Theorem sectcan 16462

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 2651	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
3		eqid 2651	. . . 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 2651	. . . . . . 7 ⊢ (Id‘𝐶) = (Id‘𝐶)
9		sectcan.s	. . . . . . 7 ⊢ 𝑆 = (Sect‘𝐶)
10	1, 2, 3, 8, 9, 4, 5, 6	issect 16460	. . . . . 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 222	. . . . 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 1093	. . . 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 16460	. . . . . 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 222	. . . . 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 1093	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝑌(Hom ‘𝐶)𝑋))
17	15	simp2d 1094	. . . 4 ⊢ (𝜑 → 𝐻 ∈ (𝑋(Hom ‘𝐶)𝑌))
18	1, 2, 3, 4, 5, 6, 5, 12, 16, 6, 17	catass 16394	. . 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 1095	. . . 4 ⊢ (𝜑 → (𝐻(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐹) = ((Id‘𝐶)‘𝑌))
20	19	oveq1d 6705	. . 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 1095	. . . 4 ⊢ (𝜑 → (𝐹(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐺) = ((Id‘𝐶)‘𝑋))
22	21	oveq2d 6706	. . 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 2693	. 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 16391	. 2 ⊢ (𝜑 → (((Id‘𝐶)‘𝑌)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)𝐺) = 𝐺)
25	1, 2, 8, 4, 5, 3, 6, 17	catrid 16392	. 2 ⊢ (𝜑 → (𝐻(⟨𝑋, 𝑋⟩(comp‘𝐶)𝑌)((Id‘𝐶)‘𝑋)) = 𝐻)
26	23, 24, 25	3eqtr3d 2693	1 ⊢ (𝜑 → 𝐺 = 𝐻)

Syntax hints:   → wi 4   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030  ⟨cop 4216   class class class wbr 4685  ‘cfv 5926  (class class class)co 6690  Basecbs 15904  Hom chom 15999  compcco 16000  Catccat 16372  Idccid 16373  Sectcsect 16451

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-cat 16376  df-cid 16377  df-sect 16454

This theorem is referenced by:  invfun  16471  inveq  16481