Theorem subcidcl 17559

Description: The identity of the original category is contained in each subcategory. (Contributed by Mario Carneiro, 4-Jan-2017.)

Hypotheses

Ref	Expression
subcidcl.j	⊢ (𝜑 → 𝐽 ∈ (Subcat‘𝐶))
subcidcl.2	⊢ (𝜑 → 𝐽 Fn (𝑆 × 𝑆))
subcidcl.x	⊢ (𝜑 → 𝑋 ∈ 𝑆)
subcidcl.1	⊢ 1 = (Id‘𝐶)

Assertion

Ref	Expression
subcidcl	⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋𝐽𝑋))

Proof of Theorem subcidcl

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

Step	Hyp	Ref	Expression
1		fveq2 6774	. . 3 ⊢ (𝑥 = 𝑋 → ( 1 ‘𝑥) = ( 1 ‘𝑋))
2		id 22	. . . 4 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋)
3	2, 2	oveq12d 7293	. . 3 ⊢ (𝑥 = 𝑋 → (𝑥𝐽𝑥) = (𝑋𝐽𝑋))
4	1, 3	eleq12d 2833	. 2 ⊢ (𝑥 = 𝑋 → (( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ↔ ( 1 ‘𝑋) ∈ (𝑋𝐽𝑋)))
5		subcidcl.j	. . . 4 ⊢ (𝜑 → 𝐽 ∈ (Subcat‘𝐶))
6		eqid 2738	. . . . 5 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶)
7		subcidcl.1	. . . . 5 ⊢ 1 = (Id‘𝐶)
8		eqid 2738	. . . . 5 ⊢ (comp‘𝐶) = (comp‘𝐶)
9		subcrcl 17528	. . . . . 6 ⊢ (𝐽 ∈ (Subcat‘𝐶) → 𝐶 ∈ Cat)
10	5, 9	syl 17	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ Cat)
11		subcidcl.2	. . . . 5 ⊢ (𝜑 → 𝐽 Fn (𝑆 × 𝑆))
12	6, 7, 8, 10, 11	issubc2 17551	. . . 4 ⊢ (𝜑 → (𝐽 ∈ (Subcat‘𝐶) ↔ (𝐽 ⊆cat (Homf ‘𝐶) ∧ ∀𝑥 ∈ 𝑆 (( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐽𝑧)))))
13	5, 12	mpbid 231	. . 3 ⊢ (𝜑 → (𝐽 ⊆cat (Homf ‘𝐶) ∧ ∀𝑥 ∈ 𝑆 (( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐽𝑧))))
14		simpl 483	. . . 4 ⊢ ((( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐽𝑧)) → ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥))
15	14	ralimi 3087	. . 3 ⊢ (∀𝑥 ∈ 𝑆 (( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐽𝑧)) → ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥))
16	13, 15	simpl2im 504	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥))
17		subcidcl.x	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝑆)
18	4, 16, 17	rspcdva 3562	1 ⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋𝐽𝑋))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ⟨cop 4567   class class class wbr 5074   × cxp 5587   Fn wfn 6428  ‘cfv 6433  (class class class)co 7275  compcco 16974  Catccat 17373  Idccid 17374  Hom_f chomf 17375   ⊆_cat cssc 17519  Subcatcsubc 17521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-pm 8618  df-ixp 8686  df-ssc 17522  df-subc 17524

This theorem is referenced by:  subccatid  17561  issubc3  17564  funcres  17611