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Mirrors > Home > MPE Home > Th. List > issect2 | Structured version Visualization version GIF version |
Description: Property of being a section. (Contributed by Mario Carneiro, 2-Jan-2017.) |
Ref | Expression |
---|---|
issect.b | ⊢ 𝐵 = (Base‘𝐶) |
issect.h | ⊢ 𝐻 = (Hom ‘𝐶) |
issect.o | ⊢ · = (comp‘𝐶) |
issect.i | ⊢ 1 = (Id‘𝐶) |
issect.s | ⊢ 𝑆 = (Sect‘𝐶) |
issect.c | ⊢ (𝜑 → 𝐶 ∈ Cat) |
issect.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
issect.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
issect.f | ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) |
issect.g | ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑋)) |
Ref | Expression |
---|---|
issect2 | ⊢ (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐺(〈𝑋, 𝑌〉 · 𝑋)𝐹) = ( 1 ‘𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | issect.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) | |
2 | issect.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑋)) | |
3 | 1, 2 | jca 514 | . 2 ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋))) |
4 | issect.b | . . . 4 ⊢ 𝐵 = (Base‘𝐶) | |
5 | issect.h | . . . 4 ⊢ 𝐻 = (Hom ‘𝐶) | |
6 | issect.o | . . . 4 ⊢ · = (comp‘𝐶) | |
7 | issect.i | . . . 4 ⊢ 1 = (Id‘𝐶) | |
8 | issect.s | . . . 4 ⊢ 𝑆 = (Sect‘𝐶) | |
9 | issect.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
10 | issect.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
11 | issect.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
12 | 4, 5, 6, 7, 8, 9, 10, 11 | issect 17025 | . . 3 ⊢ (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋) ∧ (𝐺(〈𝑋, 𝑌〉 · 𝑋)𝐹) = ( 1 ‘𝑋)))) |
13 | df-3an 1085 | . . 3 ⊢ ((𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋) ∧ (𝐺(〈𝑋, 𝑌〉 · 𝑋)𝐹) = ( 1 ‘𝑋)) ↔ ((𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋)) ∧ (𝐺(〈𝑋, 𝑌〉 · 𝑋)𝐹) = ( 1 ‘𝑋))) | |
14 | 12, 13 | syl6bb 289 | . 2 ⊢ (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ ((𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋)) ∧ (𝐺(〈𝑋, 𝑌〉 · 𝑋)𝐹) = ( 1 ‘𝑋)))) |
15 | 3, 14 | mpbirand 705 | 1 ⊢ (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐺(〈𝑋, 𝑌〉 · 𝑋)𝐹) = ( 1 ‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 〈cop 4575 class class class wbr 5068 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 Hom chom 16578 compcco 16579 Catccat 16937 Idccid 16938 Sectcsect 17016 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-sect 17019 |
This theorem is referenced by: sectco 17028 dfiso3 17045 monsect 17055 sectid 17058 invcoisoid 17064 isocoinvid 17065 funcsect 17144 fthsect 17197 fucsect 17244 2initoinv 17272 2termoinv 17279 catcisolem 17368 |
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