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Theorem 0ssc 16263
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 3920 . . 3 ∅ ⊆ (Base‘𝐶)
21a1i 11 . 2 (𝐶 ∈ Cat → ∅ ⊆ (Base‘𝐶))
3 ral0 4024 . . 3 𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥𝑦) ⊆ (𝑥(Homf𝐶)𝑦)
43a1i 11 . 2 (𝐶 ∈ Cat → ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥𝑦) ⊆ (𝑥(Homf𝐶)𝑦))
5 f0 5981 . . . . . 6 ∅:∅⟶∅
6 ffn 5941 . . . . . 6 (∅:∅⟶∅ → ∅ Fn ∅)
75, 6ax-mp 5 . . . . 5 ∅ Fn ∅
8 xp0 5454 . . . . . 6 (∅ × ∅) = ∅
98fneq2i 5883 . . . . 5 (∅ Fn (∅ × ∅) ↔ ∅ Fn ∅)
107, 9mpbir 219 . . . 4 ∅ Fn (∅ × ∅)
1110a1i 11 . . 3 (𝐶 ∈ Cat → ∅ Fn (∅ × ∅))
12 eqid 2606 . . . . 5 (Homf𝐶) = (Homf𝐶)
13 eqid 2606 . . . . 5 (Base‘𝐶) = (Base‘𝐶)
1412, 13homffn 16119 . . . 4 (Homf𝐶) Fn ((Base‘𝐶) × (Base‘𝐶))
1514a1i 11 . . 3 (𝐶 ∈ Cat → (Homf𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)))
16 fvex 6095 . . . 4 (Base‘𝐶) ∈ V
1716a1i 11 . . 3 (𝐶 ∈ Cat → (Base‘𝐶) ∈ V)
1811, 15, 17isssc 16246 . 2 (𝐶 ∈ Cat → (∅ ⊆cat (Homf𝐶) ↔ (∅ ⊆ (Base‘𝐶) ∧ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥𝑦) ⊆ (𝑥(Homf𝐶)𝑦))))
192, 4, 18mpbir2and 958 1 (𝐶 ∈ Cat → ∅ ⊆cat (Homf𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1976  wral 2892  Vcvv 3169  wss 3536  c0 3870   class class class wbr 4574   × cxp 5023   Fn wfn 5782  wf 5783  cfv 5787  (class class class)co 6524  Basecbs 15638  Catccat 16091  Homf chomf 16093  cat cssc 16233
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-rep 4690  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825  ax-un 6821
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-ral 2897  df-rex 2898  df-reu 2899  df-rab 2901  df-v 3171  df-sbc 3399  df-csb 3496  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-nul 3871  df-if 4033  df-pw 4106  df-sn 4122  df-pr 4124  df-op 4128  df-uni 4364  df-iun 4448  df-br 4575  df-opab 4635  df-mpt 4636  df-id 4940  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-res 5037  df-ima 5038  df-iota 5751  df-fun 5789  df-fn 5790  df-f 5791  df-f1 5792  df-fo 5793  df-f1o 5794  df-fv 5795  df-ov 6527  df-oprab 6528  df-mpt2 6529  df-1st 7033  df-2nd 7034  df-ixp 7769  df-homf 16097  df-ssc 16236
This theorem is referenced by:  0subcat  16264
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