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Theorem sectcan 17573
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 2738 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
3 eqid 2738 . . . 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 2738 . . . . . . 7 (Id‘𝐶) = (Id‘𝐶)
9 sectcan.s . . . . . . 7 𝑆 = (Sect‘𝐶)
101, 2, 3, 8, 9, 4, 5, 6issect 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))))
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 1143 . . . 4 (𝜑𝐺 ∈ (𝑋(Hom ‘𝐶)𝑌))
13 sectcan.2 . . . . . 6 (𝜑𝐹(𝑌𝑆𝑋)𝐻)
141, 2, 3, 8, 9, 4, 6, 5issect 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))))
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 1143 . . . 4 (𝜑𝐹 ∈ (𝑌(Hom ‘𝐶)𝑋))
1715simp2d 1144 . . . 4 (𝜑𝐻 ∈ (𝑋(Hom ‘𝐶)𝑌))
181, 2, 3, 4, 5, 6, 5, 12, 16, 6, 17catass 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)))
1915simp3d 1145 . . . 4 (𝜑 → (𝐻(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐹) = ((Id‘𝐶)‘𝑌))
2019oveq1d 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))
2111simp3d 1145 . . . 4 (𝜑 → (𝐹(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐺) = ((Id‘𝐶)‘𝑋))
2221oveq2d 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)))
2318, 20, 223eqtr3d 2786 . 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 17498 . 2 (𝜑 → (((Id‘𝐶)‘𝑌)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)𝐺) = 𝐺)
251, 2, 8, 4, 5, 3, 6, 17catrid 17499 . 2 (𝜑 → (𝐻(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)((Id‘𝐶)‘𝑋)) = 𝐻)
2623, 24, 253eqtr3d 2786 1 (𝜑𝐺 = 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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