Theorem subcssc 17169

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 2758	. . . 4 ⊢ (Id‘𝐶) = (Id‘𝐶)
4		eqid 2758	. . . 4 ⊢ (comp‘𝐶) = (comp‘𝐶)
5		subcrcl 17145	. . . . 5 ⊢ (𝐽 ∈ (Subcat‘𝐶) → 𝐶 ∈ Cat)
6	1, 5	syl 17	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
7		eqidd 2759	. . . 4 ⊢ (𝜑 → dom dom 𝐽 = dom dom 𝐽)
8	2, 3, 4, 6, 7	issubc 17164	. . 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 235	. 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 498	1 ⊢ (𝜑 → 𝐽 ⊆cat 𝐻)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3070  ⟨cop 4528   class class class wbr 5032  dom cdm 5524  ‘cfv 6335  (class class class)co 7150  compcco 16635  Catccat 16993  Idccid 16994  Hom_f chomf 16995   ⊆_cat cssc 17136  Subcatcsubc 17138

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-ov 7153  df-oprab 7154  df-mpo 7155  df-pm 8419  df-ixp 8480  df-ssc 17139  df-subc 17141

This theorem is referenced by:  subcfn  17170  subcss1  17171  subcss2  17172  issubc3  17178  subsubc  17182