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Theorem 0subcat 16262
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.
StepHypRef Expression
1 0ssc 16261 . 2 (𝐶 ∈ Cat → ∅ ⊆cat (Homf𝐶))
2 ral0 4022 . . 3 𝑥 ∈ ∅ (((Id‘𝐶)‘𝑥) ∈ (𝑥𝑥) ∧ ∀𝑦 ∈ ∅ ∀𝑧 ∈ ∅ ∀𝑓 ∈ (𝑥𝑦)∀𝑔 ∈ (𝑦𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝑧))
32a1i 11 . 2 (𝐶 ∈ Cat → ∀𝑥 ∈ ∅ (((Id‘𝐶)‘𝑥) ∈ (𝑥𝑥) ∧ ∀𝑦 ∈ ∅ ∀𝑧 ∈ ∅ ∀𝑓 ∈ (𝑥𝑦)∀𝑔 ∈ (𝑦𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝑧)))
4 eqid 2604 . . 3 (Homf𝐶) = (Homf𝐶)
5 eqid 2604 . . 3 (Id‘𝐶) = (Id‘𝐶)
6 eqid 2604 . . 3 (comp‘𝐶) = (comp‘𝐶)
7 id 22 . . 3 (𝐶 ∈ Cat → 𝐶 ∈ Cat)
8 f0 5979 . . . . . 6 ∅:∅⟶∅
9 ffn 5939 . . . . . 6 (∅:∅⟶∅ → ∅ Fn ∅)
108, 9ax-mp 5 . . . . 5 ∅ Fn ∅
11 0xp 5107 . . . . . 6 (∅ × ∅) = ∅
1211fneq2i 5881 . . . . 5 (∅ Fn (∅ × ∅) ↔ ∅ Fn ∅)
1310, 12mpbir 219 . . . 4 ∅ Fn (∅ × ∅)
1413a1i 11 . . 3 (𝐶 ∈ Cat → ∅ Fn (∅ × ∅))
154, 5, 6, 7, 14issubc2 16260 . 2 (𝐶 ∈ Cat → (∅ ∈ (Subcat‘𝐶) ↔ (∅ ⊆cat (Homf𝐶) ∧ ∀𝑥 ∈ ∅ (((Id‘𝐶)‘𝑥) ∈ (𝑥𝑥) ∧ ∀𝑦 ∈ ∅ ∀𝑧 ∈ ∅ ∀𝑓 ∈ (𝑥𝑦)∀𝑔 ∈ (𝑦𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝑧)))))
161, 3, 15mpbir2and 958 1 (𝐶 ∈ Cat → ∅ ∈ (Subcat‘𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wcel 1975  wral 2890  c0 3868  cop 4125   class class class wbr 4572   × cxp 5021   Fn wfn 5780  wf 5781  cfv 5785  (class class class)co 6522  compcco 15721  Catccat 16089  Idccid 16090  Homf chomf 16091  cat cssc 16231  Subcatcsubc 16233
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584  ax-rep 4688  ax-sep 4698  ax-nul 4707  ax-pow 4759  ax-pr 4823  ax-un 6819
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2456  df-mo 2457  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-ne 2776  df-ral 2895  df-rex 2896  df-reu 2897  df-rab 2899  df-v 3169  df-sbc 3397  df-csb 3494  df-dif 3537  df-un 3539  df-in 3541  df-ss 3548  df-nul 3869  df-if 4031  df-pw 4104  df-sn 4120  df-pr 4122  df-op 4126  df-uni 4362  df-iun 4446  df-br 4573  df-opab 4633  df-mpt 4634  df-id 4938  df-xp 5029  df-rel 5030  df-cnv 5031  df-co 5032  df-dm 5033  df-rn 5034  df-res 5035  df-ima 5036  df-iota 5749  df-fun 5787  df-fn 5788  df-f 5789  df-f1 5790  df-fo 5791  df-f1o 5792  df-fv 5793  df-ov 6525  df-oprab 6526  df-mpt2 6527  df-1st 7031  df-2nd 7032  df-pm 7719  df-ixp 7767  df-homf 16095  df-ssc 16234  df-subc 16236
This theorem is referenced by: (None)
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