Theorem dfiso3 17734

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 2737	. . 3 ⊢ (Id‘𝐶) = (Id‘𝐶)
9		eqid 2737	. . 3 ⊢ (⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋) = (⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)
10		eqid 2737	. . 3 ⊢ (⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌) = (⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)
11	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	dfiso2 17733	. 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 2737	. . . . . 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 17715	. . . . 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 17715	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑌𝐻𝑋)) → (𝐹(𝑋𝑆𝑌)𝑔 ↔ (𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋)))
21	19, 20	anbi12d 633	. . . 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 288	. . 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 3160	. 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 279	1 ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ ∃𝑔 ∈ (𝑌𝐻𝑋)(𝑔(𝑌𝑆𝑋)𝐹 ∧ 𝐹(𝑋𝑆𝑌)𝑔)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∃wrex 3062  ⟨cop 4574   class class class wbr 5086  ‘cfv 6493  (class class class)co 7361  Basecbs 17173  Hom chom 17225  compcco 17226  Catccat 17624  Idccid 17625  Sectcsect 17705  Isociso 17707

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7364  df-oprab 7365  df-mpo 7366  df-1st 7936  df-2nd 7937  df-sect 17708  df-inv 17709  df-iso 17710

This theorem is referenced by:  thinciso  49960