Theorem nelsubc 49431

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 49430	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 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052   ⊆ wss 3903  ∅c0 4287  {csn 4582  ⟨cop 4588   class class class wbr 5100   × cxp 5630   Fn wfn 6495  ‘cfv 6500  (class class class)co 7368  Basecbs 17148  compcco 17201  Idccid 17600  Hom_f chomf 17601   ⊆_cat cssc 17743

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944  df-ixp 8848  df-homf 17605  df-ssc 17746

This theorem is referenced by:  nelsubc2  49432