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Theorem 0subcat 17889
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 17888 . 2 (𝐶 ∈ Cat → ∅ ⊆cat (Homf𝐶))
2 ral0 4519 . . 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 2735 . . 3 (Homf𝐶) = (Homf𝐶)
5 eqid 2735 . . 3 (Id‘𝐶) = (Id‘𝐶)
6 eqid 2735 . . 3 (comp‘𝐶) = (comp‘𝐶)
7 id 22 . . 3 (𝐶 ∈ Cat → 𝐶 ∈ Cat)
8 f0 6790 . . . . . 6 ∅:∅⟶∅
9 ffn 6737 . . . . . 6 (∅:∅⟶∅ → ∅ Fn ∅)
108, 9ax-mp 5 . . . . 5 ∅ Fn ∅
11 0xp 5787 . . . . . 6 (∅ × ∅) = ∅
1211fneq2i 6667 . . . . 5 (∅ Fn (∅ × ∅) ↔ ∅ Fn ∅)
1310, 12mpbir 231 . . . 4 ∅ Fn (∅ × ∅)
1413a1i 11 . . 3 (𝐶 ∈ Cat → ∅ Fn (∅ × ∅))
154, 5, 6, 7, 14issubc2 17887 . 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 713 1 (𝐶 ∈ Cat → ∅ ∈ (Subcat‘𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2106  wral 3059  c0 4339  cop 4637   class class class wbr 5148   × cxp 5687   Fn wfn 6558  wf 6559  cfv 6563  (class class class)co 7431  compcco 17310  Catccat 17709  Idccid 17710  Homf chomf 17711  cat cssc 17855  Subcatcsubc 17857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-pm 8868  df-ixp 8937  df-homf 17715  df-ssc 17858  df-subc 17860
This theorem is referenced by: (None)
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