Theorem fixssrn 33372

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 33370	. 2 ⊢ Fix 𝐴 = ran (𝐴 ∩ I )
2		inss1 4208	. . 3 ⊢ (𝐴 ∩ I ) ⊆ 𝐴
3	2	rnssi 5813	. 2 ⊢ ran (𝐴 ∩ I ) ⊆ ran 𝐴
4	1, 3	eqsstri 4004	1 ⊢ Fix 𝐴 ⊆ ran 𝐴

Syntax hints:   ∩ cin 3938   ⊆ wss 3939   I cid 5462  ran crn 5559   Fix cfix 33300

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pr 5333

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-br 5070  df-opab 5132  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-dm 5568  df-rn 5569  df-fix 33324

This theorem is referenced by: (None)