Theorem thincsect 49722

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 2736	. . . 4 ⊢ (comp‘𝐶) = (comp‘𝐶)
4		eqid 2736	. . . 4 ⊢ (Id‘𝐶) = (Id‘𝐶)
5		thincsect.s	. . . 4 ⊢ 𝑆 = (Sect‘𝐶)
6		thincsect.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ ThinCat)
7	6	thinccd 49678	. . . 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 17677	. . 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 17608	. . 3 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋))) → (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) ∈ (𝑋𝐻𝑋))
20	13, 1, 2, 14, 4, 19	thincid 49687	. 2 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋))) → (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋))
21	12, 20	mpbiran3d 49052	1 ⊢ (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ⟨cop 4586   class class class wbr 5098  ‘cfv 6492  (class class class)co 7358  Basecbs 17136  Hom chom 17188  compcco 17189  Catccat 17587  Idccid 17588  Sectcsect 17668  ThinCatcthinc 49672

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 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-cat 17591  df-cid 17592  df-sect 17671  df-thinc 49673

This theorem is referenced by:  thincsect2  49723  thinciso  49725