Users' Mathboxes Mathbox for Zhi Wang < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  thincn0eu Structured version   Visualization version   GIF version

Theorem thincn0eu 45836
Description: In a thin category, a hom-set being non-empty is equivalent to having a unique element. (Contributed by Zhi Wang, 21-Sep-2024.)
Hypotheses
Ref Expression
thincmo.c (𝜑𝐶 ∈ ThinCat)
thincmo.x (𝜑𝑋𝐵)
thincmo.y (𝜑𝑌𝐵)
thincn0eu.b (𝜑𝐵 = (Base‘𝐶))
thincn0eu.h (𝜑𝐻 = (Hom ‘𝐶))
Assertion
Ref Expression
thincn0eu (𝜑 → ((𝑋𝐻𝑌) ≠ ∅ ↔ ∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌)))
Distinct variable groups:   𝐵,𝑓   𝐶,𝑓   𝑓,𝐻   𝑓,𝑋   𝑓,𝑌   𝜑,𝑓

Proof of Theorem thincn0eu
StepHypRef Expression
1 n0 4245 . . . . . 6 ((𝑋𝐻𝑌) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝑋𝐻𝑌))
21biimpi 219 . . . . 5 ((𝑋𝐻𝑌) ≠ ∅ → ∃𝑓 𝑓 ∈ (𝑋𝐻𝑌))
3 thincmo.c . . . . . 6 (𝜑𝐶 ∈ ThinCat)
4 thincmo.x . . . . . 6 (𝜑𝑋𝐵)
5 thincmo.y . . . . . 6 (𝜑𝑌𝐵)
6 thincn0eu.b . . . . . 6 (𝜑𝐵 = (Base‘𝐶))
7 thincn0eu.h . . . . . 6 (𝜑𝐻 = (Hom ‘𝐶))
83, 4, 5, 6, 7thincmod 45835 . . . . 5 (𝜑 → ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌))
92, 8anim12i 616 . . . 4 (((𝑋𝐻𝑌) ≠ ∅ ∧ 𝜑) → (∃𝑓 𝑓 ∈ (𝑋𝐻𝑌) ∧ ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌)))
10 df-eu 2571 . . . 4 (∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌) ↔ (∃𝑓 𝑓 ∈ (𝑋𝐻𝑌) ∧ ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌)))
119, 10sylibr 237 . . 3 (((𝑋𝐻𝑌) ≠ ∅ ∧ 𝜑) → ∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌))
1211expcom 417 . 2 (𝜑 → ((𝑋𝐻𝑌) ≠ ∅ → ∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌)))
13 euex 2579 . . 3 (∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌) → ∃𝑓 𝑓 ∈ (𝑋𝐻𝑌))
1413, 1sylibr 237 . 2 (∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌) → (𝑋𝐻𝑌) ≠ ∅)
1512, 14impbid1 228 1 (𝜑 → ((𝑋𝐻𝑌) ≠ ∅ ↔ ∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1542  wex 1786  wcel 2114  ∃*wmo 2539  ∃!weu 2570  wne 2935  c0 4221  cfv 6349  (class class class)co 7182  Basecbs 16598  Hom chom 16691  ThinCatcthinc 45826
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 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-nul 5184
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3402  df-sbc 3686  df-csb 3801  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4222  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4807  df-br 5041  df-iota 6307  df-fv 6357  df-ov 7185  df-thinc 45827
This theorem is referenced by:  prstchom2  45863
  Copyright terms: Public domain W3C validator