Theorem dfiso3 17045

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 2823	. . 3 ⊢ (Id‘𝐶) = (Id‘𝐶)
9		eqid 2823	. . 3 ⊢ (⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋) = (⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)
10		eqid 2823	. . 3 ⊢ (⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌) = (⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)
11	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	dfiso2 17044	. 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 2823	. . . . . 6 ⊢ (comp‘𝐶) = (comp‘𝐶)
13		dfiso3.s	. . . . . 6 ⊢ 𝑆 = (Sect‘𝐶)
14	3	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑌𝐻𝑋)) → 𝐶 ∈ Cat)
15	6	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑌𝐻𝑋)) → 𝑌 ∈ 𝐵)
16	5	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑌𝐻𝑋)) → 𝑋 ∈ 𝐵)
17		simpr 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑌𝐻𝑋)) → 𝑔 ∈ (𝑌𝐻𝑋))
18	7	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑌𝐻𝑋)) → 𝐹 ∈ (𝑋𝐻𝑌))
19	1, 2, 12, 8, 13, 14, 15, 16, 17, 18	issect2 17026	. . . . 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 17026	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑌𝐻𝑋)) → (𝐹(𝑋𝑆𝑌)𝑔 ↔ (𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋)))
21	19, 20	anbi12d 632	. . . 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 463	. . . 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	syl6rbb 290	. . 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 3298	. 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 281	1 ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ ∃𝑔 ∈ (𝑌𝐻𝑋)(𝑔(𝑌𝑆𝑋)𝐹 ∧ 𝐹(𝑋𝑆𝑌)𝑔)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∃wrex 3141  ⟨cop 4575   class class class wbr 5068  ‘cfv 6357  (class class class)co 7158  Basecbs 16485  Hom chom 16578  compcco 16579  Catccat 16937  Idccid 16938  Sectcsect 17016  Isociso 17018

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163  df-1st 7691  df-2nd 7692  df-sect 17019  df-inv 17020  df-iso 17021

This theorem is referenced by: (None)