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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 5068 | . 2 ⊢ (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ 𝐹{〈𝑓, 𝑔〉 ∣ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑔(〈𝑋, 𝑌〉 · 𝑋)𝑓) = ( 1 ‘𝑋))}𝐺)) |
11 | oveq12 7157 | . . . . . 6 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝐹) → (𝑔(〈𝑋, 𝑌〉 · 𝑋)𝑓) = (𝐺(〈𝑋, 𝑌〉 · 𝑋)𝐹)) | |
12 | 11 | ancoms 461 | . . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑔(〈𝑋, 𝑌〉 · 𝑋)𝑓) = (𝐺(〈𝑋, 𝑌〉 · 𝑋)𝐹)) |
13 | 12 | eqeq1d 2821 | . . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((𝑔(〈𝑋, 𝑌〉 · 𝑋)𝑓) = ( 1 ‘𝑋) ↔ (𝐺(〈𝑋, 𝑌〉 · 𝑋)𝐹) = ( 1 ‘𝑋))) |
14 | eqid 2819 | . . . 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 5637 | . . 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 1084 | . . 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 280 | . 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 289 | 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 208 ∧ wa 398 ∧ w3a 1082 = wceq 1531 ∈ wcel 2108 〈cop 4565 class class class wbr 5057 {copab 5119 ‘cfv 6348 (class class class)co 7148 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 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-ral 3141 df-rex 3142 df-reu 3143 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7151 df-oprab 7152 df-mpo 7153 df-1st 7681 df-2nd 7682 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 44242 rngcsectALTV 44254 ringcsect 44293 ringcsectALTV 44317 |
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