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Theorem 0subcat 16809
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 16808 . 2 (𝐶 ∈ Cat → ∅ ⊆cat (Homf𝐶))
2 ral0 4267 . . 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 2797 . . 3 (Homf𝐶) = (Homf𝐶)
5 eqid 2797 . . 3 (Id‘𝐶) = (Id‘𝐶)
6 eqid 2797 . . 3 (comp‘𝐶) = (comp‘𝐶)
7 id 22 . . 3 (𝐶 ∈ Cat → 𝐶 ∈ Cat)
8 f0 6299 . . . . . 6 ∅:∅⟶∅
9 ffn 6254 . . . . . 6 (∅:∅⟶∅ → ∅ Fn ∅)
108, 9ax-mp 5 . . . . 5 ∅ Fn ∅
11 0xp 5402 . . . . . 6 (∅ × ∅) = ∅
1211fneq2i 6195 . . . . 5 (∅ Fn (∅ × ∅) ↔ ∅ Fn ∅)
1310, 12mpbir 223 . . . 4 ∅ Fn (∅ × ∅)
1413a1i 11 . . 3 (𝐶 ∈ Cat → ∅ Fn (∅ × ∅))
154, 5, 6, 7, 14issubc2 16807 . 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 705 1 (𝐶 ∈ Cat → ∅ ∈ (Subcat‘𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  wcel 2157  wral 3087  c0 4113  cop 4372   class class class wbr 4841   × cxp 5308   Fn wfn 6094  wf 6095  cfv 6099  (class class class)co 6876  compcco 16276  Catccat 16636  Idccid 16637  Homf chomf 16638  cat cssc 16778  Subcatcsubc 16780
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-rep 4962  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095  ax-un 7181
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-fal 1667  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-rex 3093  df-reu 3094  df-rab 3096  df-v 3385  df-sbc 3632  df-csb 3727  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-pw 4349  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-iun 4710  df-br 4842  df-opab 4904  df-mpt 4921  df-id 5218  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-f1 6104  df-fo 6105  df-f1o 6106  df-fv 6107  df-ov 6879  df-oprab 6880  df-mpt2 6881  df-1st 7399  df-2nd 7400  df-pm 8096  df-ixp 8147  df-homf 16642  df-ssc 16781  df-subc 16783
This theorem is referenced by: (None)
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