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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 515 | . 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 17128 | . . 3 ⊢ (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋) ∧ (𝐺(〈𝑋, 𝑌〉 · 𝑋)𝐹) = ( 1 ‘𝑋)))) |
13 | df-3an 1090 | . . 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 | bitrdi 290 | . 2 ⊢ (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ ((𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋)) ∧ (𝐺(〈𝑋, 𝑌〉 · 𝑋)𝐹) = ( 1 ‘𝑋)))) |
15 | 3, 14 | mpbirand 707 | 1 ⊢ (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐺(〈𝑋, 𝑌〉 · 𝑋)𝐹) = ( 1 ‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 〈cop 4522 class class class wbr 5030 ‘cfv 6339 (class class class)co 7170 Basecbs 16586 Hom chom 16679 compcco 16680 Catccat 17038 Idccid 17039 Sectcsect 17119 |
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 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 |
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 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5429 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-ov 7173 df-oprab 7174 df-mpo 7175 df-1st 7714 df-2nd 7715 df-sect 17122 |
This theorem is referenced by: sectco 17131 dfiso3 17148 monsect 17158 sectid 17161 invcoisoid 17167 isocoinvid 17168 funcsect 17247 fthsect 17300 fucsect 17347 2initoinv 17382 2termoinv 17389 catcisolem 17482 |
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