Theorem catcone0 17144

Description: Composition of non-empty hom-sets is non-empty. (Contributed by Zhi Wang, 18-Sep-2024.)

Hypotheses

Ref	Expression
catcocl.b	⊢ 𝐵 = (Base‘𝐶)
catcocl.h	⊢ 𝐻 = (Hom ‘𝐶)
catcocl.o	⊢ · = (comp‘𝐶)
catcocl.c	⊢ (𝜑 → 𝐶 ∈ Cat)
catcocl.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
catcocl.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
catcocl.z	⊢ (𝜑 → 𝑍 ∈ 𝐵)
catcone0.f	⊢ (𝜑 → (𝑋𝐻𝑌) ≠ ∅)
catcone0.g	⊢ (𝜑 → (𝑌𝐻𝑍) ≠ ∅)

Assertion

Ref	Expression
catcone0	⊢ (𝜑 → (𝑋𝐻𝑍) ≠ ∅)

Proof of Theorem catcone0

Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		catcone0.f	. . . . 5 ⊢ (𝜑 → (𝑋𝐻𝑌) ≠ ∅)
2		catcone0.g	. . . . 5 ⊢ (𝜑 → (𝑌𝐻𝑍) ≠ ∅)
3		n0 4247	. . . . . . 7 ⊢ ((𝑋𝐻𝑌) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝑋𝐻𝑌))
4		n0 4247	. . . . . . 7 ⊢ ((𝑌𝐻𝑍) ≠ ∅ ↔ ∃𝑔 𝑔 ∈ (𝑌𝐻𝑍))
5	3, 4	anbi12i 630	. . . . . 6 ⊢ (((𝑋𝐻𝑌) ≠ ∅ ∧ (𝑌𝐻𝑍) ≠ ∅) ↔ (∃𝑓 𝑓 ∈ (𝑋𝐻𝑌) ∧ ∃𝑔 𝑔 ∈ (𝑌𝐻𝑍)))
6		exdistrv 1964	. . . . . 6 ⊢ (∃𝑓∃𝑔(𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑍)) ↔ (∃𝑓 𝑓 ∈ (𝑋𝐻𝑌) ∧ ∃𝑔 𝑔 ∈ (𝑌𝐻𝑍)))
7	5, 6	sylbb2 241	. . . . 5 ⊢ (((𝑋𝐻𝑌) ≠ ∅ ∧ (𝑌𝐻𝑍) ≠ ∅) → ∃𝑓∃𝑔(𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑍)))
8	1, 2, 7	syl2anc 587	. . . 4 ⊢ (𝜑 → ∃𝑓∃𝑔(𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑍)))
9	8	ancli 552	. . 3 ⊢ (𝜑 → (𝜑 ∧ ∃𝑓∃𝑔(𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑍))))
10		19.42vv 1966	. . . 4 ⊢ (∃𝑓∃𝑔(𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑍))) ↔ (𝜑 ∧ ∃𝑓∃𝑔(𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑍))))
11	10	biimpri 231	. . 3 ⊢ ((𝜑 ∧ ∃𝑓∃𝑔(𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑍))) → ∃𝑓∃𝑔(𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑍))))
12		catcocl.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐶)
13		catcocl.h	. . . . 5 ⊢ 𝐻 = (Hom ‘𝐶)
14		catcocl.o	. . . . 5 ⊢ · = (comp‘𝐶)
15		catcocl.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ Cat)
16	15	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑍))) → 𝐶 ∈ Cat)
17		catcocl.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
18	17	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑍))) → 𝑋 ∈ 𝐵)
19		catcocl.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
20	19	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑍))) → 𝑌 ∈ 𝐵)
21		catcocl.z	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
22	21	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑍))) → 𝑍 ∈ 𝐵)
23		simprl 771	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑍))) → 𝑓 ∈ (𝑋𝐻𝑌))
24		simprr 773	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑍))) → 𝑔 ∈ (𝑌𝐻𝑍))
25	12, 13, 14, 16, 18, 20, 22, 23, 24	catcocl 17142	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑍))) → (𝑔(⟨𝑋, 𝑌⟩ · 𝑍)𝑓) ∈ (𝑋𝐻𝑍))
26	25	2eximi 1843	. . 3 ⊢ (∃𝑓∃𝑔(𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑍))) → ∃𝑓∃𝑔(𝑔(⟨𝑋, 𝑌⟩ · 𝑍)𝑓) ∈ (𝑋𝐻𝑍))
27	9, 11, 26	3syl 18	. 2 ⊢ (𝜑 → ∃𝑓∃𝑔(𝑔(⟨𝑋, 𝑌⟩ · 𝑍)𝑓) ∈ (𝑋𝐻𝑍))
28		ne0i 4235	. . 3 ⊢ ((𝑔(⟨𝑋, 𝑌⟩ · 𝑍)𝑓) ∈ (𝑋𝐻𝑍) → (𝑋𝐻𝑍) ≠ ∅)
29	28	exlimivv 1940	. 2 ⊢ (∃𝑓∃𝑔(𝑔(⟨𝑋, 𝑌⟩ · 𝑍)𝑓) ∈ (𝑋𝐻𝑍) → (𝑋𝐻𝑍) ≠ ∅)
30	27, 29	syl 17	1 ⊢ (𝜑 → (𝑋𝐻𝑍) ≠ ∅)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1543  ∃wex 1787   ∈ wcel 2112   ≠ wne 2932  ∅c0 4223  ⟨cop 4533  ‘cfv 6358  (class class class)co 7191  Basecbs 16666  Hom chom 16760  compcco 16761  Catccat 17121

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-nul 5184

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-ne 2933  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3400  df-sbc 3684  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-br 5040  df-iota 6316  df-fv 6366  df-ov 7194  df-cat 17125

This theorem is referenced by:  catprs  45908