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Mirrors > Home > MPE Home > Th. List > Mathboxes > thincmo2 | Structured version Visualization version GIF version |
Description: Morphisms in the same hom-set are identical. (Contributed by Zhi Wang, 17-Sep-2024.) |
Ref | Expression |
---|---|
isthincd2lem1.1 | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
isthincd2lem1.2 | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
isthincd2lem1.3 | ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) |
isthincd2lem1.4 | ⊢ (𝜑 → 𝐺 ∈ (𝑋𝐻𝑌)) |
thincmo2.b | ⊢ 𝐵 = (Base‘𝐶) |
thincmo2.h | ⊢ 𝐻 = (Hom ‘𝐶) |
thincmo2.c | ⊢ (𝜑 → 𝐶 ∈ ThinCat) |
Ref | Expression |
---|---|
thincmo2 | ⊢ (𝜑 → 𝐹 = 𝐺) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isthincd2lem1.1 | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
2 | isthincd2lem1.2 | . 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
3 | isthincd2lem1.3 | . 2 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) | |
4 | isthincd2lem1.4 | . 2 ⊢ (𝜑 → 𝐺 ∈ (𝑋𝐻𝑌)) | |
5 | thincmo2.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ ThinCat) | |
6 | thincmo2.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐶) | |
7 | thincmo2.h | . . . . 5 ⊢ 𝐻 = (Hom ‘𝐶) | |
8 | 6, 7 | isthinc 48210 | . . . 4 ⊢ (𝐶 ∈ ThinCat ↔ (𝐶 ∈ Cat ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦))) |
9 | 8 | simprbi 495 | . . 3 ⊢ (𝐶 ∈ ThinCat → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦)) |
10 | 5, 9 | syl 17 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦)) |
11 | 1, 2, 3, 4, 10 | isthincd2lem1 48216 | 1 ⊢ (𝜑 → 𝐹 = 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ∃*wmo 2526 ∀wral 3050 ‘cfv 6549 (class class class)co 7419 Basecbs 17183 Hom chom 17247 Catccat 17647 ThinCatcthinc 48208 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-nul 5307 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-iota 6501 df-fv 6557 df-ov 7422 df-thinc 48209 |
This theorem is referenced by: thincmo 48218 thincid 48222 thincmon 48223 thincepi 48224 functhinclem4 48233 |
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