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| Mirrors > Home > MPE Home > Th. List > sectcan | Structured version Visualization version GIF version | ||
| Description: If 𝐺 is a section of 𝐹 and 𝐹 is a section of 𝐻, then 𝐺 = 𝐻. Proposition 3.10 of [Adamek] p. 28. (Contributed by Mario Carneiro, 2-Jan-2017.) | 
| Ref | Expression | 
|---|---|
| sectcan.b | ⊢ 𝐵 = (Base‘𝐶) | 
| sectcan.s | ⊢ 𝑆 = (Sect‘𝐶) | 
| sectcan.c | ⊢ (𝜑 → 𝐶 ∈ Cat) | 
| sectcan.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) | 
| sectcan.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) | 
| sectcan.1 | ⊢ (𝜑 → 𝐺(𝑋𝑆𝑌)𝐹) | 
| sectcan.2 | ⊢ (𝜑 → 𝐹(𝑌𝑆𝑋)𝐻) | 
| Ref | Expression | 
|---|---|
| sectcan | ⊢ (𝜑 → 𝐺 = 𝐻) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | sectcan.b | . . . 4 ⊢ 𝐵 = (Base‘𝐶) | |
| 2 | eqid 2737 | . . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
| 3 | eqid 2737 | . . . 4 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
| 4 | sectcan.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
| 5 | sectcan.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
| 6 | sectcan.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
| 7 | sectcan.1 | . . . . . 6 ⊢ (𝜑 → 𝐺(𝑋𝑆𝑌)𝐹) | |
| 8 | eqid 2737 | . . . . . . 7 ⊢ (Id‘𝐶) = (Id‘𝐶) | |
| 9 | sectcan.s | . . . . . . 7 ⊢ 𝑆 = (Sect‘𝐶) | |
| 10 | 1, 2, 3, 8, 9, 4, 5, 6 | issect 17797 | . . . . . 6 ⊢ (𝜑 → (𝐺(𝑋𝑆𝑌)𝐹 ↔ (𝐺 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐹 ∈ (𝑌(Hom ‘𝐶)𝑋) ∧ (𝐹(〈𝑋, 𝑌〉(comp‘𝐶)𝑋)𝐺) = ((Id‘𝐶)‘𝑋)))) | 
| 11 | 7, 10 | mpbid 232 | . . . . 5 ⊢ (𝜑 → (𝐺 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐹 ∈ (𝑌(Hom ‘𝐶)𝑋) ∧ (𝐹(〈𝑋, 𝑌〉(comp‘𝐶)𝑋)𝐺) = ((Id‘𝐶)‘𝑋))) | 
| 12 | 11 | simp1d 1143 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ (𝑋(Hom ‘𝐶)𝑌)) | 
| 13 | sectcan.2 | . . . . . 6 ⊢ (𝜑 → 𝐹(𝑌𝑆𝑋)𝐻) | |
| 14 | 1, 2, 3, 8, 9, 4, 6, 5 | issect 17797 | . . . . . 6 ⊢ (𝜑 → (𝐹(𝑌𝑆𝑋)𝐻 ↔ (𝐹 ∈ (𝑌(Hom ‘𝐶)𝑋) ∧ 𝐻 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ (𝐻(〈𝑌, 𝑋〉(comp‘𝐶)𝑌)𝐹) = ((Id‘𝐶)‘𝑌)))) | 
| 15 | 13, 14 | mpbid 232 | . . . . 5 ⊢ (𝜑 → (𝐹 ∈ (𝑌(Hom ‘𝐶)𝑋) ∧ 𝐻 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ (𝐻(〈𝑌, 𝑋〉(comp‘𝐶)𝑌)𝐹) = ((Id‘𝐶)‘𝑌))) | 
| 16 | 15 | simp1d 1143 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝑌(Hom ‘𝐶)𝑋)) | 
| 17 | 15 | simp2d 1144 | . . . 4 ⊢ (𝜑 → 𝐻 ∈ (𝑋(Hom ‘𝐶)𝑌)) | 
| 18 | 1, 2, 3, 4, 5, 6, 5, 12, 16, 6, 17 | catass 17729 | . . 3 ⊢ (𝜑 → ((𝐻(〈𝑌, 𝑋〉(comp‘𝐶)𝑌)𝐹)(〈𝑋, 𝑌〉(comp‘𝐶)𝑌)𝐺) = (𝐻(〈𝑋, 𝑋〉(comp‘𝐶)𝑌)(𝐹(〈𝑋, 𝑌〉(comp‘𝐶)𝑋)𝐺))) | 
| 19 | 15 | simp3d 1145 | . . . 4 ⊢ (𝜑 → (𝐻(〈𝑌, 𝑋〉(comp‘𝐶)𝑌)𝐹) = ((Id‘𝐶)‘𝑌)) | 
| 20 | 19 | oveq1d 7446 | . . 3 ⊢ (𝜑 → ((𝐻(〈𝑌, 𝑋〉(comp‘𝐶)𝑌)𝐹)(〈𝑋, 𝑌〉(comp‘𝐶)𝑌)𝐺) = (((Id‘𝐶)‘𝑌)(〈𝑋, 𝑌〉(comp‘𝐶)𝑌)𝐺)) | 
| 21 | 11 | simp3d 1145 | . . . 4 ⊢ (𝜑 → (𝐹(〈𝑋, 𝑌〉(comp‘𝐶)𝑋)𝐺) = ((Id‘𝐶)‘𝑋)) | 
| 22 | 21 | oveq2d 7447 | . . 3 ⊢ (𝜑 → (𝐻(〈𝑋, 𝑋〉(comp‘𝐶)𝑌)(𝐹(〈𝑋, 𝑌〉(comp‘𝐶)𝑋)𝐺)) = (𝐻(〈𝑋, 𝑋〉(comp‘𝐶)𝑌)((Id‘𝐶)‘𝑋))) | 
| 23 | 18, 20, 22 | 3eqtr3d 2785 | . 2 ⊢ (𝜑 → (((Id‘𝐶)‘𝑌)(〈𝑋, 𝑌〉(comp‘𝐶)𝑌)𝐺) = (𝐻(〈𝑋, 𝑋〉(comp‘𝐶)𝑌)((Id‘𝐶)‘𝑋))) | 
| 24 | 1, 2, 8, 4, 5, 3, 6, 12 | catlid 17726 | . 2 ⊢ (𝜑 → (((Id‘𝐶)‘𝑌)(〈𝑋, 𝑌〉(comp‘𝐶)𝑌)𝐺) = 𝐺) | 
| 25 | 1, 2, 8, 4, 5, 3, 6, 17 | catrid 17727 | . 2 ⊢ (𝜑 → (𝐻(〈𝑋, 𝑋〉(comp‘𝐶)𝑌)((Id‘𝐶)‘𝑋)) = 𝐻) | 
| 26 | 23, 24, 25 | 3eqtr3d 2785 | 1 ⊢ (𝜑 → 𝐺 = 𝐻) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ 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-rmo 3380 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-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-cat 17711 df-cid 17712 df-sect 17791 | 
| This theorem is referenced by: invfun 17808 inveq 17818 | 
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