Theorem subcrcl 17868

Description: Reverse closure for the subcategory predicate. (Contributed by Mario Carneiro, 6-Jan-2017.)

Assertion

Ref	Expression
subcrcl	⊢ (𝐻 ∈ (Subcat‘𝐶) → 𝐶 ∈ Cat)

Proof of Theorem subcrcl

Dummy variables 𝑓 𝑐 𝑔 ℎ 𝑠 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-subc 17864	. 2 ⊢ Subcat = (𝑐 ∈ Cat ↦ {ℎ ∣ (ℎ ⊆cat (Homf ‘𝑐) ∧ [dom dom ℎ / 𝑠]∀𝑥 ∈ 𝑠 (((Id‘𝑐)‘𝑥) ∈ (𝑥ℎ𝑥) ∧ ∀𝑦 ∈ 𝑠 ∀𝑧 ∈ 𝑠 ∀𝑓 ∈ (𝑥ℎ𝑦)∀𝑔 ∈ (𝑦ℎ𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥ℎ𝑧)))})
2	1	mptrcl 6999	1 ⊢ (𝐻 ∈ (Subcat‘𝐶) → 𝐶 ∈ Cat)

Syntax hints:   → wi 4   ∧ wa 400   ∈ wcel 2143  {cab 2741  ∀wral 3079  [wsbc 3744  ⟨cop 4595   class class class wbr 5109  dom cdm 5661  ‘cfv 6536  (class class class)co 7410  compcco 17317  Catccat 17715  Idccid 17716  Hom_f chomf 17717   ⊆_cat cssc 17859  Subcatcsubc 17861

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-mpt 5193  df-xp 5667  df-rel 5668  df-cnv 5669  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fv 6544  df-subc 17864

This theorem is referenced by:  subcssc  17892  subcidcl  17896  subccocl  17897  subccatid  17898  subsubc  17905  funcres2b  17949  funcres2  17950  idfusubc  17952  iinfsubc  49836