Theorem sectcan 17720

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	⊢ (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)

Assertion

Ref	Expression
sectcan	⊢ (𝜑 → 𝐺 = 𝐻)

Proof of Theorem sectcan

Step	Hyp	Ref	Expression
1		sectcan.b	. . . 4 ⊢ 𝐵 = (Base‘𝐶)
2		eqid 2740	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
3		eqid 2740	. . . 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 2740	. . . . . . 7 ⊢ (Id‘𝐶) = (Id‘𝐶)
9		sectcan.s	. . . . . . 7 ⊢ 𝑆 = (Sect‘𝐶)
10	1, 2, 3, 8, 9, 4, 5, 6	issect 17718	. . . . . 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 233	. . . . 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 1148	. . . 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 17718	. . . . . 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 233	. . . . 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 1148	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝑌(Hom ‘𝐶)𝑋))
17	15	simp2d 1149	. . . 4 ⊢ (𝜑 → 𝐻 ∈ (𝑋(Hom ‘𝐶)𝑌))
18	1, 2, 3, 4, 5, 6, 5, 12, 16, 6, 17	catass 17650	. . 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 1150	. . . 4 ⊢ (𝜑 → (𝐻(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐹) = ((Id‘𝐶)‘𝑌))
20	19	oveq1d 7378	. . 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 1150	. . . 4 ⊢ (𝜑 → (𝐹(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐺) = ((Id‘𝐶)‘𝑋))
22	21	oveq2d 7379	. . 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 2783	. 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 17647	. 2 ⊢ (𝜑 → (((Id‘𝐶)‘𝑌)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)𝐺) = 𝐺)
25	1, 2, 8, 4, 5, 3, 6, 17	catrid 17648	. 2 ⊢ (𝜑 → (𝐻(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)((Id‘𝐶)‘𝑋)) = 𝐻)
26	23, 24, 25	3eqtr3d 2783	1 ⊢ (𝜑 → 𝐺 = 𝐻)

Syntax hints:   → wi 4   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ⟨cop 4568   class class class wbr 5079  ‘cfv 6492  (class class class)co 7363  Basecbs 17177  Hom chom 17229  compcco 17230  Catccat 17628  Idccid 17629  Sectcsect 17709

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-1st 7938  df-2nd 7939  df-cat 17632  df-cid 17633  df-sect 17712

This theorem is referenced by:  invfun  17729  inveq  17739