Theorem catcone0 17644

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 4281	. . . . . 6 ⊢ ((𝑋𝐻𝑌) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝑋𝐻𝑌))
4		n0 4281	. . . . . 6 ⊢ ((𝑌𝐻𝑍) ≠ ∅ ↔ ∃𝑔 𝑔 ∈ (𝑌𝐻𝑍))
5	3, 4	anbi12i 634	. . . . 5 ⊢ (((𝑋𝐻𝑌) ≠ ∅ ∧ (𝑌𝐻𝑍) ≠ ∅) ↔ (∃𝑓 𝑓 ∈ (𝑋𝐻𝑌) ∧ ∃𝑔 𝑔 ∈ (𝑌𝐻𝑍)))
6		exdistrv 1962	. . . . 5 ⊢ (∃𝑓∃𝑔(𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑍)) ↔ (∃𝑓 𝑓 ∈ (𝑋𝐻𝑌) ∧ ∃𝑔 𝑔 ∈ (𝑌𝐻𝑍)))
7	5, 6	sylbb2 239	. . . 4 ⊢ (((𝑋𝐻𝑌) ≠ ∅ ∧ (𝑌𝐻𝑍) ≠ ∅) → ∃𝑓∃𝑔(𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑍)))
8	1, 2, 7	syl2anc 590	. . 3 ⊢ (𝜑 → ∃𝑓∃𝑔(𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑍)))
9	8	ancli 553	. 2 ⊢ (𝜑 → (𝜑 ∧ ∃𝑓∃𝑔(𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑍))))
10		19.42vv 1964	. . 3 ⊢ (∃𝑓∃𝑔(𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑍))) ↔ (𝜑 ∧ ∃𝑓∃𝑔(𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑍))))
11	10	biimpri 229	. 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 481	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑍))) → 𝐶 ∈ Cat)
17		catcocl.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
18	17	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑍))) → 𝑋 ∈ 𝐵)
19		catcocl.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
20	19	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑍))) → 𝑌 ∈ 𝐵)
21		catcocl.z	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
22	21	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑍))) → 𝑍 ∈ 𝐵)
23		simprl 776	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑍))) → 𝑓 ∈ (𝑋𝐻𝑌))
24		simprr 778	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑍))) → 𝑔 ∈ (𝑌𝐻𝑍))
25	12, 13, 14, 16, 18, 20, 22, 23, 24	catcocl 17642	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑍))) → (𝑔(⟨𝑋, 𝑌⟩ · 𝑍)𝑓) ∈ (𝑋𝐻𝑍))
26	25	2eximi 1843	. 2 ⊢ (∃𝑓∃𝑔(𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑍))) → ∃𝑓∃𝑔(𝑔(⟨𝑋, 𝑌⟩ · 𝑍)𝑓) ∈ (𝑋𝐻𝑍))
27		ne0i 4269	. . 3 ⊢ ((𝑔(⟨𝑋, 𝑌⟩ · 𝑍)𝑓) ∈ (𝑋𝐻𝑍) → (𝑋𝐻𝑍) ≠ ∅)
28	27	exlimivv 1939	. 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 396   = wceq 1547  ∃wex 1786   ∈ wcel 2119   ≠ wne 2934  ∅c0 4261  ⟨cop 4561  ‘cfv 6485  (class class class)co 7356  Basecbs 17170  Hom chom 17222  compcco 17223  Catccat 17621

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-nul 5228

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-dif 3886  df-un 3888  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-iota 6441  df-fv 6493  df-ov 7359  df-cat 17625

This theorem is referenced by:  catprs  49501