MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfiso3 Structured version   Visualization version   GIF version

Theorem dfiso3 17759
Description: Alternate definition of an isomorphism of a category as a section in both directions. (Contributed by AV, 11-Apr-2020.)
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 2725 . . 3 (Id‘𝐶) = (Id‘𝐶)
9 eqid 2725 . . 3 (⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋) = (⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)
10 eqid 2725 . . 3 (⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌) = (⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)
111, 2, 3, 4, 5, 6, 7, 8, 9, 10dfiso2 17758 . 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 2725 . . . . . 6 (comp‘𝐶) = (comp‘𝐶)
13 dfiso3.s . . . . . 6 𝑆 = (Sect‘𝐶)
143adantr 479 . . . . . 6 ((𝜑𝑔 ∈ (𝑌𝐻𝑋)) → 𝐶 ∈ Cat)
156adantr 479 . . . . . 6 ((𝜑𝑔 ∈ (𝑌𝐻𝑋)) → 𝑌𝐵)
165adantr 479 . . . . . 6 ((𝜑𝑔 ∈ (𝑌𝐻𝑋)) → 𝑋𝐵)
17 simpr 483 . . . . . 6 ((𝜑𝑔 ∈ (𝑌𝐻𝑋)) → 𝑔 ∈ (𝑌𝐻𝑋))
187adantr 479 . . . . . 6 ((𝜑𝑔 ∈ (𝑌𝐻𝑋)) → 𝐹 ∈ (𝑋𝐻𝑌))
191, 2, 12, 8, 13, 14, 15, 16, 17, 18issect2 17740 . . . . 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 17740 . . . . 5 ((𝜑𝑔 ∈ (𝑌𝐻𝑋)) → (𝐹(𝑋𝑆𝑌)𝑔 ↔ (𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋)))
2119, 20anbi12d 630 . . . 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 459 . . . 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, 22bitr2di 287 . . 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 3166 . 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 278 1 (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ ∃𝑔 ∈ (𝑌𝐻𝑋)(𝑔(𝑌𝑆𝑋)𝐹𝐹(𝑋𝑆𝑌)𝑔)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  wrex 3059  cop 4636   class class class wbr 5149  cfv 6549  (class class class)co 7419  Basecbs 17183  Hom chom 17247  compcco 17248  Catccat 17647  Idccid 17648  Sectcsect 17730  Isociso 17732
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 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  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 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7994  df-2nd 7995  df-sect 17733  df-inv 17734  df-iso 17735
This theorem is referenced by:  thinciso  48252
  Copyright terms: Public domain W3C validator