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| Mirrors > Home > MPE Home > Th. List > Mathboxes > thincmod | Structured version Visualization version GIF version | ||
| Description: At most one morphism in each hom-set (deduction form). (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 |
|---|---|
| thincmod | ⊢ (𝜑 → ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | thincmo.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ ThinCat) | |
| 2 | thincmo.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
| 3 | thincn0eu.b | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) | |
| 4 | 2, 3 | eleqtrd 2843 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶)) |
| 5 | thincmo.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
| 6 | 5, 3 | eleqtrd 2843 | . . 3 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶)) |
| 7 | eqid 2737 | . . 3 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
| 8 | eqid 2737 | . . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
| 9 | 1, 4, 6, 7, 8 | thincmo 49077 | . 2 ⊢ (𝜑 → ∃*𝑓 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌)) |
| 10 | thincn0eu.h | . . . . 5 ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶)) | |
| 11 | 10 | oveqd 7448 | . . . 4 ⊢ (𝜑 → (𝑋𝐻𝑌) = (𝑋(Hom ‘𝐶)𝑌)) |
| 12 | 11 | eleq2d 2827 | . . 3 ⊢ (𝜑 → (𝑓 ∈ (𝑋𝐻𝑌) ↔ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌))) |
| 13 | 12 | mobidv 2549 | . 2 ⊢ (𝜑 → (∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌) ↔ ∃*𝑓 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌))) |
| 14 | 9, 13 | mpbird 257 | 1 ⊢ (𝜑 → ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ∃*wmo 2538 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 Hom chom 17308 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-10 2141 ax-11 2157 ax-12 2177 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-nf 1784 df-sb 2065 df-mo 2540 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 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: thincn0eu 49080 |
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