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Theorem subcrcl 14016
 Description: Reverse closure for the subcategory predicate. (Contributed by Mario Carneiro, 6-Jan-2017.)
Assertion
Ref Expression
subcrcl Subcat

Proof of Theorem subcrcl
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-subc 14012 . . 3 Subcat cat f comp
21dmmptss 5366 . 2 Subcat
3 elfvdm 5757 . 2 Subcat Subcat
42, 3sseldi 3346 1 Subcat
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   wcel 1725  cab 2422  wral 2705  wsbc 3161  cop 3817   class class class wbr 4212   cdm 4878  cfv 5454  (class class class)co 6081  compcco 13541  ccat 13889  ccid 13890   f chomf 13891   cat cssc 14007  Subcatcsubc 14009 This theorem is referenced by:  subcssc  14037  subcidcl  14041  subccocl  14042  subccatid  14043  subsubc  14050  funcres2b  14094  funcres2  14095 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-13 1727  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-pow 4377  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-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-xp 4884  df-rel 4885  df-cnv 4886  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fv 5462  df-subc 14012
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