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Theorem 0subcat 17551
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 17550 . 2 (𝐶 ∈ Cat → ∅ ⊆cat (Homf𝐶))
2 ral0 4449 . . 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 2740 . . 3 (Homf𝐶) = (Homf𝐶)
5 eqid 2740 . . 3 (Id‘𝐶) = (Id‘𝐶)
6 eqid 2740 . . 3 (comp‘𝐶) = (comp‘𝐶)
7 id 22 . . 3 (𝐶 ∈ Cat → 𝐶 ∈ Cat)
8 f0 6653 . . . . . 6 ∅:∅⟶∅
9 ffn 6598 . . . . . 6 (∅:∅⟶∅ → ∅ Fn ∅)
108, 9ax-mp 5 . . . . 5 ∅ Fn ∅
11 0xp 5685 . . . . . 6 (∅ × ∅) = ∅
1211fneq2i 6529 . . . . 5 (∅ Fn (∅ × ∅) ↔ ∅ Fn ∅)
1310, 12mpbir 230 . . . 4 ∅ Fn (∅ × ∅)
1413a1i 11 . . 3 (𝐶 ∈ Cat → ∅ Fn (∅ × ∅))
154, 5, 6, 7, 14issubc2 17549 . 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 710 1 (𝐶 ∈ Cat → ∅ ∈ (Subcat‘𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2110  wral 3066  c0 4262  cop 4573   class class class wbr 5079   × cxp 5588   Fn wfn 6427  wf 6428  cfv 6432  (class class class)co 7271  compcco 16972  Catccat 17371  Idccid 17372  Homf chomf 17373  cat cssc 17517  Subcatcsubc 17519
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-rep 5214  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7582
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-reu 3073  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-id 5490  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-ov 7274  df-oprab 7275  df-mpo 7276  df-1st 7824  df-2nd 7825  df-pm 8601  df-ixp 8669  df-homf 17377  df-ssc 17520  df-subc 17522
This theorem is referenced by: (None)
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