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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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