Theorem thincsect 50805

Description: In a thin category, one morphism is a section of another iff they are pointing towards each other. (Contributed by Zhi Wang, 24-Sep-2024.)

Hypotheses

Ref	Expression
thincsect.c	⊢ (𝜑 → 𝐶 ∈ ThinCat)
thincsect.b	⊢ 𝐵 = (Base‘𝐶)
thincsect.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
thincsect.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
thincsect.s	⊢ 𝑆 = (Sect‘𝐶)
thincsect.h	⊢ 𝐻 = (Hom ‘𝐶)

Assertion

Ref	Expression
thincsect	⊢ (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋))))

Proof of Theorem thincsect

Step	Hyp	Ref	Expression
1		thincsect.b	. . . 4 ⊢ 𝐵 = (Base‘𝐶)
2		thincsect.h	. . . 4 ⊢ 𝐻 = (Hom ‘𝐶)
3		eqid 2763	. . . 4 ⊢ (comp‘𝐶) = (comp‘𝐶)
4		eqid 2763	. . . 4 ⊢ (Id‘𝐶) = (Id‘𝐶)
5		thincsect.s	. . . 4 ⊢ 𝑆 = (Sect‘𝐶)
6		thincsect.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ ThinCat)
7	6	thinccatd 50761	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
8		thincsect.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
9		thincsect.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
10	1, 2, 3, 4, 5, 7, 8, 9	issect 17954	. . 3 ⊢ (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋) ∧ (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋))))
11		df-3an 1105	. . 3 ⊢ ((𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋) ∧ (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋)) ↔ ((𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋)) ∧ (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋)))
12	10, 11	bitrdi 290	. 2 ⊢ (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ ((𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋)) ∧ (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋))))
13	6	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋))) → 𝐶 ∈ ThinCat)
14	8	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋))) → 𝑋 ∈ 𝐵)
15	7	adantr 486	. . . 4 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋))) → 𝐶 ∈ Cat)
16	9	adantr 486	. . . 4 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋))) → 𝑌 ∈ 𝐵)
17		simprl 783	. . . 4 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋))) → 𝐹 ∈ (𝑋𝐻𝑌))
18		simprr 785	. . . 4 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋))) → 𝐺 ∈ (𝑌𝐻𝑋))
19	1, 2, 3, 15, 14, 16, 14, 17, 18	catcocl 17885	. . 3 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋))) → (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) ∈ (𝑋𝐻𝑋))
20	13, 1, 2, 14, 4, 19	thincid 50770	. 2 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋))) → (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋))
21	12, 20	mpbiran3d 50110	1 ⊢ (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ⟨cop 4838   class class class wbr 5106  ‘cfv 6555  (class class class)co 7440  Basecbs 17413  Hom chom 17465  compcco 17466  Catccat 17864  Idccid 17865  Sectcsect 17945  ThinCatcthinc 50755

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-riota 7397  df-ov 7443  df-oprab 7444  df-mpo 7445  df-1st 8022  df-2nd 8023  df-cat 17868  df-cid 17869  df-sect 17948  df-thinc 50756

This theorem is used by:  thincsect2  50806  thinciso  50808