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Theorem dfiso3 16480
Description: Alternate definition of an isomorphism of a category as a section in both directions. (Contributed by AV, 11-Apr-2017.)
Hypotheses
Ref Expression
dfiso3.b 𝐵 = (Base‘𝐶)
dfiso3.h 𝐻 = (Hom ‘𝐶)
dfiso3.i 𝐼 = (Iso‘𝐶)
dfiso3.s 𝑆 = (Sect‘𝐶)
dfiso3.c (𝜑𝐶 ∈ Cat)
dfiso3.x (𝜑𝑋𝐵)
dfiso3.y (𝜑𝑌𝐵)
dfiso3.f (𝜑𝐹 ∈ (𝑋𝐻𝑌))
Assertion
Ref Expression
dfiso3 (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ ∃𝑔 ∈ (𝑌𝐻𝑋)(𝑔(𝑌𝑆𝑋)𝐹𝐹(𝑋𝑆𝑌)𝑔)))
Distinct variable groups:   𝐶,𝑔   𝑔,𝐹   𝑔,𝐻   𝑔,𝐼   𝑔,𝑋   𝑔,𝑌   𝜑,𝑔
Allowed substitution hints:   𝐵(𝑔)   𝑆(𝑔)

Proof of Theorem dfiso3
StepHypRef Expression
1 dfiso3.b . . 3 𝐵 = (Base‘𝐶)
2 dfiso3.h . . 3 𝐻 = (Hom ‘𝐶)
3 dfiso3.c . . 3 (𝜑𝐶 ∈ Cat)
4 dfiso3.i . . 3 𝐼 = (Iso‘𝐶)
5 dfiso3.x . . 3 (𝜑𝑋𝐵)
6 dfiso3.y . . 3 (𝜑𝑌𝐵)
7 dfiso3.f . . 3 (𝜑𝐹 ∈ (𝑋𝐻𝑌))
8 eqid 2651 . . 3 (Id‘𝐶) = (Id‘𝐶)
9 eqid 2651 . . 3 (⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋) = (⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)
10 eqid 2651 . . 3 (⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌) = (⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)
111, 2, 3, 4, 5, 6, 7, 8, 9, 10dfiso2 16479 . 2 (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ ∃𝑔 ∈ (𝑌𝐻𝑋)((𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋) ∧ (𝐹(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = ((Id‘𝐶)‘𝑌))))
12 eqid 2651 . . . . . 6 (comp‘𝐶) = (comp‘𝐶)
13 dfiso3.s . . . . . 6 𝑆 = (Sect‘𝐶)
143adantr 480 . . . . . 6 ((𝜑𝑔 ∈ (𝑌𝐻𝑋)) → 𝐶 ∈ Cat)
156adantr 480 . . . . . 6 ((𝜑𝑔 ∈ (𝑌𝐻𝑋)) → 𝑌𝐵)
165adantr 480 . . . . . 6 ((𝜑𝑔 ∈ (𝑌𝐻𝑋)) → 𝑋𝐵)
17 simpr 476 . . . . . 6 ((𝜑𝑔 ∈ (𝑌𝐻𝑋)) → 𝑔 ∈ (𝑌𝐻𝑋))
187adantr 480 . . . . . 6 ((𝜑𝑔 ∈ (𝑌𝐻𝑋)) → 𝐹 ∈ (𝑋𝐻𝑌))
191, 2, 12, 8, 13, 14, 15, 16, 17, 18issect2 16461 . . . . 5 ((𝜑𝑔 ∈ (𝑌𝐻𝑋)) → (𝑔(𝑌𝑆𝑋)𝐹 ↔ (𝐹(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = ((Id‘𝐶)‘𝑌)))
201, 2, 12, 8, 13, 14, 16, 15, 18, 17issect2 16461 . . . . 5 ((𝜑𝑔 ∈ (𝑌𝐻𝑋)) → (𝐹(𝑋𝑆𝑌)𝑔 ↔ (𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋)))
2119, 20anbi12d 747 . . . 4 ((𝜑𝑔 ∈ (𝑌𝐻𝑋)) → ((𝑔(𝑌𝑆𝑋)𝐹𝐹(𝑋𝑆𝑌)𝑔) ↔ ((𝐹(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = ((Id‘𝐶)‘𝑌) ∧ (𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋))))
22 ancom 465 . . . 4 (((𝐹(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = ((Id‘𝐶)‘𝑌) ∧ (𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋)) ↔ ((𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋) ∧ (𝐹(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = ((Id‘𝐶)‘𝑌)))
2321, 22syl6rbb 277 . . 3 ((𝜑𝑔 ∈ (𝑌𝐻𝑋)) → (((𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋) ∧ (𝐹(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = ((Id‘𝐶)‘𝑌)) ↔ (𝑔(𝑌𝑆𝑋)𝐹𝐹(𝑋𝑆𝑌)𝑔)))
2423rexbidva 3078 . 2 (𝜑 → (∃𝑔 ∈ (𝑌𝐻𝑋)((𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋) ∧ (𝐹(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = ((Id‘𝐶)‘𝑌)) ↔ ∃𝑔 ∈ (𝑌𝐻𝑋)(𝑔(𝑌𝑆𝑋)𝐹𝐹(𝑋𝑆𝑌)𝑔)))
2511, 24bitrd 268 1 (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ ∃𝑔 ∈ (𝑌𝐻𝑋)(𝑔(𝑌𝑆𝑋)𝐹𝐹(𝑋𝑆𝑌)𝑔)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wrex 2942  cop 4216   class class class wbr 4685  cfv 5926  (class class class)co 6690  Basecbs 15904  Hom chom 15999  compcco 16000  Catccat 16372  Idccid 16373  Sectcsect 16451  Isociso 16453
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-sect 16454  df-inv 16455  df-iso 16456
This theorem is referenced by: (None)
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