Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  subcrcl Structured version   Unicode version

Theorem subcrcl 15232
 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 15228 . . 3 Subcat cat f comp
21dmmptss 5509 . 2 Subcat
3 elfvdm 5898 . 2 Subcat Subcat
42, 3sseldi 3497 1 Subcat
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 369   wcel 1819  cab 2442  wral 2807  wsbc 3327  cop 4038   class class class wbr 4456   cdm 5008  cfv 5594  (class class class)co 6296  compcco 14724  ccat 15081  ccid 15082   f chomf 15083   cat cssc 15223  Subcatcsubc 15225 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-xp 5014  df-rel 5015  df-cnv 5016  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fv 5602  df-subc 15228 This theorem is referenced by:  subcssc  15256  subcidcl  15260  subccocl  15261  subccatid  15262  subsubc  15269  funcres2b  15313  funcres2  15314  idfusubc  32816
 Copyright terms: Public domain W3C validator