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Theorem fixssrn 31184
 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
StepHypRef Expression
1 dffix2 31182 . 2 Fix 𝐴 = ran (𝐴 ∩ I )
2 inss1 3795 . . 3 (𝐴 ∩ I ) ⊆ 𝐴
3 rnss 5275 . . 3 ((𝐴 ∩ I ) ⊆ 𝐴 → ran (𝐴 ∩ I ) ⊆ ran 𝐴)
42, 3ax-mp 5 . 2 ran (𝐴 ∩ I ) ⊆ ran 𝐴
51, 4eqsstri 3598 1 Fix 𝐴 ⊆ ran 𝐴
 Colors of variables: wff setvar class Syntax hints:   ∩ cin 3539   ⊆ wss 3540   I cid 4948  ran crn 5039   Fix cfix 31111 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-dm 5048  df-rn 5049  df-fix 31135 This theorem is referenced by: (None)
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