Theorem ismfs 35555

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	⊢ (𝑇 ∈ 𝑊 → (𝑇 ∈ mFS ↔ (((𝐶 ∩ 𝑉) = ∅ ∧ 𝑌:𝑉⟶𝐾) ∧ (𝐴 ⊆ 𝑆 ∧ ∀𝑣 ∈ 𝐹 ¬ (◡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.

Step	Hyp	Ref	Expression
1		fveq2 6905	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (mCN‘𝑡) = (mCN‘𝑇))
2		ismfs.c	. . . . . . 7 ⊢ 𝐶 = (mCN‘𝑇)
3	1, 2	eqtr4di 2794	. . . . . 6 ⊢ (𝑡 = 𝑇 → (mCN‘𝑡) = 𝐶)
4		fveq2 6905	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (mVR‘𝑡) = (mVR‘𝑇))
5		ismfs.v	. . . . . . 7 ⊢ 𝑉 = (mVR‘𝑇)
6	4, 5	eqtr4di 2794	. . . . . 6 ⊢ (𝑡 = 𝑇 → (mVR‘𝑡) = 𝑉)
7	3, 6	ineq12d 4220	. . . . 5 ⊢ (𝑡 = 𝑇 → ((mCN‘𝑡) ∩ (mVR‘𝑡)) = (𝐶 ∩ 𝑉))
8	7	eqeq1d 2738	. . . 4 ⊢ (𝑡 = 𝑇 → (((mCN‘𝑡) ∩ (mVR‘𝑡)) = ∅ ↔ (𝐶 ∩ 𝑉) = ∅))
9		fveq2 6905	. . . . . 6 ⊢ (𝑡 = 𝑇 → (mType‘𝑡) = (mType‘𝑇))
10		ismfs.y	. . . . . 6 ⊢ 𝑌 = (mType‘𝑇)
11	9, 10	eqtr4di 2794	. . . . 5 ⊢ (𝑡 = 𝑇 → (mType‘𝑡) = 𝑌)
12		fveq2 6905	. . . . . 6 ⊢ (𝑡 = 𝑇 → (mTC‘𝑡) = (mTC‘𝑇))
13		ismfs.k	. . . . . 6 ⊢ 𝐾 = (mTC‘𝑇)
14	12, 13	eqtr4di 2794	. . . . 5 ⊢ (𝑡 = 𝑇 → (mTC‘𝑡) = 𝐾)
15	11, 6, 14	feq123d 6724	. . . 4 ⊢ (𝑡 = 𝑇 → ((mType‘𝑡):(mVR‘𝑡)⟶(mTC‘𝑡) ↔ 𝑌:𝑉⟶𝐾))
16	8, 15	anbi12d 632	. . 3 ⊢ (𝑡 = 𝑇 → ((((mCN‘𝑡) ∩ (mVR‘𝑡)) = ∅ ∧ (mType‘𝑡):(mVR‘𝑡)⟶(mTC‘𝑡)) ↔ ((𝐶 ∩ 𝑉) = ∅ ∧ 𝑌:𝑉⟶𝐾)))
17		fveq2 6905	. . . . . 6 ⊢ (𝑡 = 𝑇 → (mAx‘𝑡) = (mAx‘𝑇))
18		ismfs.a	. . . . . 6 ⊢ 𝐴 = (mAx‘𝑇)
19	17, 18	eqtr4di 2794	. . . . 5 ⊢ (𝑡 = 𝑇 → (mAx‘𝑡) = 𝐴)
20		fveq2 6905	. . . . . 6 ⊢ (𝑡 = 𝑇 → (mStat‘𝑡) = (mStat‘𝑇))
21		ismfs.s	. . . . . 6 ⊢ 𝑆 = (mStat‘𝑇)
22	20, 21	eqtr4di 2794	. . . . 5 ⊢ (𝑡 = 𝑇 → (mStat‘𝑡) = 𝑆)
23	19, 22	sseq12d 4016	. . . 4 ⊢ (𝑡 = 𝑇 → ((mAx‘𝑡) ⊆ (mStat‘𝑡) ↔ 𝐴 ⊆ 𝑆))
24		fveq2 6905	. . . . . 6 ⊢ (𝑡 = 𝑇 → (mVT‘𝑡) = (mVT‘𝑇))
25		ismfs.f	. . . . . 6 ⊢ 𝐹 = (mVT‘𝑇)
26	24, 25	eqtr4di 2794	. . . . 5 ⊢ (𝑡 = 𝑇 → (mVT‘𝑡) = 𝐹)
27	11	cnveqd 5885	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → ◡(mType‘𝑡) = ◡𝑌)
28	27	imaeq1d 6076	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (◡(mType‘𝑡) “ {𝑣}) = (◡𝑌 “ {𝑣}))
29	28	eleq1d 2825	. . . . . 6 ⊢ (𝑡 = 𝑇 → ((◡(mType‘𝑡) “ {𝑣}) ∈ Fin ↔ (◡𝑌 “ {𝑣}) ∈ Fin))
30	29	notbid 318	. . . . 5 ⊢ (𝑡 = 𝑇 → (¬ (◡(mType‘𝑡) “ {𝑣}) ∈ Fin ↔ ¬ (◡𝑌 “ {𝑣}) ∈ Fin))
31	26, 30	raleqbidv 3345	. . . 4 ⊢ (𝑡 = 𝑇 → (∀𝑣 ∈ (mVT‘𝑡) ¬ (◡(mType‘𝑡) “ {𝑣}) ∈ Fin ↔ ∀𝑣 ∈ 𝐹 ¬ (◡𝑌 “ {𝑣}) ∈ Fin))
32	23, 31	anbi12d 632	. . 3 ⊢ (𝑡 = 𝑇 → (((mAx‘𝑡) ⊆ (mStat‘𝑡) ∧ ∀𝑣 ∈ (mVT‘𝑡) ¬ (◡(mType‘𝑡) “ {𝑣}) ∈ Fin) ↔ (𝐴 ⊆ 𝑆 ∧ ∀𝑣 ∈ 𝐹 ¬ (◡𝑌 “ {𝑣}) ∈ Fin)))
33	16, 32	anbi12d 632	. 2 ⊢ (𝑡 = 𝑇 → (((((mCN‘𝑡) ∩ (mVR‘𝑡)) = ∅ ∧ (mType‘𝑡):(mVR‘𝑡)⟶(mTC‘𝑡)) ∧ ((mAx‘𝑡) ⊆ (mStat‘𝑡) ∧ ∀𝑣 ∈ (mVT‘𝑡) ¬ (◡(mType‘𝑡) “ {𝑣}) ∈ Fin)) ↔ (((𝐶 ∩ 𝑉) = ∅ ∧ 𝑌:𝑉⟶𝐾) ∧ (𝐴 ⊆ 𝑆 ∧ ∀𝑣 ∈ 𝐹 ¬ (◡𝑌 “ {𝑣}) ∈ Fin))))
34		df-mfs 35502	. 2 ⊢ mFS = {𝑡 ∣ ((((mCN‘𝑡) ∩ (mVR‘𝑡)) = ∅ ∧ (mType‘𝑡):(mVR‘𝑡)⟶(mTC‘𝑡)) ∧ ((mAx‘𝑡) ⊆ (mStat‘𝑡) ∧ ∀𝑣 ∈ (mVT‘𝑡) ¬ (◡(mType‘𝑡) “ {𝑣}) ∈ Fin))}
35	33, 34	elab2g 3679	1 ⊢ (𝑇 ∈ 𝑊 → (𝑇 ∈ mFS ↔ (((𝐶 ∩ 𝑉) = ∅ ∧ 𝑌:𝑉⟶𝐾) ∧ (𝐴 ⊆ 𝑆 ∧ ∀𝑣 ∈ 𝐹 ¬ (◡𝑌 “ {𝑣}) ∈ Fin))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539   ∈ wcel 2107  ∀wral 3060   ∩ cin 3949   ⊆ wss 3950  ∅c0 4332  {csn 4625  ^◡ccnv 5683   “ cima 5687  ⟶wf 6556  ‘cfv 6560  Fincfn 8986  mCNcmcn 35466  mVRcmvar 35467  mTypecmty 35468  mVTcmvt 35469  mTCcmtc 35470  mAxcmax 35471  mStatcmsta 35481  mFScmfs 35482

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fv 6568  df-mfs 35502

This theorem is referenced by:  mfsdisj  35556  mtyf2  35557  maxsta  35560  mvtinf  35561