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Mathbox for Zhi Wang |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > isthincd | Structured version Visualization version GIF version |
Description: The predicate "is a thin category" (deduction form). (Contributed by Zhi Wang, 17-Sep-2024.) |
Ref | Expression |
---|---|
isthincd.b | ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) |
isthincd.h | ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶)) |
isthincd.t | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦)) |
isthincd.c | ⊢ (𝜑 → 𝐶 ∈ Cat) |
Ref | Expression |
---|---|
isthincd | ⊢ (𝜑 → 𝐶 ∈ ThinCat) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isthincd.c | . 2 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
2 | isthincd.t | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦)) | |
3 | 2 | ralrimivva 3200 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦)) |
4 | isthincd.b | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) | |
5 | isthincd.h | . . . . . . . 8 ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶)) | |
6 | 5 | oveqd 7448 | . . . . . . 7 ⊢ (𝜑 → (𝑥𝐻𝑦) = (𝑥(Hom ‘𝐶)𝑦)) |
7 | 6 | eleq2d 2825 | . . . . . 6 ⊢ (𝜑 → (𝑓 ∈ (𝑥𝐻𝑦) ↔ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) |
8 | 7 | mobidv 2547 | . . . . 5 ⊢ (𝜑 → (∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦) ↔ ∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) |
9 | 4, 8 | raleqbidv 3344 | . . . 4 ⊢ (𝜑 → (∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦) ↔ ∀𝑦 ∈ (Base‘𝐶)∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) |
10 | 4, 9 | raleqbidv 3344 | . . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦) ↔ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) |
11 | 3, 10 | mpbid 232 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) |
12 | eqid 2735 | . . 3 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
13 | eqid 2735 | . . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
14 | 12, 13 | isthinc 48821 | . 2 ⊢ (𝐶 ∈ ThinCat ↔ (𝐶 ∈ Cat ∧ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) |
15 | 1, 11, 14 | sylanbrc 583 | 1 ⊢ (𝜑 → 𝐶 ∈ ThinCat) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∃*wmo 2536 ∀wral 3059 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 Hom chom 17309 Catccat 17709 ThinCatcthinc 48819 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-nul 5312 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-mo 2538 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-iota 6516 df-fv 6571 df-ov 7434 df-thinc 48820 |
This theorem is referenced by: isthincd2 48838 oppcthin 48839 subthinc 48840 setcthin 48856 |
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