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 49053
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 4352 . . . . . 6 ((𝑋𝐻𝑌) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝑋𝐻𝑌))
21biimpi 216 . . . . 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 49052 . . . . 5 (𝜑 → ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌))
92, 8anim12i 613 . . . 4 (((𝑋𝐻𝑌) ≠ ∅ ∧ 𝜑) → (∃𝑓 𝑓 ∈ (𝑋𝐻𝑌) ∧ ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌)))
10 df-eu 2568 . . . 4 (∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌) ↔ (∃𝑓 𝑓 ∈ (𝑋𝐻𝑌) ∧ ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌)))
119, 10sylibr 234 . . 3 (((𝑋𝐻𝑌) ≠ ∅ ∧ 𝜑) → ∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌))
1211expcom 413 . 2 (𝜑 → ((𝑋𝐻𝑌) ≠ ∅ → ∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌)))
13 euex 2576 . . 3 (∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌) → ∃𝑓 𝑓 ∈ (𝑋𝐻𝑌))
1413, 1sylibr 234 . 2 (∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌) → (𝑋𝐻𝑌) ≠ ∅)
1512, 14impbid1 225 1 (𝜑 → ((𝑋𝐻𝑌) ≠ ∅ ↔ ∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wex 1779  wcel 2108  ∃*wmo 2537  ∃!weu 2567  wne 2939  c0 4332  cfv 6559  (class class class)co 7429  Basecbs 17243  Hom chom 17304  ThinCatcthinc 49040
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-nul 5304
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-br 5142  df-iota 6512  df-fv 6567  df-ov 7432  df-thinc 49041
This theorem is referenced by:  fullthinc  49072  prstchom2  49140
  Copyright terms: Public domain W3C validator