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Theorem sectcan 17467
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 17465 . . . . . 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 1141 . . . 4 (𝜑𝐺 ∈ (𝑋(Hom ‘𝐶)𝑌))
13 sectcan.2 . . . . . 6 (𝜑𝐹(𝑌𝑆𝑋)𝐻)
141, 2, 3, 8, 9, 4, 6, 5issect 17465 . . . . . 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 1141 . . . 4 (𝜑𝐹 ∈ (𝑌(Hom ‘𝐶)𝑋))
1715simp2d 1142 . . . 4 (𝜑𝐻 ∈ (𝑋(Hom ‘𝐶)𝑌))
181, 2, 3, 4, 5, 6, 5, 12, 16, 6, 17catass 17395 . . 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 1143 . . . 4 (𝜑 → (𝐻(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐹) = ((Id‘𝐶)‘𝑌))
2019oveq1d 7290 . . 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 1143 . . . 4 (𝜑 → (𝐹(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐺) = ((Id‘𝐶)‘𝑋))
2221oveq2d 7291 . . 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 17392 . 2 (𝜑 → (((Id‘𝐶)‘𝑌)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)𝐺) = 𝐺)
251, 2, 8, 4, 5, 3, 6, 17catrid 17393 . 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 1086   = wceq 1539  wcel 2106  cop 4567   class class class wbr 5074  cfv 6433  (class class class)co 7275  Basecbs 16912  Hom chom 16973  compcco 16974  Catccat 17373  Idccid 17374  Sectcsect 17456
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-cat 17377  df-cid 17378  df-sect 17459
This theorem is referenced by:  invfun  17476  inveq  17486
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