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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 4342 | . . . . . 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 47960 | . . . . 5 ⊢ (𝜑 → ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌)) |
9 | 2, 8 | anim12i 612 | . . . 4 ⊢ (((𝑋𝐻𝑌) ≠ ∅ ∧ 𝜑) → (∃𝑓 𝑓 ∈ (𝑋𝐻𝑌) ∧ ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌))) |
10 | df-eu 2558 | . . . 4 ⊢ (∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌) ↔ (∃𝑓 𝑓 ∈ (𝑋𝐻𝑌) ∧ ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌))) | |
11 | 9, 10 | sylibr 233 | . . 3 ⊢ (((𝑋𝐻𝑌) ≠ ∅ ∧ 𝜑) → ∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌)) |
12 | 11 | expcom 413 | . 2 ⊢ (𝜑 → ((𝑋𝐻𝑌) ≠ ∅ → ∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌))) |
13 | euex 2566 | . . 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 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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