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Mathbox for Zhi Wang |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > thincn0eu | Structured version Visualization version GIF version |
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.) |
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)) |
Ref | Expression |
---|---|
thincn0eu | ⊢ (𝜑 → ((𝑋𝐻𝑌) ≠ ∅ ↔ ∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | n0 4311 | . . . . . 6 ⊢ ((𝑋𝐻𝑌) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝑋𝐻𝑌)) | |
2 | 1 | biimpi 215 | . . . . 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 47171 | . . . . 5 ⊢ (𝜑 → ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌)) |
9 | 2, 8 | anim12i 613 | . . . 4 ⊢ (((𝑋𝐻𝑌) ≠ ∅ ∧ 𝜑) → (∃𝑓 𝑓 ∈ (𝑋𝐻𝑌) ∧ ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌))) |
10 | df-eu 2562 | . . . 4 ⊢ (∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌) ↔ (∃𝑓 𝑓 ∈ (𝑋𝐻𝑌) ∧ ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌))) | |
11 | 9, 10 | sylibr 233 | . . 3 ⊢ (((𝑋𝐻𝑌) ≠ ∅ ∧ 𝜑) → ∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌)) |
12 | 11 | expcom 414 | . 2 ⊢ (𝜑 → ((𝑋𝐻𝑌) ≠ ∅ → ∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌))) |
13 | euex 2570 | . . 3 ⊢ (∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌) → ∃𝑓 𝑓 ∈ (𝑋𝐻𝑌)) | |
14 | 13, 1 | sylibr 233 | . 2 ⊢ (∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌) → (𝑋𝐻𝑌) ≠ ∅) |
15 | 12, 14 | impbid1 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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