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Theorem subcss2 16550
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 2651 . . . 4 (Homf𝐶) = (Homf𝐶)
42, 3subcssc 16547 . . 3 (𝜑𝐽cat (Homf𝐶))
5 subcss2.x . . 3 (𝜑𝑋𝑆)
6 subcss2.y . . 3 (𝜑𝑌𝑆)
71, 4, 5, 6ssc2 16529 . 2 (𝜑 → (𝑋𝐽𝑌) ⊆ (𝑋(Homf𝐶)𝑌))
8 eqid 2651 . . 3 (Base‘𝐶) = (Base‘𝐶)
9 subcss2.h . . 3 𝐻 = (Hom ‘𝐶)
102, 1, 8subcss1 16549 . . . 4 (𝜑𝑆 ⊆ (Base‘𝐶))
1110, 5sseldd 3637 . . 3 (𝜑𝑋 ∈ (Base‘𝐶))
1210, 6sseldd 3637 . . 3 (𝜑𝑌 ∈ (Base‘𝐶))
133, 8, 9, 11, 12homfval 16399 . 2 (𝜑 → (𝑋(Homf𝐶)𝑌) = (𝑋𝐻𝑌))
147, 13sseqtrd 3674 1 (𝜑 → (𝑋𝐽𝑌) ⊆ (𝑋𝐻𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wcel 2030  wss 3607   × cxp 5141   Fn wfn 5921  cfv 5926  (class class class)co 6690  Basecbs 15904  Hom chom 15999  Homf chomf 16374  Subcatcsubc 16516
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-pm 7902  df-ixp 7951  df-homf 16378  df-ssc 16517  df-subc 16519
This theorem is referenced by:  subccatid  16553  funcres  16603  funcres2b  16604
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