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Theorem dfiso3 16785
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 2825 . . 3 (Id‘𝐶) = (Id‘𝐶)
9 eqid 2825 . . 3 (⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋) = (⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)
10 eqid 2825 . . 3 (⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌) = (⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)
111, 2, 3, 4, 5, 6, 7, 8, 9, 10dfiso2 16784 . 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 2825 . . . . . 6 (comp‘𝐶) = (comp‘𝐶)
13 dfiso3.s . . . . . 6 𝑆 = (Sect‘𝐶)
143adantr 474 . . . . . 6 ((𝜑𝑔 ∈ (𝑌𝐻𝑋)) → 𝐶 ∈ Cat)
156adantr 474 . . . . . 6 ((𝜑𝑔 ∈ (𝑌𝐻𝑋)) → 𝑌𝐵)
165adantr 474 . . . . . 6 ((𝜑𝑔 ∈ (𝑌𝐻𝑋)) → 𝑋𝐵)
17 simpr 479 . . . . . 6 ((𝜑𝑔 ∈ (𝑌𝐻𝑋)) → 𝑔 ∈ (𝑌𝐻𝑋))
187adantr 474 . . . . . 6 ((𝜑𝑔 ∈ (𝑌𝐻𝑋)) → 𝐹 ∈ (𝑋𝐻𝑌))
191, 2, 12, 8, 13, 14, 15, 16, 17, 18issect2 16766 . . . . 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 16766 . . . . 5 ((𝜑𝑔 ∈ (𝑌𝐻𝑋)) → (𝐹(𝑋𝑆𝑌)𝑔 ↔ (𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋)))
2119, 20anbi12d 626 . . . 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 454 . . . 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 280 . . 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 3259 . 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 271 1 (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ ∃𝑔 ∈ (𝑌𝐻𝑋)(𝑔(𝑌𝑆𝑋)𝐹𝐹(𝑋𝑆𝑌)𝑔)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1658  wcel 2166  wrex 3118  cop 4403   class class class wbr 4873  cfv 6123  (class class class)co 6905  Basecbs 16222  Hom chom 16316  compcco 16317  Catccat 16677  Idccid 16678  Sectcsect 16756  Isociso 16758
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-1st 7428  df-2nd 7429  df-sect 16759  df-inv 16760  df-iso 16761
This theorem is referenced by: (None)
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