Theorem catcone0 17730

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	. . . 4 ⊢ (𝜑 → (𝑋𝐻𝑌) ≠ ∅)
2		catcone0.g	. . . 4 ⊢ (𝜑 → (𝑌𝐻𝑍) ≠ ∅)
3		n0 4353	. . . . . 6 ⊢ ((𝑋𝐻𝑌) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝑋𝐻𝑌))
4		n0 4353	. . . . . 6 ⊢ ((𝑌𝐻𝑍) ≠ ∅ ↔ ∃𝑔 𝑔 ∈ (𝑌𝐻𝑍))
5	3, 4	anbi12i 628	. . . . 5 ⊢ (((𝑋𝐻𝑌) ≠ ∅ ∧ (𝑌𝐻𝑍) ≠ ∅) ↔ (∃𝑓 𝑓 ∈ (𝑋𝐻𝑌) ∧ ∃𝑔 𝑔 ∈ (𝑌𝐻𝑍)))
6		exdistrv 1955	. . . . 5 ⊢ (∃𝑓∃𝑔(𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑍)) ↔ (∃𝑓 𝑓 ∈ (𝑋𝐻𝑌) ∧ ∃𝑔 𝑔 ∈ (𝑌𝐻𝑍)))
7	5, 6	sylbb2 238	. . . 4 ⊢ (((𝑋𝐻𝑌) ≠ ∅ ∧ (𝑌𝐻𝑍) ≠ ∅) → ∃𝑓∃𝑔(𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑍)))
8	1, 2, 7	syl2anc 584	. . 3 ⊢ (𝜑 → ∃𝑓∃𝑔(𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑍)))
9	8	ancli 548	. 2 ⊢ (𝜑 → (𝜑 ∧ ∃𝑓∃𝑔(𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑍))))
10		19.42vv 1957	. . 3 ⊢ (∃𝑓∃𝑔(𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑍))) ↔ (𝜑 ∧ ∃𝑓∃𝑔(𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑍))))
11	10	biimpri 228	. 2 ⊢ ((𝜑 ∧ ∃𝑓∃𝑔(𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑍))) → ∃𝑓∃𝑔(𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑍))))
12		catcocl.b	. . . 4 ⊢ 𝐵 = (Base‘𝐶)
13		catcocl.h	. . . 4 ⊢ 𝐻 = (Hom ‘𝐶)
14		catcocl.o	. . . 4 ⊢ · = (comp‘𝐶)
15		catcocl.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ Cat)
16	15	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑍))) → 𝐶 ∈ Cat)
17		catcocl.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
18	17	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑍))) → 𝑋 ∈ 𝐵)
19		catcocl.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
20	19	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑍))) → 𝑌 ∈ 𝐵)
21		catcocl.z	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
22	21	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑍))) → 𝑍 ∈ 𝐵)
23		simprl 771	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑍))) → 𝑓 ∈ (𝑋𝐻𝑌))
24		simprr 773	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑍))) → 𝑔 ∈ (𝑌𝐻𝑍))
25	12, 13, 14, 16, 18, 20, 22, 23, 24	catcocl 17728	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑍))) → (𝑔(⟨𝑋, 𝑌⟩ · 𝑍)𝑓) ∈ (𝑋𝐻𝑍))
26	25	2eximi 1836	. 2 ⊢ (∃𝑓∃𝑔(𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑍))) → ∃𝑓∃𝑔(𝑔(⟨𝑋, 𝑌⟩ · 𝑍)𝑓) ∈ (𝑋𝐻𝑍))
27		ne0i 4341	. . 3 ⊢ ((𝑔(⟨𝑋, 𝑌⟩ · 𝑍)𝑓) ∈ (𝑋𝐻𝑍) → (𝑋𝐻𝑍) ≠ ∅)
28	27	exlimivv 1932	. 2 ⊢ (∃𝑓∃𝑔(𝑔(⟨𝑋, 𝑌⟩ · 𝑍)𝑓) ∈ (𝑋𝐻𝑍) → (𝑋𝐻𝑍) ≠ ∅)
29	9, 11, 26, 28	4syl 19	1 ⊢ (𝜑 → (𝑋𝐻𝑍) ≠ ∅)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2108   ≠ wne 2940  ∅c0 4333  ⟨cop 4632  ‘cfv 6561  (class class class)co 7431  Basecbs 17247  Hom chom 17308  compcco 17309  Catccat 17707

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-nul 5306

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-ov 7434  df-cat 17711

This theorem is referenced by:  catprs  48900