Theorem dfiso3 17402

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	⊢ (𝜑 → 𝐶 ∈ Cat)
dfiso3.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
dfiso3.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
dfiso3.f	⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))

Assertion

Ref	Expression
dfiso3	⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ ∃𝑔 ∈ (𝑌𝐻𝑋)(𝑔(𝑌𝑆𝑋)𝐹 ∧ 𝐹(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

Step	Hyp	Ref	Expression
1		dfiso3.b	. . 3 ⊢ 𝐵 = (Base‘𝐶)
2		dfiso3.h	. . 3 ⊢ 𝐻 = (Hom ‘𝐶)
3		dfiso3.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
4		dfiso3.i	. . 3 ⊢ 𝐼 = (Iso‘𝐶)
5		dfiso3.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
6		dfiso3.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
7		dfiso3.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
8		eqid 2738	. . 3 ⊢ (Id‘𝐶) = (Id‘𝐶)
9		eqid 2738	. . 3 ⊢ (⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋) = (⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)
10		eqid 2738	. . 3 ⊢ (⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌) = (⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)
11	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	dfiso2 17401	. 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 2738	. . . . . 6 ⊢ (comp‘𝐶) = (comp‘𝐶)
13		dfiso3.s	. . . . . 6 ⊢ 𝑆 = (Sect‘𝐶)
14	3	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑌𝐻𝑋)) → 𝐶 ∈ Cat)
15	6	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑌𝐻𝑋)) → 𝑌 ∈ 𝐵)
16	5	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑌𝐻𝑋)) → 𝑋 ∈ 𝐵)
17		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑌𝐻𝑋)) → 𝑔 ∈ (𝑌𝐻𝑋))
18	7	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑌𝐻𝑋)) → 𝐹 ∈ (𝑋𝐻𝑌))
19	1, 2, 12, 8, 13, 14, 15, 16, 17, 18	issect2 17383	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑌𝐻𝑋)) → (𝑔(𝑌𝑆𝑋)𝐹 ↔ (𝐹(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = ((Id‘𝐶)‘𝑌)))
20	1, 2, 12, 8, 13, 14, 16, 15, 18, 17	issect2 17383	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑌𝐻𝑋)) → (𝐹(𝑋𝑆𝑌)𝑔 ↔ (𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋)))
21	19, 20	anbi12d 630	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑌𝐻𝑋)) → ((𝑔(𝑌𝑆𝑋)𝐹 ∧ 𝐹(𝑋𝑆𝑌)𝑔) ↔ ((𝐹(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = ((Id‘𝐶)‘𝑌) ∧ (𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋))))
22		ancom 460	. . . 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)))
23	21, 22	bitr2di 287	. . 3 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑌𝐻𝑋)) → (((𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋) ∧ (𝐹(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = ((Id‘𝐶)‘𝑌)) ↔ (𝑔(𝑌𝑆𝑋)𝐹 ∧ 𝐹(𝑋𝑆𝑌)𝑔)))
24	23	rexbidva 3224	. 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)𝐹 ∧ 𝐹(𝑋𝑆𝑌)𝑔)))
25	11, 24	bitrd 278	1 ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ ∃𝑔 ∈ (𝑌𝐻𝑋)(𝑔(𝑌𝑆𝑋)𝐹 ∧ 𝐹(𝑋𝑆𝑌)𝑔)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∃wrex 3064  ⟨cop 4564   class class class wbr 5070  ‘cfv 6418  (class class class)co 7255  Basecbs 16840  Hom chom 16899  compcco 16900  Catccat 17290  Idccid 17291  Sectcsect 17373  Isociso 17375

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-1st 7804  df-2nd 7805  df-sect 17376  df-inv 17377  df-iso 17378

This theorem is referenced by:  thinciso  46229