Theorem maxsta 30705

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 2610	. . . . 5 ⊢ (mCN‘𝑇) = (mCN‘𝑇)
2		eqid 2610	. . . . 5 ⊢ (mVR‘𝑇) = (mVR‘𝑇)
3		eqid 2610	. . . . 5 ⊢ (mType‘𝑇) = (mType‘𝑇)
4		eqid 2610	. . . . 5 ⊢ (mVT‘𝑇) = (mVT‘𝑇)
5		eqid 2610	. . . . 5 ⊢ (mTC‘𝑇) = (mTC‘𝑇)
6		maxsta.a	. . . . 5 ⊢ 𝐴 = (mAx‘𝑇)
7		maxsta.s	. . . . 5 ⊢ 𝑆 = (mStat‘𝑇)
8	1, 2, 3, 4, 5, 6, 7	ismfs 30700	. . . 4 ⊢ (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 255	. . 3 ⊢ (𝑇 ∈ mFS → ((((mCN‘𝑇) ∩ (mVR‘𝑇)) = ∅ ∧ (mType‘𝑇):(mVR‘𝑇)⟶(mTC‘𝑇)) ∧ (𝐴 ⊆ 𝑆 ∧ ∀𝑣 ∈ (mVT‘𝑇) ¬ (◡(mType‘𝑇) “ {𝑣}) ∈ Fin)))
10	9	simprd 478	. 2 ⊢ (𝑇 ∈ mFS → (𝐴 ⊆ 𝑆 ∧ ∀𝑣 ∈ (mVT‘𝑇) ¬ (◡(mType‘𝑇) “ {𝑣}) ∈ Fin))
11	10	simpld 474	1 ⊢ (𝑇 ∈ mFS → 𝐴 ⊆ 𝑆)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  {csn 4125  ^◡ccnv 5037   “ cima 5041  ⟶wf 5800  ‘cfv 5804  Fincfn 7841  mCNcmcn 30611  mVRcmvar 30612  mTypecmty 30613  mVTcmvt 30614  mTCcmtc 30615  mAxcmax 30616  mStatcmsta 30626  mFScmfs 30627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-mfs 30647

This theorem is referenced by:  mclsssvlem  30713  mclsax  30720  mclsind  30721  mclsppslem  30734