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Theorem sectcan 17674
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.)
Hypotheses
Ref Expression
sectcan.b 𝐵 = (Base‘𝐶)
sectcan.s 𝑆 = (Sect‘𝐶)
sectcan.c (𝜑𝐶 ∈ Cat)
sectcan.x (𝜑𝑋𝐵)
sectcan.y (𝜑𝑌𝐵)
sectcan.1 (𝜑𝐺(𝑋𝑆𝑌)𝐹)
sectcan.2 (𝜑𝐹(𝑌𝑆𝑋)𝐻)
Assertion
Ref Expression
sectcan (𝜑𝐺 = 𝐻)

Proof of Theorem sectcan
StepHypRef Expression
1 sectcan.b . . . 4 𝐵 = (Base‘𝐶)
2 eqid 2731 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
3 eqid 2731 . . . 4 (comp‘𝐶) = (comp‘𝐶)
4 sectcan.c . . . 4 (𝜑𝐶 ∈ Cat)
5 sectcan.x . . . 4 (𝜑𝑋𝐵)
6 sectcan.y . . . 4 (𝜑𝑌𝐵)
7 sectcan.1 . . . . . 6 (𝜑𝐺(𝑋𝑆𝑌)𝐹)
8 eqid 2731 . . . . . . 7 (Id‘𝐶) = (Id‘𝐶)
9 sectcan.s . . . . . . 7 𝑆 = (Sect‘𝐶)
101, 2, 3, 8, 9, 4, 5, 6issect 17672 . . . . . 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))))
117, 10mpbid 231 . . . . 5 (𝜑 → (𝐺 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐹 ∈ (𝑌(Hom ‘𝐶)𝑋) ∧ (𝐹(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐺) = ((Id‘𝐶)‘𝑋)))
1211simp1d 1142 . . . 4 (𝜑𝐺 ∈ (𝑋(Hom ‘𝐶)𝑌))
13 sectcan.2 . . . . . 6 (𝜑𝐹(𝑌𝑆𝑋)𝐻)
141, 2, 3, 8, 9, 4, 6, 5issect 17672 . . . . . 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))))
1513, 14mpbid 231 . . . . 5 (𝜑 → (𝐹 ∈ (𝑌(Hom ‘𝐶)𝑋) ∧ 𝐻 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ (𝐻(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐹) = ((Id‘𝐶)‘𝑌)))
1615simp1d 1142 . . . 4 (𝜑𝐹 ∈ (𝑌(Hom ‘𝐶)𝑋))
1715simp2d 1143 . . . 4 (𝜑𝐻 ∈ (𝑋(Hom ‘𝐶)𝑌))
181, 2, 3, 4, 5, 6, 5, 12, 16, 6, 17catass 17602 . . 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)))
1915simp3d 1144 . . . 4 (𝜑 → (𝐻(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐹) = ((Id‘𝐶)‘𝑌))
2019oveq1d 7399 . . 3 (𝜑 → ((𝐻(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐹)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)𝐺) = (((Id‘𝐶)‘𝑌)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)𝐺))
2111simp3d 1144 . . . 4 (𝜑 → (𝐹(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐺) = ((Id‘𝐶)‘𝑋))
2221oveq2d 7400 . . 3 (𝜑 → (𝐻(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)(𝐹(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐺)) = (𝐻(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)((Id‘𝐶)‘𝑋)))
2318, 20, 223eqtr3d 2779 . 2 (𝜑 → (((Id‘𝐶)‘𝑌)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)𝐺) = (𝐻(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)((Id‘𝐶)‘𝑋)))
241, 2, 8, 4, 5, 3, 6, 12catlid 17599 . 2 (𝜑 → (((Id‘𝐶)‘𝑌)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)𝐺) = 𝐺)
251, 2, 8, 4, 5, 3, 6, 17catrid 17600 . 2 (𝜑 → (𝐻(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)((Id‘𝐶)‘𝑋)) = 𝐻)
2623, 24, 253eqtr3d 2779 1 (𝜑𝐺 = 𝐻)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1087   = wceq 1541  wcel 2106  cop 4619   class class class wbr 5132  cfv 6523  (class class class)co 7384  Basecbs 17116  Hom chom 17180  compcco 17181  Catccat 17580  Idccid 17581  Sectcsect 17663
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5269  ax-sep 5283  ax-nul 5290  ax-pow 5347  ax-pr 5411  ax-un 7699
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3371  df-reu 3372  df-rab 3426  df-v 3468  df-sbc 3765  df-csb 3881  df-dif 3938  df-un 3940  df-in 3942  df-ss 3952  df-nul 4310  df-if 4514  df-pw 4589  df-sn 4614  df-pr 4616  df-op 4620  df-uni 4893  df-iun 4983  df-br 5133  df-opab 5195  df-mpt 5216  df-id 5558  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-iota 6475  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-riota 7340  df-ov 7387  df-oprab 7388  df-mpo 7389  df-1st 7948  df-2nd 7949  df-cat 17584  df-cid 17585  df-sect 17666
This theorem is referenced by:  invfun  17683  inveq  17693
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