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Mirrors > Home > MPE Home > Th. List > issect | Structured version Visualization version GIF version |
Description: The property "𝐹 is a section of 𝐺". (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) |
Ref | Expression |
---|---|
issect | ⊢ (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋) ∧ (𝐺(〈𝑋, 𝑌〉 · 𝑋)𝐹) = ( 1 ‘𝑋)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | issect.b | . . . 4 ⊢ 𝐵 = (Base‘𝐶) | |
2 | issect.h | . . . 4 ⊢ 𝐻 = (Hom ‘𝐶) | |
3 | issect.o | . . . 4 ⊢ · = (comp‘𝐶) | |
4 | issect.i | . . . 4 ⊢ 1 = (Id‘𝐶) | |
5 | issect.s | . . . 4 ⊢ 𝑆 = (Sect‘𝐶) | |
6 | issect.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
7 | issect.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
8 | issect.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | sectfval 17013 | . . 3 ⊢ (𝜑 → (𝑋𝑆𝑌) = {〈𝑓, 𝑔〉 ∣ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑔(〈𝑋, 𝑌〉 · 𝑋)𝑓) = ( 1 ‘𝑋))}) |
10 | 9 | breqd 5041 | . 2 ⊢ (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ 𝐹{〈𝑓, 𝑔〉 ∣ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑔(〈𝑋, 𝑌〉 · 𝑋)𝑓) = ( 1 ‘𝑋))}𝐺)) |
11 | oveq12 7144 | . . . . . 6 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝐹) → (𝑔(〈𝑋, 𝑌〉 · 𝑋)𝑓) = (𝐺(〈𝑋, 𝑌〉 · 𝑋)𝐹)) | |
12 | 11 | ancoms 462 | . . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑔(〈𝑋, 𝑌〉 · 𝑋)𝑓) = (𝐺(〈𝑋, 𝑌〉 · 𝑋)𝐹)) |
13 | 12 | eqeq1d 2800 | . . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((𝑔(〈𝑋, 𝑌〉 · 𝑋)𝑓) = ( 1 ‘𝑋) ↔ (𝐺(〈𝑋, 𝑌〉 · 𝑋)𝐹) = ( 1 ‘𝑋))) |
14 | eqid 2798 | . . . 4 ⊢ {〈𝑓, 𝑔〉 ∣ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑔(〈𝑋, 𝑌〉 · 𝑋)𝑓) = ( 1 ‘𝑋))} = {〈𝑓, 𝑔〉 ∣ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑔(〈𝑋, 𝑌〉 · 𝑋)𝑓) = ( 1 ‘𝑋))} | |
15 | 13, 14 | brab2a 5608 | . . 3 ⊢ (𝐹{〈𝑓, 𝑔〉 ∣ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑔(〈𝑋, 𝑌〉 · 𝑋)𝑓) = ( 1 ‘𝑋))}𝐺 ↔ ((𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋)) ∧ (𝐺(〈𝑋, 𝑌〉 · 𝑋)𝐹) = ( 1 ‘𝑋))) |
16 | df-3an 1086 | . . 3 ⊢ ((𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋) ∧ (𝐺(〈𝑋, 𝑌〉 · 𝑋)𝐹) = ( 1 ‘𝑋)) ↔ ((𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋)) ∧ (𝐺(〈𝑋, 𝑌〉 · 𝑋)𝐹) = ( 1 ‘𝑋))) | |
17 | 15, 16 | bitr4i 281 | . 2 ⊢ (𝐹{〈𝑓, 𝑔〉 ∣ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑔(〈𝑋, 𝑌〉 · 𝑋)𝑓) = ( 1 ‘𝑋))}𝐺 ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋) ∧ (𝐺(〈𝑋, 𝑌〉 · 𝑋)𝐹) = ( 1 ‘𝑋))) |
18 | 10, 17 | syl6bb 290 | 1 ⊢ (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋) ∧ (𝐺(〈𝑋, 𝑌〉 · 𝑋)𝐹) = ( 1 ‘𝑋)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 〈cop 4531 class class class wbr 5030 {copab 5092 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 Hom chom 16568 compcco 16569 Catccat 16927 Idccid 16928 Sectcsect 17006 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-sect 17009 |
This theorem is referenced by: issect2 17016 sectcan 17017 sectco 17018 oppcsect 17040 sectmon 17044 monsect 17045 funcsect 17134 fucsect 17234 invfuc 17236 setcsect 17341 catciso 17359 rngcsect 44604 rngcsectALTV 44616 ringcsect 44655 ringcsectALTV 44679 |
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