Theorem subcidcl 16711

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 6333	. . 3 ⊢ (𝑥 = 𝑋 → ( 1 ‘𝑥) = ( 1 ‘𝑋))
2		id 22	. . . 4 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋)
3	2, 2	oveq12d 6814	. . 3 ⊢ (𝑥 = 𝑋 → (𝑥𝐽𝑥) = (𝑋𝐽𝑋))
4	1, 3	eleq12d 2844	. 2 ⊢ (𝑥 = 𝑋 → (( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ↔ ( 1 ‘𝑋) ∈ (𝑋𝐽𝑋)))
5		subcidcl.j	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ (Subcat‘𝐶))
6		eqid 2771	. . . . . 6 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶)
7		subcidcl.1	. . . . . 6 ⊢ 1 = (Id‘𝐶)
8		eqid 2771	. . . . . 6 ⊢ (comp‘𝐶) = (comp‘𝐶)
9		subcrcl 16683	. . . . . . 7 ⊢ (𝐽 ∈ (Subcat‘𝐶) → 𝐶 ∈ Cat)
10	5, 9	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ Cat)
11		subcidcl.2	. . . . . 6 ⊢ (𝜑 → 𝐽 Fn (𝑆 × 𝑆))
12	6, 7, 8, 10, 11	issubc2 16703	. . . . 5 ⊢ (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 222	. . . 4 ⊢ (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	13	simprd 483	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 (( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐽𝑧)))
15		simpl 468	. . . 4 ⊢ ((( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐽𝑧)) → ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥))
16	15	ralimi 3101	. . 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))
17	14, 16	syl 17	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥))
18		subcidcl.x	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝑆)
19	4, 17, 18	rspcdva 3466	1 ⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋𝐽𝑋))

Syntax hints:   → wi 4   ∧ wa 382   = wceq 1631   ∈ wcel 2145  ∀wral 3061  ⟨cop 4323   class class class wbr 4787   × cxp 5248   Fn wfn 6025  ‘cfv 6030  (class class class)co 6796  compcco 16161  Catccat 16532  Idccid 16533  Hom_f chomf 16534   ⊆_cat cssc 16674  Subcatcsubc 16676

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-pm 8016  df-ixp 8067  df-ssc 16677  df-subc 16679

This theorem is referenced by:  subccatid  16713  issubc3  16716  funcres  16763