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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 45918 | . . . 4 ⊢ (𝐶 ∈ ThinCat ↔ (𝐶 ∈ Cat ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦))) |
9 | 8 | simprbi 500 | . . 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 45924 | 1 ⊢ (𝜑 → 𝐹 = 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2112 ∃*wmo 2537 ∀wral 3051 ‘cfv 6358 (class class class)co 7191 Basecbs 16666 Hom chom 16760 Catccat 17121 ThinCatcthinc 45916 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-nul 5184 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-iota 6316 df-fv 6366 df-ov 7194 df-thinc 45917 |
This theorem is referenced by: thincmo 45926 thincid 45930 thincmon 45931 thincepi 45932 functhinclem4 45941 |
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