Theorem thincsect 49654

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 2734	. . . 4 ⊢ (comp‘𝐶) = (comp‘𝐶)
4		eqid 2734	. . . 4 ⊢ (Id‘𝐶) = (Id‘𝐶)
5		thincsect.s	. . . 4 ⊢ 𝑆 = (Sect‘𝐶)
6		thincsect.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ ThinCat)
7	6	thinccd 49610	. . . 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 17675	. . 3 ⊢ (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋) ∧ (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋))))
11		df-3an 1088	. . 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 287	. 2 ⊢ (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ ((𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋)) ∧ (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋))))
13	6	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋))) → 𝐶 ∈ ThinCat)
14	8	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋))) → 𝑋 ∈ 𝐵)
15	7	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋))) → 𝐶 ∈ Cat)
16	9	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋))) → 𝑌 ∈ 𝐵)
17		simprl 770	. . . 4 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋))) → 𝐹 ∈ (𝑋𝐻𝑌))
18		simprr 772	. . . 4 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋))) → 𝐺 ∈ (𝑌𝐻𝑋))
19	1, 2, 3, 15, 14, 16, 14, 17, 18	catcocl 17606	. . 3 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋))) → (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) ∈ (𝑋𝐻𝑋))
20	13, 1, 2, 14, 4, 19	thincid 49619	. 2 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋))) → (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋))
21	12, 20	mpbiran3d 48984	1 ⊢ (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ⟨cop 4584   class class class wbr 5096  ‘cfv 6490  (class class class)co 7356  Basecbs 17134  Hom chom 17186  compcco 17187  Catccat 17585  Idccid 17586  Sectcsect 17666  ThinCatcthinc 49604

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-cat 17589  df-cid 17590  df-sect 17669  df-thinc 49605

This theorem is referenced by:  thincsect2  49655  thinciso  49657