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Theorem thincn0eu 47172
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 4311 . . . . . 6 ((𝑋𝐻𝑌) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝑋𝐻𝑌))
21biimpi 215 . . . . 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 47171 . . . . 5 (𝜑 → ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌))
92, 8anim12i 613 . . . 4 (((𝑋𝐻𝑌) ≠ ∅ ∧ 𝜑) → (∃𝑓 𝑓 ∈ (𝑋𝐻𝑌) ∧ ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌)))
10 df-eu 2562 . . . 4 (∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌) ↔ (∃𝑓 𝑓 ∈ (𝑋𝐻𝑌) ∧ ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌)))
119, 10sylibr 233 . . 3 (((𝑋𝐻𝑌) ≠ ∅ ∧ 𝜑) → ∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌))
1211expcom 414 . 2 (𝜑 → ((𝑋𝐻𝑌) ≠ ∅ → ∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌)))
13 euex 2570 . . 3 (∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌) → ∃𝑓 𝑓 ∈ (𝑋𝐻𝑌))
1413, 1sylibr 233 . 2 (∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌) → (𝑋𝐻𝑌) ≠ ∅)
1512, 14impbid1 224 1 (𝜑 → ((𝑋𝐻𝑌) ≠ ∅ ↔ ∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wex 1781  wcel 2106  ∃*wmo 2531  ∃!weu 2561  wne 2939  c0 4287  cfv 6501  (class class class)co 7362  Basecbs 17094  Hom chom 17158  ThinCatcthinc 47159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-nul 5268
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rab 3406  df-v 3448  df-sbc 3743  df-csb 3859  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-iota 6453  df-fv 6509  df-ov 7365  df-thinc 47160
This theorem is referenced by:  fullthinc  47186  prstchom2  47218
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