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Theorem 0ssc 17095
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 4347 . . 3 ∅ ⊆ (Base‘𝐶)
21a1i 11 . 2 (𝐶 ∈ Cat → ∅ ⊆ (Base‘𝐶))
3 ral0 4452 . . 3 𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥𝑦) ⊆ (𝑥(Homf𝐶)𝑦)
43a1i 11 . 2 (𝐶 ∈ Cat → ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥𝑦) ⊆ (𝑥(Homf𝐶)𝑦))
5 f0 6553 . . . . . 6 ∅:∅⟶∅
6 ffn 6507 . . . . . 6 (∅:∅⟶∅ → ∅ Fn ∅)
75, 6ax-mp 5 . . . . 5 ∅ Fn ∅
8 xp0 6008 . . . . . 6 (∅ × ∅) = ∅
98fneq2i 6444 . . . . 5 (∅ Fn (∅ × ∅) ↔ ∅ Fn ∅)
107, 9mpbir 232 . . . 4 ∅ Fn (∅ × ∅)
1110a1i 11 . . 3 (𝐶 ∈ Cat → ∅ Fn (∅ × ∅))
12 eqid 2818 . . . . 5 (Homf𝐶) = (Homf𝐶)
13 eqid 2818 . . . . 5 (Base‘𝐶) = (Base‘𝐶)
1412, 13homffn 16951 . . . 4 (Homf𝐶) Fn ((Base‘𝐶) × (Base‘𝐶))
1514a1i 11 . . 3 (𝐶 ∈ Cat → (Homf𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)))
16 fvexd 6678 . . 3 (𝐶 ∈ Cat → (Base‘𝐶) ∈ V)
1711, 15, 16isssc 17078 . 2 (𝐶 ∈ Cat → (∅ ⊆cat (Homf𝐶) ↔ (∅ ⊆ (Base‘𝐶) ∧ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥𝑦) ⊆ (𝑥(Homf𝐶)𝑦))))
182, 4, 17mpbir2and 709 1 (𝐶 ∈ Cat → ∅ ⊆cat (Homf𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2105  wral 3135  Vcvv 3492  wss 3933  c0 4288   class class class wbr 5057   × cxp 5546   Fn wfn 6343  wf 6344  cfv 6348  (class class class)co 7145  Basecbs 16471  Catccat 16923  Homf chomf 16925  cat cssc 17065
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679  df-ixp 8450  df-homf 16929  df-ssc 17068
This theorem is referenced by:  0subcat  17096
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