β—‘ccnv 5666   β€œ cima 5670  βŸΆwf 6530  β€˜cfv 6534  Fincfn 8936  mCNcmcn 34968  mVRcmvar 34969  mTypecmty 34970  mVTcmvt 34971  mTCcmtc 34972  mAxcmax 34973  mStatcmsta 34983  mFScmfs 34984
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-fv 6542  df-mfs 35004
This theorem is referenced by:  mfsdisj  35058  mtyf2  35059  maxsta  35062  mvtinf  35063
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