Theorem maxsta 34841

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 2730	. . . 4 ⊢ (mCN‘𝑇) = (mCN‘𝑇)
2		eqid 2730	. . . 4 ⊢ (mVR‘𝑇) = (mVR‘𝑇)
3		eqid 2730	. . . 4 ⊢ (mType‘𝑇) = (mType‘𝑇)
4		eqid 2730	. . . 4 ⊢ (mVT‘𝑇) = (mVT‘𝑇)
5		eqid 2730	. . . 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 34836	. . 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 266	. 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 768	1 ⊢ (𝑇 ∈ mFS → 𝐴 ⊆ 𝑆)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 394   = wceq 1539   ∈ wcel 2104  ∀wral 3059   ∩ cin 3948   ⊆ wss 3949  ∅c0 4323  {csn 4629  ^◡ccnv 5676   “ cima 5680  ⟶wf 6540  ‘cfv 6544  Fincfn 8943  mCNcmcn 34747  mVRcmvar 34748  mTypecmty 34749  mVTcmvt 34750  mTCcmtc 34751  mAxcmax 34752  mStatcmsta 34762  mFScmfs 34763

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-rab 3431  df-v 3474  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-mfs 34783

This theorem is referenced by:  mclsssvlem  34849  mclsax  34856  mclsind  34857  mclsppslem  34870