Theorem maxsta 35586

Description: An axiom is a statement. (Contributed by Mario Carneiro, 18-Jul-2016.)

Hypotheses

Ref	Expression
maxsta.a	⊢ 𝐴 = (mAx‘𝑇)
maxsta.s	⊢ 𝑆 = (mStat‘𝑇)

Assertion

Ref	Expression
maxsta	⊢ (𝑇 ∈ mFS → 𝐴 ⊆ 𝑆)

Proof of Theorem maxsta

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2731	. . . 4 ⊢ (mCN‘𝑇) = (mCN‘𝑇)
2		eqid 2731	. . . 4 ⊢ (mVR‘𝑇) = (mVR‘𝑇)
3		eqid 2731	. . . 4 ⊢ (mType‘𝑇) = (mType‘𝑇)
4		eqid 2731	. . . 4 ⊢ (mVT‘𝑇) = (mVT‘𝑇)
5		eqid 2731	. . . 4 ⊢ (mTC‘𝑇) = (mTC‘𝑇)
6		maxsta.a	. . . 4 ⊢ 𝐴 = (mAx‘𝑇)
7		maxsta.s	. . . 4 ⊢ 𝑆 = (mStat‘𝑇)
8	1, 2, 3, 4, 5, 6, 7	ismfs 35581	. . 3 ⊢ (𝑇 ∈ mFS → (𝑇 ∈ mFS ↔ ((((mCN‘𝑇) ∩ (mVR‘𝑇)) = ∅ ∧ (mType‘𝑇):(mVR‘𝑇)⟶(mTC‘𝑇)) ∧ (𝐴 ⊆ 𝑆 ∧ ∀𝑣 ∈ (mVT‘𝑇) ¬ (◡(mType‘𝑇) “ {𝑣}) ∈ Fin))))
9	8	ibi 267	. 2 ⊢ (𝑇 ∈ mFS → ((((mCN‘𝑇) ∩ (mVR‘𝑇)) = ∅ ∧ (mType‘𝑇):(mVR‘𝑇)⟶(mTC‘𝑇)) ∧ (𝐴 ⊆ 𝑆 ∧ ∀𝑣 ∈ (mVT‘𝑇) ¬ (◡(mType‘𝑇) “ {𝑣}) ∈ Fin)))
10	9	simprld 771	1 ⊢ (𝑇 ∈ mFS → 𝐴 ⊆ 𝑆)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∀wral 3047   ∩ cin 3901   ⊆ wss 3902  ∅c0 4283  {csn 4576  ^◡ccnv 5615   “ cima 5619  ⟶wf 6477  ‘cfv 6481  Fincfn 8869  mCNcmcn 35492  mVRcmvar 35493  mTypecmty 35494  mVTcmvt 35495  mTCcmtc 35496  mAxcmax 35497  mStatcmsta 35507  mFScmfs 35508

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-fv 6489  df-mfs 35528

This theorem is referenced by:  mclsssvlem  35594  mclsax  35601  mclsind  35602  mclsppslem  35615