Theorem subcssc 17471

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 2738	. . . 4 ⊢ (Id‘𝐶) = (Id‘𝐶)
4		eqid 2738	. . . 4 ⊢ (comp‘𝐶) = (comp‘𝐶)
5		subcrcl 17445	. . . . 5 ⊢ (𝐽 ∈ (Subcat‘𝐶) → 𝐶 ∈ Cat)
6	1, 5	syl 17	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
7		eqidd 2739	. . . 4 ⊢ (𝜑 → dom dom 𝐽 = dom dom 𝐽)
8	2, 3, 4, 6, 7	issubc 17466	. . 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 494	1 ⊢ (𝜑 → 𝐽 ⊆cat 𝐻)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063  ⟨cop 4564   class class class wbr 5070  dom cdm 5580  ‘cfv 6418  (class class class)co 7255  compcco 16900  Catccat 17290  Idccid 17291  Hom_f chomf 17292   ⊆_cat cssc 17436  Subcatcsubc 17438

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-pm 8576  df-ixp 8644  df-ssc 17439  df-subc 17441

This theorem is referenced by:  subcfn  17472  subcss1  17473  subcss2  17474  issubc3  17480  subsubc  17484