Theorem subcssc 17765

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 2729	. . . 4 ⊢ (Id‘𝐶) = (Id‘𝐶)
4		eqid 2729	. . . 4 ⊢ (comp‘𝐶) = (comp‘𝐶)
5		subcrcl 17741	. . . . 5 ⊢ (𝐽 ∈ (Subcat‘𝐶) → 𝐶 ∈ Cat)
6	1, 5	syl 17	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
7		eqidd 2730	. . . 4 ⊢ (𝜑 → dom dom 𝐽 = dom dom 𝐽)
8	2, 3, 4, 6, 7	issubc 17760	. . 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 232	. 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 1540   ∈ wcel 2109  ∀wral 3044  ⟨cop 4585   class class class wbr 5095  dom cdm 5623  ‘cfv 6486  (class class class)co 7353  compcco 17191  Catccat 17588  Idccid 17589  Hom_f chomf 17590   ⊆_cat cssc 17732  Subcatcsubc 17734

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-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

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-ne 2926  df-ral 3045  df-rex 3054  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7356  df-oprab 7357  df-mpo 7358  df-pm 8763  df-ixp 8832  df-ssc 17735  df-subc 17737

This theorem is referenced by:  subcfn  17766  subcss1  17767  subcss2  17768  issubc3  17774  subsubc  17778  iinfsubc  49044  infsubc2  49047  iinfconstbas  49052