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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 4352 | . . . . . 6 ⊢ ((𝑋𝐻𝑌) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝑋𝐻𝑌)) | |
2 | 1 | biimpi 216 | . . . . 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 49052 | . . . . 5 ⊢ (𝜑 → ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌)) |
9 | 2, 8 | anim12i 613 | . . . 4 ⊢ (((𝑋𝐻𝑌) ≠ ∅ ∧ 𝜑) → (∃𝑓 𝑓 ∈ (𝑋𝐻𝑌) ∧ ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌))) |
10 | df-eu 2568 | . . . 4 ⊢ (∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌) ↔ (∃𝑓 𝑓 ∈ (𝑋𝐻𝑌) ∧ ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌))) | |
11 | 9, 10 | sylibr 234 | . . 3 ⊢ (((𝑋𝐻𝑌) ≠ ∅ ∧ 𝜑) → ∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌)) |
12 | 11 | expcom 413 | . 2 ⊢ (𝜑 → ((𝑋𝐻𝑌) ≠ ∅ → ∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌))) |
13 | euex 2576 | . . 3 ⊢ (∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌) → ∃𝑓 𝑓 ∈ (𝑋𝐻𝑌)) | |
14 | 13, 1 | sylibr 234 | . 2 ⊢ (∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌) → (𝑋𝐻𝑌) ≠ ∅) |
15 | 12, 14 | impbid1 225 | 1 ⊢ (𝜑 → ((𝑋𝐻𝑌) ≠ ∅ ↔ ∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2108 ∃*wmo 2537 ∃!weu 2567 ≠ wne 2939 ∅c0 4332 ‘cfv 6559 (class class class)co 7429 Basecbs 17243 Hom chom 17304 ThinCatcthinc 49040 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-nul 5304 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-br 5142 df-iota 6512 df-fv 6567 df-ov 7432 df-thinc 49041 |
This theorem is referenced by: fullthinc 49072 prstchom2 49140 |
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