Theorem 0subcat 17831

Description: For any category 𝐶, the empty set is a (full) subcategory of 𝐶, see example 4.3(1.a) in [Adamek] p. 48. (Contributed by AV, 23-Apr-2020.)

Assertion

Ref	Expression
0subcat	⊢ (𝐶 ∈ Cat → ∅ ∈ (Subcat‘𝐶))

Proof of Theorem 0subcat

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

Step	Hyp	Ref	Expression
1		0ssc 17830	. 2 ⊢ (𝐶 ∈ Cat → ∅ ⊆cat (Homf ‘𝐶))
2		ral0 4516	. . 3 ⊢ ∀𝑥 ∈ ∅ (((Id‘𝐶)‘𝑥) ∈ (𝑥∅𝑥) ∧ ∀𝑦 ∈ ∅ ∀𝑧 ∈ ∅ ∀𝑓 ∈ (𝑥∅𝑦)∀𝑔 ∈ (𝑦∅𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥∅𝑧))
3	2	a1i 11	. 2 ⊢ (𝐶 ∈ Cat → ∀𝑥 ∈ ∅ (((Id‘𝐶)‘𝑥) ∈ (𝑥∅𝑥) ∧ ∀𝑦 ∈ ∅ ∀𝑧 ∈ ∅ ∀𝑓 ∈ (𝑥∅𝑦)∀𝑔 ∈ (𝑦∅𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥∅𝑧)))
4		eqid 2728	. . 3 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶)
5		eqid 2728	. . 3 ⊢ (Id‘𝐶) = (Id‘𝐶)
6		eqid 2728	. . 3 ⊢ (comp‘𝐶) = (comp‘𝐶)
7		id 22	. . 3 ⊢ (𝐶 ∈ Cat → 𝐶 ∈ Cat)
8		f0 6783	. . . . . 6 ⊢ ∅:∅⟶∅
9		ffn 6727	. . . . . 6 ⊢ (∅:∅⟶∅ → ∅ Fn ∅)
10	8, 9	ax-mp 5	. . . . 5 ⊢ ∅ Fn ∅
11		0xp 5780	. . . . . 6 ⊢ (∅ × ∅) = ∅
12	11	fneq2i 6657	. . . . 5 ⊢ (∅ Fn (∅ × ∅) ↔ ∅ Fn ∅)
13	10, 12	mpbir 230	. . . 4 ⊢ ∅ Fn (∅ × ∅)
14	13	a1i 11	. . 3 ⊢ (𝐶 ∈ Cat → ∅ Fn (∅ × ∅))
15	4, 5, 6, 7, 14	issubc2 17829	. 2 ⊢ (𝐶 ∈ Cat → (∅ ∈ (Subcat‘𝐶) ↔ (∅ ⊆cat (Homf ‘𝐶) ∧ ∀𝑥 ∈ ∅ (((Id‘𝐶)‘𝑥) ∈ (𝑥∅𝑥) ∧ ∀𝑦 ∈ ∅ ∀𝑧 ∈ ∅ ∀𝑓 ∈ (𝑥∅𝑦)∀𝑔 ∈ (𝑦∅𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥∅𝑧)))))
16	1, 3, 15	mpbir2and 711	1 ⊢ (𝐶 ∈ Cat → ∅ ∈ (Subcat‘𝐶))

Syntax hints:   → wi 4   ∧ wa 394   ∈ wcel 2098  ∀wral 3058  ∅c0 4326  ⟨cop 4638   class class class wbr 5152   × cxp 5680   Fn wfn 6548  ⟶wf 6549  ‘cfv 6553  (class class class)co 7426  compcco 17252  Catccat 17651  Idccid 17652  Hom_f chomf 17653   ⊆_cat cssc 17797  Subcatcsubc 17799

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-1st 7999  df-2nd 8000  df-pm 8854  df-ixp 8923  df-homf 17657  df-ssc 17800  df-subc 17802

This theorem is referenced by: (None)