MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  subcss2 Structured version   Visualization version   GIF version

Theorem subcss2 17092
Description: The morphisms of a subcategory are a subset of the morphisms of the original. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
subcss1.1 (𝜑𝐽 ∈ (Subcat‘𝐶))
subcss1.2 (𝜑𝐽 Fn (𝑆 × 𝑆))
subcss2.h 𝐻 = (Hom ‘𝐶)
subcss2.x (𝜑𝑋𝑆)
subcss2.y (𝜑𝑌𝑆)
Assertion
Ref Expression
subcss2 (𝜑 → (𝑋𝐽𝑌) ⊆ (𝑋𝐻𝑌))

Proof of Theorem subcss2
StepHypRef Expression
1 subcss1.2 . . 3 (𝜑𝐽 Fn (𝑆 × 𝑆))
2 subcss1.1 . . . 4 (𝜑𝐽 ∈ (Subcat‘𝐶))
3 eqid 2820 . . . 4 (Homf𝐶) = (Homf𝐶)
42, 3subcssc 17089 . . 3 (𝜑𝐽cat (Homf𝐶))
5 subcss2.x . . 3 (𝜑𝑋𝑆)
6 subcss2.y . . 3 (𝜑𝑌𝑆)
71, 4, 5, 6ssc2 17071 . 2 (𝜑 → (𝑋𝐽𝑌) ⊆ (𝑋(Homf𝐶)𝑌))
8 eqid 2820 . . 3 (Base‘𝐶) = (Base‘𝐶)
9 subcss2.h . . 3 𝐻 = (Hom ‘𝐶)
102, 1, 8subcss1 17091 . . . 4 (𝜑𝑆 ⊆ (Base‘𝐶))
1110, 5sseldd 3947 . . 3 (𝜑𝑋 ∈ (Base‘𝐶))
1210, 6sseldd 3947 . . 3 (𝜑𝑌 ∈ (Base‘𝐶))
133, 8, 9, 11, 12homfval 16941 . 2 (𝜑 → (𝑋(Homf𝐶)𝑌) = (𝑋𝐻𝑌))
147, 13sseqtrd 3986 1 (𝜑 → (𝑋𝐽𝑌) ⊆ (𝑋𝐻𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2114  wss 3913   × cxp 5529   Fn wfn 6326  cfv 6331  (class class class)co 7133  Basecbs 16462  Hom chom 16555  Homf chomf 16916  Subcatcsubc 17058
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-reu 3132  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-iun 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5436  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-f1 6336  df-fo 6337  df-f1o 6338  df-fv 6339  df-ov 7136  df-oprab 7137  df-mpo 7138  df-1st 7667  df-2nd 7668  df-pm 8387  df-ixp 8440  df-homf 16920  df-ssc 17059  df-subc 17061
This theorem is referenced by:  subccatid  17095  funcres  17145  funcres2b  17146
  Copyright terms: Public domain W3C validator