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Theorem thinciso 48861
Description: In a thin category, 𝐹:𝑋𝑌 is an isomorphism iff there is a morphism from 𝑌 to 𝑋. (Contributed by Zhi Wang, 25-Sep-2024.)
Hypotheses
Ref Expression
thincsect.c (𝜑𝐶 ∈ ThinCat)
thincsect.b 𝐵 = (Base‘𝐶)
thincsect.x (𝜑𝑋𝐵)
thincsect.y (𝜑𝑌𝐵)
thinciso.h 𝐻 = (Hom ‘𝐶)
thinciso.i 𝐼 = (Iso‘𝐶)
thinciso.f (𝜑𝐹 ∈ (𝑋𝐻𝑌))
Assertion
Ref Expression
thinciso (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ (𝑌𝐻𝑋) ≠ ∅))

Proof of Theorem thinciso
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 thincsect.b . . 3 𝐵 = (Base‘𝐶)
2 thinciso.h . . 3 𝐻 = (Hom ‘𝐶)
3 thinciso.i . . 3 𝐼 = (Iso‘𝐶)
4 eqid 2735 . . 3 (Sect‘𝐶) = (Sect‘𝐶)
5 thincsect.c . . . 4 (𝜑𝐶 ∈ ThinCat)
65thinccd 48825 . . 3 (𝜑𝐶 ∈ Cat)
7 thincsect.x . . 3 (𝜑𝑋𝐵)
8 thincsect.y . . 3 (𝜑𝑌𝐵)
9 thinciso.f . . 3 (𝜑𝐹 ∈ (𝑋𝐻𝑌))
101, 2, 3, 4, 6, 7, 8, 9dfiso3 17821 . 2 (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ ∃𝑔 ∈ (𝑌𝐻𝑋)(𝑔(𝑌(Sect‘𝐶)𝑋)𝐹𝐹(𝑋(Sect‘𝐶)𝑌)𝑔)))
11 simprl 771 . . . . . 6 (((𝜑 ∧ (𝑌𝐻𝑋) ≠ ∅) ∧ (𝑔 ∈ (𝑌𝐻𝑋) ∧ ⊤)) → 𝑔 ∈ (𝑌𝐻𝑋))
129ad2antrr 726 . . . . . 6 (((𝜑 ∧ (𝑌𝐻𝑋) ≠ ∅) ∧ (𝑔 ∈ (𝑌𝐻𝑋) ∧ ⊤)) → 𝐹 ∈ (𝑋𝐻𝑌))
135ad2antrr 726 . . . . . . 7 (((𝜑 ∧ (𝑌𝐻𝑋) ≠ ∅) ∧ (𝑔 ∈ (𝑌𝐻𝑋) ∧ ⊤)) → 𝐶 ∈ ThinCat)
148ad2antrr 726 . . . . . . 7 (((𝜑 ∧ (𝑌𝐻𝑋) ≠ ∅) ∧ (𝑔 ∈ (𝑌𝐻𝑋) ∧ ⊤)) → 𝑌𝐵)
157ad2antrr 726 . . . . . . 7 (((𝜑 ∧ (𝑌𝐻𝑋) ≠ ∅) ∧ (𝑔 ∈ (𝑌𝐻𝑋) ∧ ⊤)) → 𝑋𝐵)
1613, 1, 14, 15, 4, 2thincsect 48858 . . . . . 6 (((𝜑 ∧ (𝑌𝐻𝑋) ≠ ∅) ∧ (𝑔 ∈ (𝑌𝐻𝑋) ∧ ⊤)) → (𝑔(𝑌(Sect‘𝐶)𝑋)𝐹 ↔ (𝑔 ∈ (𝑌𝐻𝑋) ∧ 𝐹 ∈ (𝑋𝐻𝑌))))
1711, 12, 16mpbir2and 713 . . . . 5 (((𝜑 ∧ (𝑌𝐻𝑋) ≠ ∅) ∧ (𝑔 ∈ (𝑌𝐻𝑋) ∧ ⊤)) → 𝑔(𝑌(Sect‘𝐶)𝑋)𝐹)
1813, 1, 15, 14, 4, 2thincsect 48858 . . . . . 6 (((𝜑 ∧ (𝑌𝐻𝑋) ≠ ∅) ∧ (𝑔 ∈ (𝑌𝐻𝑋) ∧ ⊤)) → (𝐹(𝑋(Sect‘𝐶)𝑌)𝑔 ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋))))
1912, 11, 18mpbir2and 713 . . . . 5 (((𝜑 ∧ (𝑌𝐻𝑋) ≠ ∅) ∧ (𝑔 ∈ (𝑌𝐻𝑋) ∧ ⊤)) → 𝐹(𝑋(Sect‘𝐶)𝑌)𝑔)
2017, 19jca 511 . . . 4 (((𝜑 ∧ (𝑌𝐻𝑋) ≠ ∅) ∧ (𝑔 ∈ (𝑌𝐻𝑋) ∧ ⊤)) → (𝑔(𝑌(Sect‘𝐶)𝑋)𝐹𝐹(𝑋(Sect‘𝐶)𝑌)𝑔))
21 trud 1547 . . . . 5 ((𝜑𝑔 ∈ (𝑌𝐻𝑋)) → ⊤)
2221reximdva0 4361 . . . 4 ((𝜑 ∧ (𝑌𝐻𝑋) ≠ ∅) → ∃𝑔 ∈ (𝑌𝐻𝑋)⊤)
2320, 22reximddv 3169 . . 3 ((𝜑 ∧ (𝑌𝐻𝑋) ≠ ∅) → ∃𝑔 ∈ (𝑌𝐻𝑋)(𝑔(𝑌(Sect‘𝐶)𝑋)𝐹𝐹(𝑋(Sect‘𝐶)𝑌)𝑔))
24 rexn0 4517 . . . 4 (∃𝑔 ∈ (𝑌𝐻𝑋)(𝑔(𝑌(Sect‘𝐶)𝑋)𝐹𝐹(𝑋(Sect‘𝐶)𝑌)𝑔) → (𝑌𝐻𝑋) ≠ ∅)
2524adantl 481 . . 3 ((𝜑 ∧ ∃𝑔 ∈ (𝑌𝐻𝑋)(𝑔(𝑌(Sect‘𝐶)𝑋)𝐹𝐹(𝑋(Sect‘𝐶)𝑌)𝑔)) → (𝑌𝐻𝑋) ≠ ∅)
2623, 25impbida 801 . 2 (𝜑 → ((𝑌𝐻𝑋) ≠ ∅ ↔ ∃𝑔 ∈ (𝑌𝐻𝑋)(𝑔(𝑌(Sect‘𝐶)𝑋)𝐹𝐹(𝑋(Sect‘𝐶)𝑌)𝑔)))
2710, 26bitr4d 282 1 (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ (𝑌𝐻𝑋) ≠ ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wtru 1538  wcel 2106  wne 2938  wrex 3068  c0 4339   class class class wbr 5148  cfv 6563  (class class class)co 7431  Basecbs 17245  Hom chom 17309  Sectcsect 17792  Isociso 17794  ThinCatcthinc 48819
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-cat 17713  df-cid 17714  df-sect 17795  df-inv 17796  df-iso 17797  df-thinc 48820
This theorem is referenced by:  thinccic  48862
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