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Theorem subcssc 16547
Description: An element in the set of subcategories is a subset of the category. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
subcixp.1 (𝜑𝐽 ∈ (Subcat‘𝐶))
subcssc.h 𝐻 = (Homf𝐶)
Assertion
Ref Expression
subcssc (𝜑𝐽cat 𝐻)

Proof of Theorem subcssc
Dummy variables 𝑓 𝑔 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 subcixp.1 . . 3 (𝜑𝐽 ∈ (Subcat‘𝐶))
2 subcssc.h . . . 4 𝐻 = (Homf𝐶)
3 eqid 2651 . . . 4 (Id‘𝐶) = (Id‘𝐶)
4 eqid 2651 . . . 4 (comp‘𝐶) = (comp‘𝐶)
5 subcrcl 16523 . . . . 5 (𝐽 ∈ (Subcat‘𝐶) → 𝐶 ∈ Cat)
61, 5syl 17 . . . 4 (𝜑𝐶 ∈ Cat)
7 eqidd 2652 . . . 4 (𝜑 → dom dom 𝐽 = dom dom 𝐽)
82, 3, 4, 6, 7issubc 16542 . . 3 (𝜑 → (𝐽 ∈ (Subcat‘𝐶) ↔ (𝐽cat 𝐻 ∧ ∀𝑥 ∈ dom dom 𝐽(((Id‘𝐶)‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦 ∈ dom dom 𝐽𝑧 ∈ dom dom 𝐽𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐽𝑧)))))
91, 8mpbid 222 . 2 (𝜑 → (𝐽cat 𝐻 ∧ ∀𝑥 ∈ dom dom 𝐽(((Id‘𝐶)‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦 ∈ dom dom 𝐽𝑧 ∈ dom dom 𝐽𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐽𝑧))))
109simpld 474 1 (𝜑𝐽cat 𝐻)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  wral 2941  cop 4216   class class class wbr 4685  dom cdm 5143  cfv 5926  (class class class)co 6690  compcco 16000  Catccat 16372  Idccid 16373  Homf chomf 16374  cat cssc 16514  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-pm 7902  df-ixp 7951  df-ssc 16517  df-subc 16519
This theorem is referenced by:  subcfn  16548  subcss1  16549  subcss2  16550  issubc3  16556  subsubc  16560
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