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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 2841 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶)) |
5 | thincmo.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
6 | 5, 3 | eleqtrd 2841 | . . 3 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶)) |
7 | eqid 2735 | . . 3 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
8 | eqid 2735 | . . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
9 | 1, 4, 6, 7, 8 | thincmo 48829 | . 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 2825 | . . 3 ⊢ (𝜑 → (𝑓 ∈ (𝑋𝐻𝑌) ↔ 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌))) |
13 | 12 | mobidv 2547 | . 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 1537 ∈ wcel 2106 ∃*wmo 2536 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 Hom chom 17309 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-10 2139 ax-11 2155 ax-12 2175 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-nf 1781 df-sb 2063 df-mo 2538 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 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: thincn0eu 48832 |
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