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Theorem fixssrn 25752
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  C_ 
ran  A

Proof of Theorem fixssrn
StepHypRef Expression
1 dffix2 25750 . 2  |-  Fix A  =  ran  ( A  i^i  _I  )
2 inss1 3561 . . 3  |-  ( A  i^i  _I  )  C_  A
3 rnss 5098 . . 3  |-  ( ( A  i^i  _I  )  C_  A  ->  ran  ( A  i^i  _I  )  C_  ran  A )
42, 3ax-mp 8 . 2  |-  ran  ( A  i^i  _I  )  C_  ran  A
51, 4eqsstri 3378 1  |-  Fix A  C_ 
ran  A
Colors of variables: wff set class
Syntax hints:    i^i cin 3319    C_ wss 3320    _I cid 4493   ran crn 4879   Fixcfix 25679
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-dm 4888  df-rn 4889  df-fix 25703
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