Theorem sectcan 17680

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 2729	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
3		eqid 2729	. . . 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 2729	. . . . . . 7 ⊢ (Id‘𝐶) = (Id‘𝐶)
9		sectcan.s	. . . . . . 7 ⊢ 𝑆 = (Sect‘𝐶)
10	1, 2, 3, 8, 9, 4, 5, 6	issect 17678	. . . . . 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 232	. . . . 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 1142	. . . 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 17678	. . . . . 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 232	. . . . 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 1142	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝑌(Hom ‘𝐶)𝑋))
17	15	simp2d 1143	. . . 4 ⊢ (𝜑 → 𝐻 ∈ (𝑋(Hom ‘𝐶)𝑌))
18	1, 2, 3, 4, 5, 6, 5, 12, 16, 6, 17	catass 17610	. . 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 1144	. . . 4 ⊢ (𝜑 → (𝐻(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐹) = ((Id‘𝐶)‘𝑌))
20	19	oveq1d 7368	. . 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 1144	. . . 4 ⊢ (𝜑 → (𝐹(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐺) = ((Id‘𝐶)‘𝑋))
22	21	oveq2d 7369	. . 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 2772	. 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 17607	. 2 ⊢ (𝜑 → (((Id‘𝐶)‘𝑌)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)𝐺) = 𝐺)
25	1, 2, 8, 4, 5, 3, 6, 17	catrid 17608	. 2 ⊢ (𝜑 → (𝐻(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)((Id‘𝐶)‘𝑋)) = 𝐻)
26	23, 24, 25	3eqtr3d 2772	1 ⊢ (𝜑 → 𝐺 = 𝐻)

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ⟨cop 4585   class class class wbr 5095  ‘cfv 6486  (class class class)co 7353  Basecbs 17138  Hom chom 17190  compcco 17191  Catccat 17588  Idccid 17589  Sectcsect 17669

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-1st 7931  df-2nd 7932  df-cat 17592  df-cid 17593  df-sect 17672

This theorem is referenced by:  invfun  17689  inveq  17699