Theorem nelsubc 49063

Description: An empty "hom-set" for non-empty base satisfies all conditions for a subcategory but the existence of identity morphisms. (Contributed by Zhi Wang, 5-Nov-2025.)

Hypotheses

Ref	Expression
nelsubc.b	⊢ 𝐵 = (Base‘𝐶)
nelsubc.s	⊢ (𝜑 → 𝑆 ⊆ 𝐵)
nelsubc.0	⊢ (𝜑 → 𝑆 ≠ ∅)
nelsubc.j	⊢ (𝜑 → 𝐽 = ((𝑆 × 𝑆) × {∅}))
nelsubc.h	⊢ 𝐻 = (Homf ‘𝐶)
nelsubc.i	⊢ 1 = (Id‘𝐶)
nelsubc.o	⊢ · = (comp‘𝐶)

Assertion

Ref	Expression
nelsubc	⊢ (𝜑 → (𝐽 Fn (𝑆 × 𝑆) ∧ (𝐽 ⊆cat 𝐻 ∧ (¬ ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐽𝑧)))))

Distinct variable groups:   𝑓,𝐽   𝑥,𝑆,𝑦,𝑧   𝑥,𝑓,𝑦   𝜑,𝑥,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑓,𝑔)   𝐵(𝑥,𝑦,𝑧,𝑓,𝑔)   𝐶(𝑥,𝑦,𝑧,𝑓,𝑔)   𝑆(𝑓,𝑔)   · (𝑥,𝑦,𝑧,𝑓,𝑔)   1 (𝑥,𝑦,𝑧,𝑓,𝑔)   𝐻(𝑥,𝑦,𝑧,𝑓,𝑔)   𝐽(𝑥,𝑦,𝑧,𝑔)

Proof of Theorem nelsubc

Step	Hyp	Ref	Expression
1		nelsubc.b	. 2 ⊢ 𝐵 = (Base‘𝐶)
2		nelsubc.s	. 2 ⊢ (𝜑 → 𝑆 ⊆ 𝐵)
3		nelsubc.0	. 2 ⊢ (𝜑 → 𝑆 ≠ ∅)
4		nelsubc.j	. 2 ⊢ (𝜑 → 𝐽 = ((𝑆 × 𝑆) × {∅}))
5		nelsubc.h	. 2 ⊢ 𝐻 = (Homf ‘𝐶)
6	1, 2, 3, 4, 5	nelsubclem 49062	1 ⊢ (𝜑 → (𝐽 Fn (𝑆 × 𝑆) ∧ (𝐽 ⊆cat 𝐻 ∧ (¬ ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐽𝑧)))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044   ⊆ wss 3903  ∅c0 4284  {csn 4577  ⟨cop 4583   class class class wbr 5092   × cxp 5617   Fn wfn 6477  ‘cfv 6482  (class class class)co 7349  Basecbs 17120  compcco 17173  Idccid 17571  Hom_f chomf 17572   ⊆_cat cssc 17714

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-1st 7924  df-2nd 7925  df-ixp 8825  df-homf 17576  df-ssc 17717

This theorem is referenced by:  nelsubc2  49064