Theorem fixssrn 35947

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 35945	. 2 ⊢ Fix 𝐴 = ran (𝐴 ∩ I )
2		inss1 4187	. . 3 ⊢ (𝐴 ∩ I ) ⊆ 𝐴
3	2	rnssi 5880	. 2 ⊢ ran (𝐴 ∩ I ) ⊆ ran 𝐴
4	1, 3	eqsstri 3981	1 ⊢ Fix 𝐴 ⊆ ran 𝐴

Syntax hints:   ∩ cin 3901   ⊆ wss 3902   I cid 5510  ran crn 5617   Fix cfix 35875

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 2113  ax-9 2121  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

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 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-br 5092  df-opab 5154  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-dm 5626  df-rn 5627  df-fix 35899

This theorem is referenced by: (None)