Theorem fixssdm 36883

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 36836	. 2 ⊢ Fix 𝐴 = dom (𝐴 ∩ I )
2		inss1 4136	. . 3 ⊢ (𝐴 ∩ I ) ⊆ 𝐴
3		dmss 5896	. . 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 3959	1 ⊢ Fix 𝐴 ⊆ dom 𝐴

This proof depends on syntax axioms:   ⊆ wss 3900   ∩ cin 4092   I cid 5552  dom cdm 5658   Fix cfix 36812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-br 5109  df-dm 5668  df-fix 36836

This theorem is used by: (None)