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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 17795 | . . 3 ⊢ (𝜑 → (𝑋𝑆𝑌) = {〈𝑓, 𝑔〉 ∣ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑔(〈𝑋, 𝑌〉 · 𝑋)𝑓) = ( 1 ‘𝑋))}) |
| 10 | 9 | breqd 5154 | . 2 ⊢ (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ 𝐹{〈𝑓, 𝑔〉 ∣ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑔(〈𝑋, 𝑌〉 · 𝑋)𝑓) = ( 1 ‘𝑋))}𝐺)) |
| 11 | oveq12 7440 | . . . . . 6 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝐹) → (𝑔(〈𝑋, 𝑌〉 · 𝑋)𝑓) = (𝐺(〈𝑋, 𝑌〉 · 𝑋)𝐹)) | |
| 12 | 11 | ancoms 458 | . . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑔(〈𝑋, 𝑌〉 · 𝑋)𝑓) = (𝐺(〈𝑋, 𝑌〉 · 𝑋)𝐹)) |
| 13 | 12 | eqeq1d 2739 | . . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((𝑔(〈𝑋, 𝑌〉 · 𝑋)𝑓) = ( 1 ‘𝑋) ↔ (𝐺(〈𝑋, 𝑌〉 · 𝑋)𝐹) = ( 1 ‘𝑋))) |
| 14 | eqid 2737 | . . . 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 5779 | . . 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 1089 | . . 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 278 | . 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 | bitrdi 287 | 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 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 〈cop 4632 class class class wbr 5143 {copab 5205 ‘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: issect2 17798 sectcan 17799 sectco 17800 oppcsect 17822 sectmon 17826 monsect 17827 funcsect 17917 fucsect 18020 invfuc 18022 setcsect 18134 catciso 18156 rngcsect 20636 ringcsect 20670 rngcsectALTV 48191 ringcsectALTV 48225 thincsect 49114 |
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