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Mirrors > Home > MPE Home > Th. List > Mathboxes > thincsect2 | Structured version Visualization version GIF version |
Description: In a thin category, 𝐹 is a section of 𝐺 iff 𝐺 is a section of 𝐹. (Contributed by Zhi Wang, 24-Sep-2024.) |
Ref | Expression |
---|---|
thincsect.c | ⊢ (𝜑 → 𝐶 ∈ ThinCat) |
thincsect.b | ⊢ 𝐵 = (Base‘𝐶) |
thincsect.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
thincsect.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
thincsect.s | ⊢ 𝑆 = (Sect‘𝐶) |
Ref | Expression |
---|---|
thincsect2 | ⊢ (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ 𝐺(𝑌𝑆𝑋)𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ancom 464 | . . 3 ⊢ ((𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋)) ↔ (𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋) ∧ 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))) | |
2 | 1 | a1i 11 | . 2 ⊢ (𝜑 → ((𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋)) ↔ (𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋) ∧ 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌)))) |
3 | thincsect.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ ThinCat) | |
4 | thincsect.b | . . 3 ⊢ 𝐵 = (Base‘𝐶) | |
5 | thincsect.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
6 | thincsect.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
7 | thincsect.s | . . 3 ⊢ 𝑆 = (Sect‘𝐶) | |
8 | eqid 2736 | . . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
9 | 3, 4, 5, 6, 7, 8 | thincsect 45954 | . 2 ⊢ (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋)))) |
10 | 3, 4, 6, 5, 7, 8 | thincsect 45954 | . 2 ⊢ (𝜑 → (𝐺(𝑌𝑆𝑋)𝐹 ↔ (𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋) ∧ 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌)))) |
11 | 2, 9, 10 | 3bitr4d 314 | 1 ⊢ (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ 𝐺(𝑌𝑆𝑋)𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2112 class class class wbr 5039 ‘cfv 6358 (class class class)co 7191 Basecbs 16666 Hom chom 16760 Sectcsect 17203 ThinCatcthinc 45916 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-1st 7739 df-2nd 7740 df-cat 17125 df-cid 17126 df-sect 17206 df-thinc 45917 |
This theorem is referenced by: thincinv 45956 |
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