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Theorem 0ssc 17099
Description: For any category 𝐶, the empty set is a subcategory subset of 𝐶. (Contributed by AV, 23-Apr-2020.)
Assertion
Ref Expression
0ssc (𝐶 ∈ Cat → ∅ ⊆cat (Homf𝐶))

Proof of Theorem 0ssc
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0ss 4304 . . 3 ∅ ⊆ (Base‘𝐶)
21a1i 11 . 2 (𝐶 ∈ Cat → ∅ ⊆ (Base‘𝐶))
3 ral0 4414 . . 3 𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥𝑦) ⊆ (𝑥(Homf𝐶)𝑦)
43a1i 11 . 2 (𝐶 ∈ Cat → ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥𝑦) ⊆ (𝑥(Homf𝐶)𝑦))
5 f0 6534 . . . . . 6 ∅:∅⟶∅
6 ffn 6487 . . . . . 6 (∅:∅⟶∅ → ∅ Fn ∅)
75, 6ax-mp 5 . . . . 5 ∅ Fn ∅
8 xp0 5982 . . . . . 6 (∅ × ∅) = ∅
98fneq2i 6421 . . . . 5 (∅ Fn (∅ × ∅) ↔ ∅ Fn ∅)
107, 9mpbir 234 . . . 4 ∅ Fn (∅ × ∅)
1110a1i 11 . . 3 (𝐶 ∈ Cat → ∅ Fn (∅ × ∅))
12 eqid 2798 . . . . 5 (Homf𝐶) = (Homf𝐶)
13 eqid 2798 . . . . 5 (Base‘𝐶) = (Base‘𝐶)
1412, 13homffn 16955 . . . 4 (Homf𝐶) Fn ((Base‘𝐶) × (Base‘𝐶))
1514a1i 11 . . 3 (𝐶 ∈ Cat → (Homf𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)))
16 fvexd 6660 . . 3 (𝐶 ∈ Cat → (Base‘𝐶) ∈ V)
1711, 15, 16isssc 17082 . 2 (𝐶 ∈ Cat → (∅ ⊆cat (Homf𝐶) ↔ (∅ ⊆ (Base‘𝐶) ∧ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥𝑦) ⊆ (𝑥(Homf𝐶)𝑦))))
182, 4, 17mpbir2and 712 1 (𝐶 ∈ Cat → ∅ ⊆cat (Homf𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2111  wral 3106  Vcvv 3441  wss 3881  c0 4243   class class class wbr 5030   × cxp 5517   Fn wfn 6319  wf 6320  cfv 6324  (class class class)co 7135  Basecbs 16475  Catccat 16927  Homf chomf 16929  cat cssc 17069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-1st 7671  df-2nd 7672  df-ixp 8445  df-homf 16933  df-ssc 17072
This theorem is referenced by:  0subcat  17100
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