Theorem subcrcl 17754

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 17750	. 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 6959	1 ⊢ (𝐻 ∈ (Subcat‘𝐶) → 𝐶 ∈ Cat)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2109  {cab 2707  ∀wral 3044  [wsbc 3750  ⟨cop 4591   class class class wbr 5102  dom cdm 5631  ‘cfv 6499  (class class class)co 7369  compcco 17208  Catccat 17601  Idccid 17602  Hom_f chomf 17603   ⊆_cat cssc 17745  Subcatcsubc 17747

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-xp 5637  df-rel 5638  df-cnv 5639  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fv 6507  df-subc 17750

This theorem is referenced by:  subcssc  17778  subcidcl  17782  subccocl  17783  subccatid  17784  subsubc  17791  funcres2b  17835  funcres2  17836  idfusubc  17838  iinfsubc  49020