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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 512 | . 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 17684 | . . 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 286 | . 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 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 〈cop 4629 class class class wbr 5142 ‘cfv 6533 (class class class)co 7394 Basecbs 17128 Hom chom 17192 compcco 17193 Catccat 17592 Idccid 17593 Sectcsect 17675 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5357 ax-pr 5421 ax-un 7709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3775 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5568 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-ov 7397 df-oprab 7398 df-mpo 7399 df-1st 7959 df-2nd 7960 df-sect 17678 |
This theorem is referenced by: sectco 17687 dfiso3 17704 monsect 17714 sectid 17717 invcoisoid 17723 isocoinvid 17724 funcsect 17806 fthsect 17860 fucsect 17909 2initoinv 17944 2termoinv 17951 catcisolem 18044 |
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