Theorem sectcan 17017

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 2798	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
3		eqid 2798	. . . 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 2798	. . . . . . 7 ⊢ (Id‘𝐶) = (Id‘𝐶)
9		sectcan.s	. . . . . . 7 ⊢ 𝑆 = (Sect‘𝐶)
10	1, 2, 3, 8, 9, 4, 5, 6	issect 17015	. . . . . 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 235	. . . . 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 1139	. . . 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 17015	. . . . . 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 235	. . . . 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 1139	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝑌(Hom ‘𝐶)𝑋))
17	15	simp2d 1140	. . . 4 ⊢ (𝜑 → 𝐻 ∈ (𝑋(Hom ‘𝐶)𝑌))
18	1, 2, 3, 4, 5, 6, 5, 12, 16, 6, 17	catass 16949	. . 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 1141	. . . 4 ⊢ (𝜑 → (𝐻(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐹) = ((Id‘𝐶)‘𝑌))
20	19	oveq1d 7150	. . 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 1141	. . . 4 ⊢ (𝜑 → (𝐹(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐺) = ((Id‘𝐶)‘𝑋))
22	21	oveq2d 7151	. . 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 2841	. 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 16946	. 2 ⊢ (𝜑 → (((Id‘𝐶)‘𝑌)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)𝐺) = 𝐺)
25	1, 2, 8, 4, 5, 3, 6, 17	catrid 16947	. 2 ⊢ (𝜑 → (𝐻(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)((Id‘𝐶)‘𝑋)) = 𝐻)
26	23, 24, 25	3eqtr3d 2841	1 ⊢ (𝜑 → 𝐺 = 𝐻)

Syntax hints:   → wi 4   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ⟨cop 4531   class class class wbr 5030  ‘cfv 6324  (class class class)co 7135  Basecbs 16475  Hom chom 16568  compcco 16569  Catccat 16927  Idccid 16928  Sectcsect 17006

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

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 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-1st 7671  df-2nd 7672  df-cat 16931  df-cid 16932  df-sect 17009

This theorem is referenced by:  invfun  17026  inveq  17036