Theorem thinciso 45957

Description: In a thin category, 𝐹:𝑋⟶𝑌 is an isomorphism iff there is a morphism from 𝑌 to 𝑋. (Contributed by Zhi Wang, 25-Sep-2024.)

Hypotheses

Ref	Expression
thincsect.c	⊢ (𝜑 → 𝐶 ∈ ThinCat)
thincsect.b	⊢ 𝐵 = (Base‘𝐶)
thincsect.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
thincsect.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
thinciso.h	⊢ 𝐻 = (Hom ‘𝐶)
thinciso.i	⊢ 𝐼 = (Iso‘𝐶)
thinciso.f	⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))

Assertion

Ref	Expression
thinciso	⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ (𝑌𝐻𝑋) ≠ ∅))

Proof of Theorem thinciso

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		thincsect.b	. . 3 ⊢ 𝐵 = (Base‘𝐶)
2		thinciso.h	. . 3 ⊢ 𝐻 = (Hom ‘𝐶)
3		thinciso.i	. . 3 ⊢ 𝐼 = (Iso‘𝐶)
4		eqid 2736	. . 3 ⊢ (Sect‘𝐶) = (Sect‘𝐶)
5		thincsect.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ ThinCat)
6	5	thinccd 45922	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
7		thincsect.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
8		thincsect.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
9		thinciso.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
10	1, 2, 3, 4, 6, 7, 8, 9	dfiso3 17232	. 2 ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ ∃𝑔 ∈ (𝑌𝐻𝑋)(𝑔(𝑌(Sect‘𝐶)𝑋)𝐹 ∧ 𝐹(𝑋(Sect‘𝐶)𝑌)𝑔)))
11		simprl 771	. . . . . 6 ⊢ (((𝜑 ∧ (𝑌𝐻𝑋) ≠ ∅) ∧ (𝑔 ∈ (𝑌𝐻𝑋) ∧ ⊤)) → 𝑔 ∈ (𝑌𝐻𝑋))
12	9	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ (𝑌𝐻𝑋) ≠ ∅) ∧ (𝑔 ∈ (𝑌𝐻𝑋) ∧ ⊤)) → 𝐹 ∈ (𝑋𝐻𝑌))
13	5	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑌𝐻𝑋) ≠ ∅) ∧ (𝑔 ∈ (𝑌𝐻𝑋) ∧ ⊤)) → 𝐶 ∈ ThinCat)
14	8	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑌𝐻𝑋) ≠ ∅) ∧ (𝑔 ∈ (𝑌𝐻𝑋) ∧ ⊤)) → 𝑌 ∈ 𝐵)
15	7	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑌𝐻𝑋) ≠ ∅) ∧ (𝑔 ∈ (𝑌𝐻𝑋) ∧ ⊤)) → 𝑋 ∈ 𝐵)
16	13, 1, 14, 15, 4, 2	thincsect 45954	. . . . . 6 ⊢ (((𝜑 ∧ (𝑌𝐻𝑋) ≠ ∅) ∧ (𝑔 ∈ (𝑌𝐻𝑋) ∧ ⊤)) → (𝑔(𝑌(Sect‘𝐶)𝑋)𝐹 ↔ (𝑔 ∈ (𝑌𝐻𝑋) ∧ 𝐹 ∈ (𝑋𝐻𝑌))))
17	11, 12, 16	mpbir2and 713	. . . . 5 ⊢ (((𝜑 ∧ (𝑌𝐻𝑋) ≠ ∅) ∧ (𝑔 ∈ (𝑌𝐻𝑋) ∧ ⊤)) → 𝑔(𝑌(Sect‘𝐶)𝑋)𝐹)
18	13, 1, 15, 14, 4, 2	thincsect 45954	. . . . . 6 ⊢ (((𝜑 ∧ (𝑌𝐻𝑋) ≠ ∅) ∧ (𝑔 ∈ (𝑌𝐻𝑋) ∧ ⊤)) → (𝐹(𝑋(Sect‘𝐶)𝑌)𝑔 ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋))))
19	12, 11, 18	mpbir2and 713	. . . . 5 ⊢ (((𝜑 ∧ (𝑌𝐻𝑋) ≠ ∅) ∧ (𝑔 ∈ (𝑌𝐻𝑋) ∧ ⊤)) → 𝐹(𝑋(Sect‘𝐶)𝑌)𝑔)
20	17, 19	jca 515	. . . 4 ⊢ (((𝜑 ∧ (𝑌𝐻𝑋) ≠ ∅) ∧ (𝑔 ∈ (𝑌𝐻𝑋) ∧ ⊤)) → (𝑔(𝑌(Sect‘𝐶)𝑋)𝐹 ∧ 𝐹(𝑋(Sect‘𝐶)𝑌)𝑔))
21		trud 1553	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑌𝐻𝑋)) → ⊤)
22	21	reximdva0 4252	. . . 4 ⊢ ((𝜑 ∧ (𝑌𝐻𝑋) ≠ ∅) → ∃𝑔 ∈ (𝑌𝐻𝑋)⊤)
23	20, 22	reximddv 3184	. . 3 ⊢ ((𝜑 ∧ (𝑌𝐻𝑋) ≠ ∅) → ∃𝑔 ∈ (𝑌𝐻𝑋)(𝑔(𝑌(Sect‘𝐶)𝑋)𝐹 ∧ 𝐹(𝑋(Sect‘𝐶)𝑌)𝑔))
24		rexn0 4408	. . . 4 ⊢ (∃𝑔 ∈ (𝑌𝐻𝑋)(𝑔(𝑌(Sect‘𝐶)𝑋)𝐹 ∧ 𝐹(𝑋(Sect‘𝐶)𝑌)𝑔) → (𝑌𝐻𝑋) ≠ ∅)
25	24	adantl 485	. . 3 ⊢ ((𝜑 ∧ ∃𝑔 ∈ (𝑌𝐻𝑋)(𝑔(𝑌(Sect‘𝐶)𝑋)𝐹 ∧ 𝐹(𝑋(Sect‘𝐶)𝑌)𝑔)) → (𝑌𝐻𝑋) ≠ ∅)
26	23, 25	impbida 801	. 2 ⊢ (𝜑 → ((𝑌𝐻𝑋) ≠ ∅ ↔ ∃𝑔 ∈ (𝑌𝐻𝑋)(𝑔(𝑌(Sect‘𝐶)𝑋)𝐹 ∧ 𝐹(𝑋(Sect‘𝐶)𝑌)𝑔)))
27	10, 26	bitr4d 285	1 ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ (𝑌𝐻𝑋) ≠ ∅))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1543  ⊤wtru 1544   ∈ wcel 2112   ≠ wne 2932  ∃wrex 3052  ∅c0 4223   class class class wbr 5039  ‘cfv 6358  (class class class)co 7191  Basecbs 16666  Hom chom 16760  Sectcsect 17203  Isociso 17205  ThinCatcthinc 45916

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pow 5243  ax-pr 5307  ax-un 7501

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-ral 3056  df-rex 3057  df-reu 3058  df-rmo 3059  df-rab 3060  df-v 3400  df-sbc 3684  df-csb 3799  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-pw 4501  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-iun 4892  df-br 5040  df-opab 5102  df-mpt 5121  df-id 5440  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-iota 6316  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-riota 7148  df-ov 7194  df-oprab 7195  df-mpo 7196  df-1st 7739  df-2nd 7740  df-cat 17125  df-cid 17126  df-sect 17206  df-inv 17207  df-iso 17208  df-thinc 45917

This theorem is referenced by:  thinccic  45958