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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 2738 | . . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
3 | eqid 2738 | . . . 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 2738 | . . . . . . 7 ⊢ (Id‘𝐶) = (Id‘𝐶) | |
9 | sectcan.s | . . . . . . 7 ⊢ 𝑆 = (Sect‘𝐶) | |
10 | 1, 2, 3, 8, 9, 4, 5, 6 | issect 17571 | . . . . . 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 231 | . . . . 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 17571 | . . . . . 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 231 | . . . . 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 17501 | . . 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 7365 | . . 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 7366 | . . 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 2786 | . 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 17498 | . 2 ⊢ (𝜑 → (((Id‘𝐶)‘𝑌)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)𝐺) = 𝐺) |
25 | 1, 2, 8, 4, 5, 3, 6, 17 | catrid 17499 | . 2 ⊢ (𝜑 → (𝐻(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)((Id‘𝐶)‘𝑋)) = 𝐻) |
26 | 23, 24, 25 | 3eqtr3d 2786 | 1 ⊢ (𝜑 → 𝐺 = 𝐻) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ⟨cop 4591 class class class wbr 5104 ‘cfv 6492 (class class class)co 7350 Basecbs 17018 Hom chom 17079 compcco 17080 Catccat 17479 Idccid 17480 Sectcsect 17562 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7663 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7306 df-ov 7353 df-oprab 7354 df-mpo 7355 df-1st 7912 df-2nd 7913 df-cat 17483 df-cid 17484 df-sect 17565 |
This theorem is referenced by: invfun 17582 inveq 17592 |
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