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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 511 | . 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 17797 | . . 3 ⊢ (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋) ∧ (𝐺(〈𝑋, 𝑌〉 · 𝑋)𝐹) = ( 1 ‘𝑋)))) |
| 13 | df-3an 1089 | . . 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 287 | . 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 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 〈cop 4632 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 Hom chom 17308 compcco 17309 Catccat 17707 Idccid 17708 Sectcsect 17788 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-sect 17791 |
| This theorem is referenced by: sectco 17800 dfiso3 17817 monsect 17827 sectid 17830 invcoisoid 17836 isocoinvid 17837 funcsect 17917 fthsect 17972 fucsect 18020 2initoinv 18055 2termoinv 18062 catcisolem 18155 |
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