Theorem thincsect 49939

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 2737	. . . 4 ⊢ (comp‘𝐶) = (comp‘𝐶)
4		eqid 2737	. . . 4 ⊢ (Id‘𝐶) = (Id‘𝐶)
5		thincsect.s	. . . 4 ⊢ 𝑆 = (Sect‘𝐶)
6		thincsect.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ ThinCat)
7	6	thinccd 49895	. . . 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 17709	. . 3 ⊢ (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋) ∧ (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋))))
11		df-3an 1089	. . 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 771	. . . 4 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋))) → 𝐹 ∈ (𝑋𝐻𝑌))
18		simprr 773	. . . 4 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋))) → 𝐺 ∈ (𝑌𝐻𝑋))
19	1, 2, 3, 15, 14, 16, 14, 17, 18	catcocl 17640	. . 3 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋))) → (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) ∈ (𝑋𝐻𝑋))
20	13, 1, 2, 14, 4, 19	thincid 49904	. 2 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋))) → (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋))
21	12, 20	mpbiran3d 49269	1 ⊢ (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ⟨cop 4574   class class class wbr 5086  ‘cfv 6490  (class class class)co 7358  Basecbs 17168  Hom chom 17220  compcco 17221  Catccat 17619  Idccid 17620  Sectcsect 17700  ThinCatcthinc 49889

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5300  ax-pr 5368  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  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 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-cat 17623  df-cid 17624  df-sect 17703  df-thinc 49890

This theorem is referenced by:  thincsect2  49940  thinciso  49942