Theorem fixssdm 35971

Description: The fixpoints of a class are a subset of its domain. (Contributed by Scott Fenton, 16-Apr-2012.)

Assertion

Ref	Expression
fixssdm	⊢ Fix 𝐴 ⊆ dom 𝐴

Proof of Theorem fixssdm

Step	Hyp	Ref	Expression
1		df-fix 35924	. 2 ⊢ Fix 𝐴 = dom (𝐴 ∩ I )
2		inss1 4186	. . 3 ⊢ (𝐴 ∩ I ) ⊆ 𝐴
3		dmss 5848	. . 3 ⊢ ((𝐴 ∩ I ) ⊆ 𝐴 → dom (𝐴 ∩ I ) ⊆ dom 𝐴)
4	2, 3	ax-mp 5	. 2 ⊢ dom (𝐴 ∩ I ) ⊆ dom 𝐴
5	1, 4	eqsstri 3977	1 ⊢ Fix 𝐴 ⊆ dom 𝐴

Syntax hints:   ∩ cin 3897   ⊆ wss 3898   I cid 5515  dom cdm 5621   Fix cfix 35900

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 2115  ax-9 2123  ax-ext 2705

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 2712  df-cleq 2725  df-clel 2808  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-br 5096  df-dm 5631  df-fix 35924

This theorem is referenced by: (None)