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Theorem subcss2 17888
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 2737 . . . 4 (Homf𝐶) = (Homf𝐶)
42, 3subcssc 17885 . . 3 (𝜑𝐽cat (Homf𝐶))
5 subcss2.x . . 3 (𝜑𝑋𝑆)
6 subcss2.y . . 3 (𝜑𝑌𝑆)
71, 4, 5, 6ssc2 17866 . 2 (𝜑 → (𝑋𝐽𝑌) ⊆ (𝑋(Homf𝐶)𝑌))
8 eqid 2737 . . 3 (Base‘𝐶) = (Base‘𝐶)
9 subcss2.h . . 3 𝐻 = (Hom ‘𝐶)
102, 1, 8subcss1 17887 . . . 4 (𝜑𝑆 ⊆ (Base‘𝐶))
1110, 5sseldd 3984 . . 3 (𝜑𝑋 ∈ (Base‘𝐶))
1210, 6sseldd 3984 . . 3 (𝜑𝑌 ∈ (Base‘𝐶))
133, 8, 9, 11, 12homfval 17735 . 2 (𝜑 → (𝑋(Homf𝐶)𝑌) = (𝑋𝐻𝑌))
147, 13sseqtrd 4020 1 (𝜑 → (𝑋𝐽𝑌) ⊆ (𝑋𝐻𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  wss 3951   × cxp 5683   Fn wfn 6556  cfv 6561  (class class class)co 7431  Basecbs 17247  Hom chom 17308  Homf chomf 17709  Subcatcsubc 17853
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-pm 8869  df-ixp 8938  df-homf 17713  df-ssc 17854  df-subc 17856
This theorem is referenced by:  subccatid  17891  funcres  17941  funcres2b  17942  subthinc  49092
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