MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  0ssc Structured version   Visualization version   GIF version

Theorem 0ssc 16544
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 4005 . . 3 ∅ ⊆ (Base‘𝐶)
21a1i 11 . 2 (𝐶 ∈ Cat → ∅ ⊆ (Base‘𝐶))
3 ral0 4109 . . 3 𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥𝑦) ⊆ (𝑥(Homf𝐶)𝑦)
43a1i 11 . 2 (𝐶 ∈ Cat → ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥𝑦) ⊆ (𝑥(Homf𝐶)𝑦))
5 f0 6124 . . . . . 6 ∅:∅⟶∅
6 ffn 6083 . . . . . 6 (∅:∅⟶∅ → ∅ Fn ∅)
75, 6ax-mp 5 . . . . 5 ∅ Fn ∅
8 xp0 5587 . . . . . 6 (∅ × ∅) = ∅
98fneq2i 6024 . . . . 5 (∅ Fn (∅ × ∅) ↔ ∅ Fn ∅)
107, 9mpbir 221 . . . 4 ∅ Fn (∅ × ∅)
1110a1i 11 . . 3 (𝐶 ∈ Cat → ∅ Fn (∅ × ∅))
12 eqid 2651 . . . . 5 (Homf𝐶) = (Homf𝐶)
13 eqid 2651 . . . . 5 (Base‘𝐶) = (Base‘𝐶)
1412, 13homffn 16400 . . . 4 (Homf𝐶) Fn ((Base‘𝐶) × (Base‘𝐶))
1514a1i 11 . . 3 (𝐶 ∈ Cat → (Homf𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)))
16 fvexd 6241 . . 3 (𝐶 ∈ Cat → (Base‘𝐶) ∈ V)
1711, 15, 16isssc 16527 . 2 (𝐶 ∈ Cat → (∅ ⊆cat (Homf𝐶) ↔ (∅ ⊆ (Base‘𝐶) ∧ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥𝑦) ⊆ (𝑥(Homf𝐶)𝑦))))
182, 4, 17mpbir2and 977 1 (𝐶 ∈ Cat → ∅ ⊆cat (Homf𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2030  wral 2941  Vcvv 3231  wss 3607  c0 3948   class class class wbr 4685   × cxp 5141   Fn wfn 5921  wf 5922  cfv 5926  (class class class)co 6690  Basecbs 15904  Catccat 16372  Homf chomf 16374  cat cssc 16514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-ixp 7951  df-homf 16378  df-ssc 16517
This theorem is referenced by:  0subcat  16545
  Copyright terms: Public domain W3C validator