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Theorem thincsect2 47952
Description: In a thin category, 𝐹 is a section of 𝐺 iff 𝐺 is a section of 𝐹. (Contributed by Zhi Wang, 24-Sep-2024.)
Hypotheses
Ref Expression
thincsect.c (𝜑𝐶 ∈ ThinCat)
thincsect.b 𝐵 = (Base‘𝐶)
thincsect.x (𝜑𝑋𝐵)
thincsect.y (𝜑𝑌𝐵)
thincsect.s 𝑆 = (Sect‘𝐶)
Assertion
Ref Expression
thincsect2 (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺𝐺(𝑌𝑆𝑋)𝐹))

Proof of Theorem thincsect2
StepHypRef Expression
1 ancom 460 . . 3 ((𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋)) ↔ (𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋) ∧ 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌)))
21a1i 11 . 2 (𝜑 → ((𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋)) ↔ (𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋) ∧ 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))))
3 thincsect.c . . 3 (𝜑𝐶 ∈ ThinCat)
4 thincsect.b . . 3 𝐵 = (Base‘𝐶)
5 thincsect.x . . 3 (𝜑𝑋𝐵)
6 thincsect.y . . 3 (𝜑𝑌𝐵)
7 thincsect.s . . 3 𝑆 = (Sect‘𝐶)
8 eqid 2726 . . 3 (Hom ‘𝐶) = (Hom ‘𝐶)
93, 4, 5, 6, 7, 8thincsect 47951 . 2 (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋))))
103, 4, 6, 5, 7, 8thincsect 47951 . 2 (𝜑 → (𝐺(𝑌𝑆𝑋)𝐹 ↔ (𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋) ∧ 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))))
112, 9, 103bitr4d 311 1 (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺𝐺(𝑌𝑆𝑋)𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098   class class class wbr 5141  cfv 6537  (class class class)co 7405  Basecbs 17153  Hom chom 17217  Sectcsect 17700  ThinCatcthinc 47913
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7974  df-2nd 7975  df-cat 17621  df-cid 17622  df-sect 17703  df-thinc 47914
This theorem is referenced by:  thincinv  47953
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