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Mirrors > Home > MPE Home > Th. List > Mathboxes > thincmo | Structured version Visualization version GIF version |
Description: There is at most one morphism in each hom-set. (Contributed by Zhi Wang, 21-Sep-2024.) |
Ref | Expression |
---|---|
thincmo.c | ⊢ (𝜑 → 𝐶 ∈ ThinCat) |
thincmo.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
thincmo.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
thincmo.b | ⊢ 𝐵 = (Base‘𝐶) |
thincmo.h | ⊢ 𝐻 = (Hom ‘𝐶) |
Ref | Expression |
---|---|
thincmo | ⊢ (𝜑 → ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | thincmo.x | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
2 | 1 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑋𝐻𝑌))) → 𝑋 ∈ 𝐵) |
3 | thincmo.y | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
4 | 3 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑋𝐻𝑌))) → 𝑌 ∈ 𝐵) |
5 | simprl 771 | . . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑋𝐻𝑌))) → 𝑓 ∈ (𝑋𝐻𝑌)) | |
6 | simprr 773 | . . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑋𝐻𝑌))) → 𝑔 ∈ (𝑋𝐻𝑌)) | |
7 | thincmo.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐶) | |
8 | thincmo.h | . . . . 5 ⊢ 𝐻 = (Hom ‘𝐶) | |
9 | thincmo.c | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ThinCat) | |
10 | 9 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑋𝐻𝑌))) → 𝐶 ∈ ThinCat) |
11 | 2, 4, 5, 6, 7, 8, 10 | thincmo2 45832 | . . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑋𝐻𝑌))) → 𝑓 = 𝑔) |
12 | 11 | ex 416 | . . 3 ⊢ (𝜑 → ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑋𝐻𝑌)) → 𝑓 = 𝑔)) |
13 | 12 | alrimivv 1935 | . 2 ⊢ (𝜑 → ∀𝑓∀𝑔((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑋𝐻𝑌)) → 𝑓 = 𝑔)) |
14 | eleq1w 2816 | . . 3 ⊢ (𝑓 = 𝑔 → (𝑓 ∈ (𝑋𝐻𝑌) ↔ 𝑔 ∈ (𝑋𝐻𝑌))) | |
15 | 14 | mo4 2567 | . 2 ⊢ (∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌) ↔ ∀𝑓∀𝑔((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑋𝐻𝑌)) → 𝑓 = 𝑔)) |
16 | 13, 15 | sylibr 237 | 1 ⊢ (𝜑 → ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∀wal 1540 = wceq 1542 ∈ wcel 2114 ∃*wmo 2539 ‘cfv 6349 (class class class)co 7182 Basecbs 16598 Hom chom 16691 ThinCatcthinc 45826 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-nul 5184 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4222 df-sn 4527 df-pr 4529 df-op 4533 df-uni 4807 df-br 5041 df-iota 6307 df-fv 6357 df-ov 7185 df-thinc 45827 |
This theorem is referenced by: thincmod 45835 |
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