|   | Mathbox for Zhi Wang | < Previous  
      Next > Nearby theorems | |
| 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 3202 | . . 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 2827 | . . . . . 6 ⊢ (𝜑 → (𝑓 ∈ (𝑥𝐻𝑦) ↔ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) | 
| 8 | 7 | mobidv 2549 | . . . . 5 ⊢ (𝜑 → (∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦) ↔ ∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) | 
| 9 | 4, 8 | raleqbidv 3346 | . . . 4 ⊢ (𝜑 → (∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦) ↔ ∀𝑦 ∈ (Base‘𝐶)∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) | 
| 10 | 4, 9 | raleqbidv 3346 | . . 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 2737 | . . 3 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
| 13 | eqid 2737 | . . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
| 14 | 12, 13 | isthinc 49069 | . 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 1540 ∈ wcel 2108 ∃*wmo 2538 ∀wral 3061 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 Hom chom 17308 Catccat 17707 ThinCatcthinc 49067 | 
| 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-ext 2708 ax-nul 5306 | 
| 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-sb 2065 df-mo 2540 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 df-thinc 49068 | 
| This theorem is referenced by: isthincd2 49086 oppcthin 49087 subthinc 49092 thincciso2 49104 setcthin 49112 | 
| Copyright terms: Public domain | W3C validator |