Theorem fixssrn 36133

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 36131	. 2 ⊢ Fix 𝐴 = ran (𝐴 ∩ I )
2		inss1 4165	. . 3 ⊢ (𝐴 ∩ I ) ⊆ 𝐴
3	2	rnssi 5882	. 2 ⊢ ran (𝐴 ∩ I ) ⊆ ran 𝐴
4	1, 3	eqsstri 3961	1 ⊢ Fix 𝐴 ⊆ ran 𝐴

Syntax hints:   ∩ cin 3882   ⊆ wss 3883   I cid 5512  ran crn 5619   Fix cfix 36061

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5218  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-br 5073  df-opab 5135  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-dm 5628  df-rn 5629  df-fix 36085

This theorem is referenced by: (None)