Theorem thincn0eu 50209

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	⊢ (𝜑 → 𝐶 ∈ ThinCat)
thincmo.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
thincmo.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
thincn0eu.b	⊢ (𝜑 → 𝐵 = (Base‘𝐶))
thincn0eu.h	⊢ (𝜑 → 𝐻 = (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

Step	Hyp	Ref	Expression
1		n0 4307	. . . . . 6 ⊢ ((𝑋𝐻𝑌) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝑋𝐻𝑌))
2	1	biimpi 219	. . . . 5 ⊢ ((𝑋𝐻𝑌) ≠ ∅ → ∃𝑓 𝑓 ∈ (𝑋𝐻𝑌))
3		thincmo.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ThinCat)
4		thincmo.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
5		thincmo.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
6		thincn0eu.b	. . . . . 6 ⊢ (𝜑 → 𝐵 = (Base‘𝐶))
7		thincn0eu.h	. . . . . 6 ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶))
8	3, 4, 5, 6, 7	thincmod 50208	. . . . 5 ⊢ (𝜑 → ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌))
9	2, 8	anim12i 624	. . . 4 ⊢ (((𝑋𝐻𝑌) ≠ ∅ ∧ 𝜑) → (∃𝑓 𝑓 ∈ (𝑋𝐻𝑌) ∧ ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌)))
10		df-eu 2597	. . . 4 ⊢ (∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌) ↔ (∃𝑓 𝑓 ∈ (𝑋𝐻𝑌) ∧ ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌)))
11	9, 10	sylibr 237	. . 3 ⊢ (((𝑋𝐻𝑌) ≠ ∅ ∧ 𝜑) → ∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌))
12	11	expcom 418	. 2 ⊢ (𝜑 → ((𝑋𝐻𝑌) ≠ ∅ → ∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌)))
13		euex 2605	. . 3 ⊢ (∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌) → ∃𝑓 𝑓 ∈ (𝑋𝐻𝑌))
14	13, 1	sylibr 237	. 2 ⊢ (∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌) → (𝑋𝐻𝑌) ≠ ∅)
15	12, 14	impbid1 228	1 ⊢ (𝜑 → ((𝑋𝐻𝑌) ≠ ∅ ↔ ∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570  ∃wex 1809   ∈ wcel 2143  ∃*wmo 2565  ∃!weu 2596   ≠ wne 2958  ∅c0 4286  ‘cfv 6536  (class class class)co 7410  Basecbs 17264  Hom chom 17316  ThinCatcthinc 50195

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-nul 5269

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-iota 6492  df-fv 6544  df-ov 7413  df-thinc 50196

This theorem is referenced by:  fullthinc  50228  prstchom2  50341