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Theorem thinciso 49142
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 2736 . . 3 (Sect‘𝐶) = (Sect‘𝐶)
5 thincsect.c . . . 4 (𝜑𝐶 ∈ ThinCat)
65thinccd 49097 . . 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 17818 . 2 (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ ∃𝑔 ∈ (𝑌𝐻𝑋)(𝑔(𝑌(Sect‘𝐶)𝑋)𝐹𝐹(𝑋(Sect‘𝐶)𝑌)𝑔)))
11 simprl 770 . . . . . 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 49139 . . . . . 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 49139 . . . . . 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 1549 . . . . 5 ((𝜑𝑔 ∈ (𝑌𝐻𝑋)) → ⊤)
2221reximdva0 4354 . . . 4 ((𝜑 ∧ (𝑌𝐻𝑋) ≠ ∅) → ∃𝑔 ∈ (𝑌𝐻𝑋)⊤)
2320, 22reximddv 3170 . . 3 ((𝜑 ∧ (𝑌𝐻𝑋) ≠ ∅) → ∃𝑔 ∈ (𝑌𝐻𝑋)(𝑔(𝑌(Sect‘𝐶)𝑋)𝐹𝐹(𝑋(Sect‘𝐶)𝑌)𝑔))
24 rexn0 4510 . . . 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 800 . 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 1539  wtru 1540  wcel 2107  wne 2939  wrex 3069  c0 4332   class class class wbr 5142  cfv 6560  (class class class)co 7432  Basecbs 17248  Hom chom 17309  Sectcsect 17789  Isociso 17791  ThinCatcthinc 49091
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-1st 8015  df-2nd 8016  df-cat 17712  df-cid 17713  df-sect 17792  df-inv 17793  df-iso 17794  df-thinc 49092
This theorem is referenced by:  thinccic  49143
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