Theorem fixssrn 35863

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 35861	. 2 ⊢ Fix 𝐴 = ran (𝐴 ∩ I )
2		inss1 4252	. . 3 ⊢ (𝐴 ∩ I ) ⊆ 𝐴
3	2	rnssi 5964	. 2 ⊢ ran (𝐴 ∩ I ) ⊆ ran 𝐴
4	1, 3	eqsstri 4037	1 ⊢ Fix 𝐴 ⊆ ran 𝐴

Syntax hints:   ∩ cin 3969   ⊆ wss 3970   I cid 5596  ran crn 5700   Fix cfix 35791

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-ext 2705  ax-sep 5320  ax-nul 5327  ax-pr 5450

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3064  df-rex 3073  df-rab 3439  df-v 3484  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-nul 4348  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5170  df-opab 5232  df-id 5597  df-xp 5705  df-rel 5706  df-cnv 5707  df-dm 5709  df-rn 5710  df-fix 35815

This theorem is referenced by: (None)