Theorem maxsta 35789

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 2740	. . . 4 ⊢ (mCN‘𝑇) = (mCN‘𝑇)
2		eqid 2740	. . . 4 ⊢ (mVR‘𝑇) = (mVR‘𝑇)
3		eqid 2740	. . . 4 ⊢ (mType‘𝑇) = (mType‘𝑇)
4		eqid 2740	. . . 4 ⊢ (mVT‘𝑇) = (mVT‘𝑇)
5		eqid 2740	. . . 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 35784	. . 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 268	. 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 777	1 ⊢ (𝑇 ∈ mFS → 𝐴 ⊆ 𝑆)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∀wral 3054   ∩ cin 3889   ⊆ wss 3890  ∅c0 4268  {csn 4562  ^◡ccnv 5624   “ cima 5628  ⟶wf 6488  ‘cfv 6492  Fincfn 8890  mCNcmcn 35695  mVRcmvar 35696  mTypecmty 35697  mVTcmvt 35698  mTCcmtc 35699  mAxcmax 35700  mStatcmsta 35710  mFScmfs 35711

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ral 3055  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-mfs 35731

This theorem is referenced by:  mclsssvlem  35797  mclsax  35804  mclsind  35805  mclsppslem  35818