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Theorem 0subcat 17827
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 17826 . 2 (𝐶 ∈ Cat → ∅ ⊆cat (Homf𝐶))
2 ral0 4514 . . 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 2725 . . 3 (Homf𝐶) = (Homf𝐶)
5 eqid 2725 . . 3 (Id‘𝐶) = (Id‘𝐶)
6 eqid 2725 . . 3 (comp‘𝐶) = (comp‘𝐶)
7 id 22 . . 3 (𝐶 ∈ Cat → 𝐶 ∈ Cat)
8 f0 6778 . . . . . 6 ∅:∅⟶∅
9 ffn 6723 . . . . . 6 (∅:∅⟶∅ → ∅ Fn ∅)
108, 9ax-mp 5 . . . . 5 ∅ Fn ∅
11 0xp 5776 . . . . . 6 (∅ × ∅) = ∅
1211fneq2i 6653 . . . . 5 (∅ Fn (∅ × ∅) ↔ ∅ Fn ∅)
1310, 12mpbir 230 . . . 4 ∅ Fn (∅ × ∅)
1413a1i 11 . . 3 (𝐶 ∈ Cat → ∅ Fn (∅ × ∅))
154, 5, 6, 7, 14issubc2 17825 . 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 711 1 (𝐶 ∈ Cat → ∅ ∈ (Subcat‘𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wcel 2098  wral 3050  c0 4322  cop 4636   class class class wbr 5149   × cxp 5676   Fn wfn 6544  wf 6545  cfv 6549  (class class class)co 7419  compcco 17248  Catccat 17647  Idccid 17648  Homf chomf 17649  cat cssc 17793  Subcatcsubc 17795
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 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
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 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7994  df-2nd 7995  df-pm 8848  df-ixp 8917  df-homf 17653  df-ssc 17796  df-subc 17798
This theorem is referenced by: (None)
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