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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 2622 | . . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
3 | eqid 2622 | . . . 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 2622 | . . . . . . 7 ⊢ (Id‘𝐶) = (Id‘𝐶) | |
9 | sectcan.s | . . . . . . 7 ⊢ 𝑆 = (Sect‘𝐶) | |
10 | 1, 2, 3, 8, 9, 4, 5, 6 | issect 16413 | . . . . . 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 222 | . . . . 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 1073 | . . . 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 16413 | . . . . . 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 222 | . . . . 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 1073 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝑌(Hom ‘𝐶)𝑋)) |
17 | 15 | simp2d 1074 | . . . 4 ⊢ (𝜑 → 𝐻 ∈ (𝑋(Hom ‘𝐶)𝑌)) |
18 | 1, 2, 3, 4, 5, 6, 5, 12, 16, 6, 17 | catass 16347 | . . 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 1075 | . . . 4 ⊢ (𝜑 → (𝐻(〈𝑌, 𝑋〉(comp‘𝐶)𝑌)𝐹) = ((Id‘𝐶)‘𝑌)) |
20 | 19 | oveq1d 6665 | . . 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 1075 | . . . 4 ⊢ (𝜑 → (𝐹(〈𝑋, 𝑌〉(comp‘𝐶)𝑋)𝐺) = ((Id‘𝐶)‘𝑋)) |
22 | 21 | oveq2d 6666 | . . 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 2664 | . 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 16344 | . 2 ⊢ (𝜑 → (((Id‘𝐶)‘𝑌)(〈𝑋, 𝑌〉(comp‘𝐶)𝑌)𝐺) = 𝐺) |
25 | 1, 2, 8, 4, 5, 3, 6, 17 | catrid 16345 | . 2 ⊢ (𝜑 → (𝐻(〈𝑋, 𝑋〉(comp‘𝐶)𝑌)((Id‘𝐶)‘𝑋)) = 𝐻) |
26 | 23, 24, 25 | 3eqtr3d 2664 | 1 ⊢ (𝜑 → 𝐺 = 𝐻) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 〈cop 4183 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 Hom chom 15952 compcco 15953 Catccat 16325 Idccid 16326 Sectcsect 16404 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-cat 16329 df-cid 16330 df-sect 16407 |
This theorem is referenced by: invfun 16424 inveq 16434 |
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