Mathbox for Frédéric Liné < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cmpsubc Unicode version

Theorem cmpsubc 25856
 Description: The composition law of a subcategory is a subset of the composition law of the supercategory. (Contributed by FL, 20-Sep-2009.)
Hypotheses
Ref Expression
cmpsubc.1
cmpsubc.2
Assertion
Ref Expression
cmpsubc

Proof of Theorem cmpsubc
StepHypRef Expression
1 besubbeca 25848 . 2
2 eqid 2283 . . . . . 6
3 eqid 2283 . . . . . 6
4 eqid 2283 . . . . . 6
5 eqid 2283 . . . . . 6
62, 3, 4, 5issubcata 25846 . . . . 5
7 simpr3 963 . . . . 5
86, 7syl6bi 219 . . . 4
98imp 418 . . 3
10 cmpsubc.2 . . 3
11 cmpsubc.1 . . 3
129, 10, 113sstr4g 3219 . 2
131, 12mpancom 650 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 358   w3a 934   wceq 1623   wcel 1684   wss 3152  cfv 5255  cdom_ 25712  ccod_ 25713  cid_ 25714  co_ 25715   ccatOLD 25752   csubcat 25843 This theorem is referenced by:  idsubfun  25858 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-13 1686  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-pow 4188  ax-pr 4214  ax-un 4512 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-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-1st 6122  df-2nd 6123  df-dom_ 25717  df-cod_ 25718  df-id_ 25719  df-cmpa 25720  df-catOLD 25753  df-subcat 25844
 Copyright terms: Public domain W3C validator