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Theorem catcone0 17732
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 (𝜑𝐶 ∈ Cat)
catcocl.x (𝜑𝑋𝐵)
catcocl.y (𝜑𝑌𝐵)
catcocl.z (𝜑𝑍𝐵)
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.
StepHypRef Expression
1 catcone0.f . . . 4 (𝜑 → (𝑋𝐻𝑌) ≠ ∅)
2 catcone0.g . . . 4 (𝜑 → (𝑌𝐻𝑍) ≠ ∅)
3 n0 4359 . . . . . 6 ((𝑋𝐻𝑌) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝑋𝐻𝑌))
4 n0 4359 . . . . . 6 ((𝑌𝐻𝑍) ≠ ∅ ↔ ∃𝑔 𝑔 ∈ (𝑌𝐻𝑍))
53, 4anbi12i 628 . . . . 5 (((𝑋𝐻𝑌) ≠ ∅ ∧ (𝑌𝐻𝑍) ≠ ∅) ↔ (∃𝑓 𝑓 ∈ (𝑋𝐻𝑌) ∧ ∃𝑔 𝑔 ∈ (𝑌𝐻𝑍)))
6 exdistrv 1953 . . . . 5 (∃𝑓𝑔(𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑍)) ↔ (∃𝑓 𝑓 ∈ (𝑋𝐻𝑌) ∧ ∃𝑔 𝑔 ∈ (𝑌𝐻𝑍)))
75, 6sylbb2 238 . . . 4 (((𝑋𝐻𝑌) ≠ ∅ ∧ (𝑌𝐻𝑍) ≠ ∅) → ∃𝑓𝑔(𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑍)))
81, 2, 7syl2anc 584 . . 3 (𝜑 → ∃𝑓𝑔(𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑍)))
98ancli 548 . 2 (𝜑 → (𝜑 ∧ ∃𝑓𝑔(𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑍))))
10 19.42vv 1955 . . 3 (∃𝑓𝑔(𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑍))) ↔ (𝜑 ∧ ∃𝑓𝑔(𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑍))))
1110biimpri 228 . 2 ((𝜑 ∧ ∃𝑓𝑔(𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑍))) → ∃𝑓𝑔(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 (𝜑𝐶 ∈ Cat)
1615adantr 480 . . . 4 ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑍))) → 𝐶 ∈ Cat)
17 catcocl.x . . . . 5 (𝜑𝑋𝐵)
1817adantr 480 . . . 4 ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑍))) → 𝑋𝐵)
19 catcocl.y . . . . 5 (𝜑𝑌𝐵)
2019adantr 480 . . . 4 ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑍))) → 𝑌𝐵)
21 catcocl.z . . . . 5 (𝜑𝑍𝐵)
2221adantr 480 . . . 4 ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑍))) → 𝑍𝐵)
23 simprl 771 . . . 4 ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑍))) → 𝑓 ∈ (𝑋𝐻𝑌))
24 simprr 773 . . . 4 ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑍))) → 𝑔 ∈ (𝑌𝐻𝑍))
2512, 13, 14, 16, 18, 20, 22, 23, 24catcocl 17730 . . 3 ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑍))) → (𝑔(⟨𝑋, 𝑌· 𝑍)𝑓) ∈ (𝑋𝐻𝑍))
26252eximi 1833 . 2 (∃𝑓𝑔(𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑍))) → ∃𝑓𝑔(𝑔(⟨𝑋, 𝑌· 𝑍)𝑓) ∈ (𝑋𝐻𝑍))
27 ne0i 4347 . . 3 ((𝑔(⟨𝑋, 𝑌· 𝑍)𝑓) ∈ (𝑋𝐻𝑍) → (𝑋𝐻𝑍) ≠ ∅)
2827exlimivv 1930 . 2 (∃𝑓𝑔(𝑔(⟨𝑋, 𝑌· 𝑍)𝑓) ∈ (𝑋𝐻𝑍) → (𝑋𝐻𝑍) ≠ ∅)
299, 11, 26, 284syl 19 1 (𝜑 → (𝑋𝐻𝑍) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wex 1776  wcel 2106  wne 2938  c0 4339  cop 4637  cfv 6563  (class class class)co 7431  Basecbs 17245  Hom chom 17309  compcco 17310  Catccat 17709
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-nul 5312
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434  df-cat 17713
This theorem is referenced by:  catprs  48800
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