Theorem fixssrn 36796

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 36794	. 2 ⊢ Fix 𝐴 = ran (𝐴 ∩ I )
2		inss1 4135	. . 3 ⊢ (𝐴 ∩ I ) ⊆ 𝐴
3	2	rnssi 5926	. 2 ⊢ ran (𝐴 ∩ I ) ⊆ ran 𝐴
4	1, 3	eqsstri 3958	1 ⊢ Fix 𝐴 ⊆ ran 𝐴

This proof depends on syntax axioms:   ⊆ wss 3899   ∩ cin 4091   I cid 5548  ran crn 5655   Fix cfix 36724

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735  ax-sep 5251  ax-pr 5394

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3416  df-v 3455  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-sn 4588  df-pr 4590  df-op 4838  df-br 5106  df-opab 5170  df-id 5549  df-xp 5660  df-rel 5661  df-cnv 5662  df-dm 5664  df-rn 5665  df-fix 36748

This theorem is used by: (None)