Theorem fixssdm 32013

Description: The fixpoints of a class are a subset of its domain. (Contributed by Scott Fenton, 16-Apr-2012.)

Assertion

Ref	Expression
fixssdm	|- Fix A C_ dom A

Proof of Theorem fixssdm

Step	Hyp	Ref	Expression
1		df-fix 31966	. 2 |- Fix A = dom ( A i^i _I )
2		inss1 3833	. . 3 |- ( A i^i _I ) C_ A
3		dmss 5323	. . 3 |- ( ( A i^i _I ) C_ A -> dom ( A i^i _I ) C_ dom A )
4	2, 3	ax-mp 5	. 2 |- dom ( A i^i _I ) C_ dom A
5	1, 4	eqsstri 3635	1 |- Fix A C_ dom A

Syntax hints:   i^i cin 3573   C_ wss 3574   _I cid 5023   dom cdm 5114   Fixcfix 31942

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-dm 5124  df-fix 31966

This theorem is referenced by: (None)