Theorem fixssdm 36254

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 36207	. 2 ⊢ Fix 𝐴 = dom (𝐴 ∩ I )
2		inss1 4188	. . 3 ⊢ (𝐴 ∩ I ) ⊆ 𝐴
3		dmss 5878	. . 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 3982	1 ⊢ Fix 𝐴 ⊆ dom 𝐴

Syntax hints:   ∩ cin 3903   ⊆ wss 3904   I cid 5541  dom cdm 5647   Fix cfix 36183

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-dm 5657  df-fix 36207

This theorem is referenced by: (None)