Theorem subcidcl 17766

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 6832	. . 3 ⊢ (𝑥 = 𝑋 → ( 1 ‘𝑥) = ( 1 ‘𝑋))
2		id 22	. . . 4 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋)
3	2, 2	oveq12d 7374	. . 3 ⊢ (𝑥 = 𝑋 → (𝑥𝐽𝑥) = (𝑋𝐽𝑋))
4	1, 3	eleq12d 2828	. 2 ⊢ (𝑥 = 𝑋 → (( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ↔ ( 1 ‘𝑋) ∈ (𝑋𝐽𝑋)))
5		subcidcl.j	. . . 4 ⊢ (𝜑 → 𝐽 ∈ (Subcat‘𝐶))
6		eqid 2734	. . . . 5 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶)
7		subcidcl.1	. . . . 5 ⊢ 1 = (Id‘𝐶)
8		eqid 2734	. . . . 5 ⊢ (comp‘𝐶) = (comp‘𝐶)
9		subcrcl 17738	. . . . . 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 17758	. . . 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 232	. . 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 482	. . . 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 3071	. . 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 503	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥))
17		subcidcl.x	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝑆)
18	4, 16, 17	rspcdva 3575	1 ⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋𝐽𝑋))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3049  ⟨cop 4584   class class class wbr 5096   × cxp 5620   Fn wfn 6485  ‘cfv 6490  (class class class)co 7356  compcco 17187  Catccat 17585  Idccid 17586  Hom_f chomf 17587   ⊆_cat cssc 17729  Subcatcsubc 17731

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ov 7359  df-oprab 7360  df-mpo 7361  df-pm 8764  df-ixp 8834  df-ssc 17732  df-subc 17734

This theorem is referenced by:  subccatid  17768  issubc3  17771  funcres  17818  iinfsubc  49245  iinfconstbas  49253