MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  subcss1 Structured version   Visualization version   GIF version

Theorem subcss1 16887
Description: The objects of a subcategory are a subset of the objects of the original. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
subcss1.1 (𝜑𝐽 ∈ (Subcat‘𝐶))
subcss1.2 (𝜑𝐽 Fn (𝑆 × 𝑆))
subcss1.b 𝐵 = (Base‘𝐶)
Assertion
Ref Expression
subcss1 (𝜑𝑆𝐵)

Proof of Theorem subcss1
StepHypRef Expression
1 subcss1.2 . 2 (𝜑𝐽 Fn (𝑆 × 𝑆))
2 eqid 2777 . . . 4 (Homf𝐶) = (Homf𝐶)
3 subcss1.b . . . 4 𝐵 = (Base‘𝐶)
42, 3homffn 16738 . . 3 (Homf𝐶) Fn (𝐵 × 𝐵)
54a1i 11 . 2 (𝜑 → (Homf𝐶) Fn (𝐵 × 𝐵))
6 subcss1.1 . . 3 (𝜑𝐽 ∈ (Subcat‘𝐶))
76, 2subcssc 16885 . 2 (𝜑𝐽cat (Homf𝐶))
81, 5, 7ssc1 16866 1 (𝜑𝑆𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1601  wcel 2106  wss 3791   × cxp 5353   Fn wfn 6130  cfv 6135  Basecbs 16255  Homf chomf 16712  Subcatcsubc 16854
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-fal 1615  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3399  df-sbc 3652  df-csb 3751  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-1st 7445  df-2nd 7446  df-pm 8143  df-ixp 8195  df-homf 16716  df-ssc 16855  df-subc 16857
This theorem is referenced by:  subcss2  16888  subccatid  16891  subsubc  16898  funcres  16941  funcres2b  16942  funcres2  16943  idfusubc  42863
  Copyright terms: Public domain W3C validator