Theorem subcssc 17740

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	⊢ (𝜑 → 𝐽 ∈ (Subcat‘𝐶))
subcssc.h	⊢ 𝐻 = (Homf ‘𝐶)

Assertion

Ref	Expression
subcssc	⊢ (𝜑 → 𝐽 ⊆cat 𝐻)

Proof of Theorem subcssc

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

Step	Hyp	Ref	Expression
1		subcixp.1	. . 3 ⊢ (𝜑 → 𝐽 ∈ (Subcat‘𝐶))
2		subcssc.h	. . . 4 ⊢ 𝐻 = (Homf ‘𝐶)
3		eqid 2731	. . . 4 ⊢ (Id‘𝐶) = (Id‘𝐶)
4		eqid 2731	. . . 4 ⊢ (comp‘𝐶) = (comp‘𝐶)
5		subcrcl 17713	. . . . 5 ⊢ (𝐽 ∈ (Subcat‘𝐶) → 𝐶 ∈ Cat)
6	1, 5	syl 17	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
7		eqidd 2732	. . . 4 ⊢ (𝜑 → dom dom 𝐽 = dom dom 𝐽)
8	2, 3, 4, 6, 7	issubc 17735	. . 3 ⊢ (𝜑 → (𝐽 ∈ (Subcat‘𝐶) ↔ (𝐽 ⊆cat 𝐻 ∧ ∀𝑥 ∈ dom dom 𝐽(((Id‘𝐶)‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦 ∈ dom dom 𝐽∀𝑧 ∈ dom dom 𝐽∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐽𝑧)))))
9	1, 8	mpbid 231	. 2 ⊢ (𝜑 → (𝐽 ⊆cat 𝐻 ∧ ∀𝑥 ∈ dom dom 𝐽(((Id‘𝐶)‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦 ∈ dom dom 𝐽∀𝑧 ∈ dom dom 𝐽∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐽𝑧))))
10	9	simpld 495	1 ⊢ (𝜑 → 𝐽 ⊆cat 𝐻)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3060  ⟨cop 4597   class class class wbr 5110  dom cdm 5638  ‘cfv 6501  (class class class)co 7362  compcco 17159  Catccat 17558  Idccid 17559  Hom_f chomf 17560   ⊆_cat cssc 17704  Subcatcsubc 17706

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3352  df-rab 3406  df-v 3448  df-sbc 3743  df-csb 3859  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-pm 8775  df-ixp 8843  df-ssc 17707  df-subc 17709

This theorem is referenced by:  subcfn  17741  subcss1  17742  subcss2  17743  issubc3  17749  subsubc  17753