Theorem fixssdm 33362

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 33315	. 2 ⊢ Fix 𝐴 = dom (𝐴 ∩ I )
2		inss1 4204	. . 3 ⊢ (𝐴 ∩ I ) ⊆ 𝐴
3		dmss 5765	. . 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 4000	1 ⊢ Fix 𝐴 ⊆ dom 𝐴

Syntax hints:   ∩ cin 3934   ⊆ wss 3935   I cid 5453  dom cdm 5549   Fix cfix 33291

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-br 5059  df-dm 5559  df-fix 33315

This theorem is referenced by: (None)