Theorem catcone0 17313

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 4277	. . . . . . 7 ⊢ ((𝑋𝐻𝑌) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝑋𝐻𝑌))
4		n0 4277	. . . . . . 7 ⊢ ((𝑌𝐻𝑍) ≠ ∅ ↔ ∃𝑔 𝑔 ∈ (𝑌𝐻𝑍))
5	3, 4	anbi12i 626	. . . . . 6 ⊢ (((𝑋𝐻𝑌) ≠ ∅ ∧ (𝑌𝐻𝑍) ≠ ∅) ↔ (∃𝑓 𝑓 ∈ (𝑋𝐻𝑌) ∧ ∃𝑔 𝑔 ∈ (𝑌𝐻𝑍)))
6		exdistrv 1960	. . . . . 6 ⊢ (∃𝑓∃𝑔(𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑍)) ↔ (∃𝑓 𝑓 ∈ (𝑋𝐻𝑌) ∧ ∃𝑔 𝑔 ∈ (𝑌𝐻𝑍)))
7	5, 6	sylbb2 237	. . . . 5 ⊢ (((𝑋𝐻𝑌) ≠ ∅ ∧ (𝑌𝐻𝑍) ≠ ∅) → ∃𝑓∃𝑔(𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑍)))
8	1, 2, 7	syl2anc 583	. . . 4 ⊢ (𝜑 → ∃𝑓∃𝑔(𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑍)))
9	8	ancli 548	. . 3 ⊢ (𝜑 → (𝜑 ∧ ∃𝑓∃𝑔(𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑍))))
10		19.42vv 1962	. . . 4 ⊢ (∃𝑓∃𝑔(𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑍))) ↔ (𝜑 ∧ ∃𝑓∃𝑔(𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑍))))
11	10	biimpri 227	. . 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 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑍))) → 𝐶 ∈ Cat)
17		catcocl.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
18	17	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑍))) → 𝑋 ∈ 𝐵)
19		catcocl.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
20	19	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑍))) → 𝑌 ∈ 𝐵)
21		catcocl.z	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
22	21	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑍))) → 𝑍 ∈ 𝐵)
23		simprl 767	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑍))) → 𝑓 ∈ (𝑋𝐻𝑌))
24		simprr 769	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑍))) → 𝑔 ∈ (𝑌𝐻𝑍))
25	12, 13, 14, 16, 18, 20, 22, 23, 24	catcocl 17311	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑍))) → (𝑔(⟨𝑋, 𝑌⟩ · 𝑍)𝑓) ∈ (𝑋𝐻𝑍))
26	25	2eximi 1839	. . 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 4265	. . 3 ⊢ ((𝑔(⟨𝑋, 𝑌⟩ · 𝑍)𝑓) ∈ (𝑋𝐻𝑍) → (𝑋𝐻𝑍) ≠ ∅)
29	28	exlimivv 1936	. 2 ⊢ (∃𝑓∃𝑔(𝑔(⟨𝑋, 𝑌⟩ · 𝑍)𝑓) ∈ (𝑋𝐻𝑍) → (𝑋𝐻𝑍) ≠ ∅)
30	27, 29	syl 17	1 ⊢ (𝜑 → (𝑋𝐻𝑍) ≠ ∅)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539  ∃wex 1783   ∈ wcel 2108   ≠ wne 2942  ∅c0 4253  ⟨cop 4564  ‘cfv 6418  (class class class)co 7255  Basecbs 16840  Hom chom 16899  compcco 16900  Catccat 17290

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-nul 5225

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-iota 6376  df-fv 6426  df-ov 7258  df-cat 17294

This theorem is referenced by:  catprs  46180