Theorem subcrcl 17738

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 17734	. 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 6948	1 ⊢ (𝐻 ∈ (Subcat‘𝐶) → 𝐶 ∈ Cat)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2113  {cab 2712  ∀wral 3049  [wsbc 3738  ⟨cop 4584   class class class wbr 5096  dom cdm 5622  ‘cfv 6490  (class class class)co 7356  compcco 17187  Catccat 17585  Idccid 17586  Hom_f chomf 17587   ⊆_cat cssc 17729  Subcatcsubc 17731

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-mpt 5178  df-xp 5628  df-rel 5629  df-cnv 5630  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fv 6498  df-subc 17734

This theorem is referenced by:  subcssc  17762  subcidcl  17766  subccocl  17767  subccatid  17768  subsubc  17775  funcres2b  17819  funcres2  17820  idfusubc  17822  iinfsubc  49245