Theorem fixssrn 32617

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

Assertion

Ref	Expression
fixssrn	⊢ Fix 𝐴 ⊆ ran 𝐴

Proof of Theorem fixssrn

Step	Hyp	Ref	Expression
1		dffix2 32615	. 2 ⊢ Fix 𝐴 = ran (𝐴 ∩ I )
2		inss1 4052	. . 3 ⊢ (𝐴 ∩ I ) ⊆ 𝐴
3		rnss 5599	. . 3 ⊢ ((𝐴 ∩ I ) ⊆ 𝐴 → ran (𝐴 ∩ I ) ⊆ ran 𝐴)
4	2, 3	ax-mp 5	. 2 ⊢ ran (𝐴 ∩ I ) ⊆ ran 𝐴
5	1, 4	eqsstri 3853	1 ⊢ Fix 𝐴 ⊆ ran 𝐴

Syntax hints:   ∩ cin 3790   ⊆ wss 3791   I cid 5260  ran crn 5356   Fix cfix 32545

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pr 5138

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-br 4887  df-opab 4949  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-dm 5365  df-rn 5366  df-fix 32569

This theorem is referenced by: (None)