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Theorem thincn0eu 47961
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 4342 . . . . . 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 47960 . . . . 5 (𝜑 → ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌))
92, 8anim12i 612 . . . 4 (((𝑋𝐻𝑌) ≠ ∅ ∧ 𝜑) → (∃𝑓 𝑓 ∈ (𝑋𝐻𝑌) ∧ ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌)))
10 df-eu 2558 . . . 4 (∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌) ↔ (∃𝑓 𝑓 ∈ (𝑋𝐻𝑌) ∧ ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌)))
119, 10sylibr 233 . . 3 (((𝑋𝐻𝑌) ≠ ∅ ∧ 𝜑) → ∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌))
1211expcom 413 . 2 (𝜑 → ((𝑋𝐻𝑌) ≠ ∅ → ∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌)))
13 euex 2566 . . 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 395   = wceq 1534  wex 1774  wcel 2099  ∃*wmo 2527  ∃!weu 2557  wne 2935  c0 4318  cfv 6542  (class class class)co 7414  Basecbs 17171  Hom chom 17235  ThinCatcthinc 47948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-nul 5300
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-iota 6494  df-fv 6550  df-ov 7417  df-thinc 47949
This theorem is referenced by:  fullthinc  47975  prstchom2  48007
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