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Theorem sectcan 17014
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 2824 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
3 eqid 2824 . . . 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 2824 . . . . . . 7 (Id‘𝐶) = (Id‘𝐶)
9 sectcan.s . . . . . . 7 𝑆 = (Sect‘𝐶)
101, 2, 3, 8, 9, 4, 5, 6issect 17012 . . . . . 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 235 . . . . 5 (𝜑 → (𝐺 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐹 ∈ (𝑌(Hom ‘𝐶)𝑋) ∧ (𝐹(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐺) = ((Id‘𝐶)‘𝑋)))
1211simp1d 1139 . . . 4 (𝜑𝐺 ∈ (𝑋(Hom ‘𝐶)𝑌))
13 sectcan.2 . . . . . 6 (𝜑𝐹(𝑌𝑆𝑋)𝐻)
141, 2, 3, 8, 9, 4, 6, 5issect 17012 . . . . . 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 235 . . . . 5 (𝜑 → (𝐹 ∈ (𝑌(Hom ‘𝐶)𝑋) ∧ 𝐻 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ (𝐻(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐹) = ((Id‘𝐶)‘𝑌)))
1615simp1d 1139 . . . 4 (𝜑𝐹 ∈ (𝑌(Hom ‘𝐶)𝑋))
1715simp2d 1140 . . . 4 (𝜑𝐻 ∈ (𝑋(Hom ‘𝐶)𝑌))
181, 2, 3, 4, 5, 6, 5, 12, 16, 6, 17catass 16946 . . 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 1141 . . . 4 (𝜑 → (𝐻(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐹) = ((Id‘𝐶)‘𝑌))
2019oveq1d 7153 . . 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 1141 . . . 4 (𝜑 → (𝐹(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐺) = ((Id‘𝐶)‘𝑋))
2221oveq2d 7154 . . 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 2867 . 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 16943 . 2 (𝜑 → (((Id‘𝐶)‘𝑌)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)𝐺) = 𝐺)
251, 2, 8, 4, 5, 3, 6, 17catrid 16944 . 2 (𝜑 → (𝐻(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)((Id‘𝐶)‘𝑋)) = 𝐻)
2623, 24, 253eqtr3d 2867 1 (𝜑𝐺 = 𝐻)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1538  wcel 2115  cop 4554   class class class wbr 5047  cfv 6336  (class class class)co 7138  Basecbs 16472  Hom chom 16565  compcco 16566  Catccat 16924  Idccid 16925  Sectcsect 17003
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5171  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311  ax-un 7444
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-ral 3137  df-rex 3138  df-reu 3139  df-rmo 3140  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4820  df-iun 4902  df-br 5048  df-opab 5110  df-mpt 5128  df-id 5441  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-riota 7096  df-ov 7141  df-oprab 7142  df-mpo 7143  df-1st 7672  df-2nd 7673  df-cat 16928  df-cid 16929  df-sect 17006
This theorem is referenced by:  invfun  17023  inveq  17033
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