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Theorem fixssrn 24447
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 24445 . 2  |-  Fix A  =  ran  ( A  i^i  _I  )
2 inss1 3389 . . 3  |-  ( A  i^i  _I  )  C_  A
3 rnss 4907 . . 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 3208 1  |-  Fix A  C_ 
ran  A
Colors of variables: wff set class
Syntax hints:    i^i cin 3151    C_ wss 3152    _I cid 4304   ran crn 4690   Fixcfix 24378
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-dm 4699  df-rn 4700  df-fix 24400
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