Theorem thinciso 49903

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 2737	. . 3 ⊢ (Sect‘𝐶) = (Sect‘𝐶)
5		thincsect.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ ThinCat)
6	5	thinccd 49856	. . 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 17698	. 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 727	. . . . . 6 ⊢ (((𝜑 ∧ (𝑌𝐻𝑋) ≠ ∅) ∧ (𝑔 ∈ (𝑌𝐻𝑋) ∧ ⊤)) → 𝐹 ∈ (𝑋𝐻𝑌))
13	5	ad2antrr 727	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑌𝐻𝑋) ≠ ∅) ∧ (𝑔 ∈ (𝑌𝐻𝑋) ∧ ⊤)) → 𝐶 ∈ ThinCat)
14	8	ad2antrr 727	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑌𝐻𝑋) ≠ ∅) ∧ (𝑔 ∈ (𝑌𝐻𝑋) ∧ ⊤)) → 𝑌 ∈ 𝐵)
15	7	ad2antrr 727	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑌𝐻𝑋) ≠ ∅) ∧ (𝑔 ∈ (𝑌𝐻𝑋) ∧ ⊤)) → 𝑋 ∈ 𝐵)
16	13, 1, 14, 15, 4, 2	thincsect 49900	. . . . . 6 ⊢ (((𝜑 ∧ (𝑌𝐻𝑋) ≠ ∅) ∧ (𝑔 ∈ (𝑌𝐻𝑋) ∧ ⊤)) → (𝑔(𝑌(Sect‘𝐶)𝑋)𝐹 ↔ (𝑔 ∈ (𝑌𝐻𝑋) ∧ 𝐹 ∈ (𝑋𝐻𝑌))))
17	11, 12, 16	mpbir2and 714	. . . . 5 ⊢ (((𝜑 ∧ (𝑌𝐻𝑋) ≠ ∅) ∧ (𝑔 ∈ (𝑌𝐻𝑋) ∧ ⊤)) → 𝑔(𝑌(Sect‘𝐶)𝑋)𝐹)
18	13, 1, 15, 14, 4, 2	thincsect 49900	. . . . . 6 ⊢ (((𝜑 ∧ (𝑌𝐻𝑋) ≠ ∅) ∧ (𝑔 ∈ (𝑌𝐻𝑋) ∧ ⊤)) → (𝐹(𝑋(Sect‘𝐶)𝑌)𝑔 ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋))))
19	12, 11, 18	mpbir2and 714	. . . . 5 ⊢ (((𝜑 ∧ (𝑌𝐻𝑋) ≠ ∅) ∧ (𝑔 ∈ (𝑌𝐻𝑋) ∧ ⊤)) → 𝐹(𝑋(Sect‘𝐶)𝑌)𝑔)
20	17, 19	jca 511	. . . 4 ⊢ (((𝜑 ∧ (𝑌𝐻𝑋) ≠ ∅) ∧ (𝑔 ∈ (𝑌𝐻𝑋) ∧ ⊤)) → (𝑔(𝑌(Sect‘𝐶)𝑋)𝐹 ∧ 𝐹(𝑋(Sect‘𝐶)𝑌)𝑔))
21		trud 1552	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑌𝐻𝑋)) → ⊤)
22	21	reximdva0 4296	. . . 4 ⊢ ((𝜑 ∧ (𝑌𝐻𝑋) ≠ ∅) → ∃𝑔 ∈ (𝑌𝐻𝑋)⊤)
23	20, 22	reximddv 3154	. . 3 ⊢ ((𝜑 ∧ (𝑌𝐻𝑋) ≠ ∅) → ∃𝑔 ∈ (𝑌𝐻𝑋)(𝑔(𝑌(Sect‘𝐶)𝑋)𝐹 ∧ 𝐹(𝑋(Sect‘𝐶)𝑌)𝑔))
24		rexn0 4437	. . . 4 ⊢ (∃𝑔 ∈ (𝑌𝐻𝑋)(𝑔(𝑌(Sect‘𝐶)𝑋)𝐹 ∧ 𝐹(𝑋(Sect‘𝐶)𝑌)𝑔) → (𝑌𝐻𝑋) ≠ ∅)
25	24	adantl 481	. . 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 282	1 ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ (𝑌𝐻𝑋) ≠ ∅))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542  ⊤wtru 1543   ∈ wcel 2114   ≠ wne 2933  ∃wrex 3062  ∅c0 4274   class class class wbr 5086  ‘cfv 6490  (class class class)co 7358  Basecbs 17137  Hom chom 17189  Sectcsect 17669  Isociso 17671  ThinCatcthinc 49850

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 5212  ax-sep 5231  ax-nul 5241  ax-pow 5300  ax-pr 5368  ax-un 7680

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-rmo 3343  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 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-cat 17592  df-cid 17593  df-sect 17672  df-inv 17673  df-iso 17674  df-thinc 49851

This theorem is referenced by:  thinccic  49904