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Theorem subcss2 16272
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 2609 . . . 4 (Homf𝐶) = (Homf𝐶)
42, 3subcssc 16269 . . 3 (𝜑𝐽cat (Homf𝐶))
5 subcss2.x . . 3 (𝜑𝑋𝑆)
6 subcss2.y . . 3 (𝜑𝑌𝑆)
71, 4, 5, 6ssc2 16251 . 2 (𝜑 → (𝑋𝐽𝑌) ⊆ (𝑋(Homf𝐶)𝑌))
8 eqid 2609 . . 3 (Base‘𝐶) = (Base‘𝐶)
9 subcss2.h . . 3 𝐻 = (Hom ‘𝐶)
102, 1, 8subcss1 16271 . . . 4 (𝜑𝑆 ⊆ (Base‘𝐶))
1110, 5sseldd 3568 . . 3 (𝜑𝑋 ∈ (Base‘𝐶))
1210, 6sseldd 3568 . . 3 (𝜑𝑌 ∈ (Base‘𝐶))
133, 8, 9, 11, 12homfval 16121 . 2 (𝜑 → (𝑋(Homf𝐶)𝑌) = (𝑋𝐻𝑌))
147, 13sseqtrd 3603 1 (𝜑 → (𝑋𝐽𝑌) ⊆ (𝑋𝐻𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1976  wss 3539   × cxp 5026   Fn wfn 5785  cfv 5790  (class class class)co 6527  Basecbs 15641  Hom chom 15725  Homf chomf 16096  Subcatcsubc 16238
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-1st 7036  df-2nd 7037  df-pm 7724  df-ixp 7772  df-homf 16100  df-ssc 16239  df-subc 16241
This theorem is referenced by:  subccatid  16275  funcres  16325  funcres2b  16326
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