Theorem nelsubclem 49904

Description: Lemma for nelsubc 49905. (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)

Assertion

Ref	Expression
nelsubclem	⊢ (𝜑 → (𝐽 Fn (𝑆 × 𝑆) ∧ (𝐽 ⊆cat 𝐻 ∧ (¬ ∀𝑥 ∈ 𝑆 𝐼 ∈ (𝑥𝐽𝑥) ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)𝜓))))

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

Allowed substitution hints:   𝜑(𝑓)   𝜓(𝑥, 𝑦, 𝑧, 𝑓)   𝐵(𝑥, 𝑦, 𝑧, 𝑓)   𝐶(𝑥, 𝑦, 𝑧, 𝑓)   𝑆(𝑓)   𝐻(𝑥, 𝑦, 𝑧, 𝑓)   𝐼(𝑥, 𝑦, 𝑧, 𝑓)   𝐽(𝑥, 𝑦, 𝑧)

Proof of Theorem nelsubclem

Dummy variables 𝑝 𝑞 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0ex 5272	. . . 4 ⊢ ∅ ∈ V
2		fnconstg 6770	. . . 4 ⊢ (∅ ∈ V → ((𝑆 × 𝑆) × {∅}) Fn (𝑆 × 𝑆))
3	1, 2	ax-mp 5	. . 3 ⊢ ((𝑆 × 𝑆) × {∅}) Fn (𝑆 × 𝑆)
4		nelsubc.j	. . . 4 ⊢ (𝜑 → 𝐽 = ((𝑆 × 𝑆) × {∅}))
5	4	fneq1d 6632	. . 3 ⊢ (𝜑 → (𝐽 Fn (𝑆 × 𝑆) ↔ ((𝑆 × 𝑆) × {∅}) Fn (𝑆 × 𝑆)))
6	3, 5	mpbiri 261	. 2 ⊢ (𝜑 → 𝐽 Fn (𝑆 × 𝑆))
7		nelsubc.s	. . 3 ⊢ (𝜑 → 𝑆 ⊆ 𝐵)
8	4	oveqd 7436	. . . . . 6 ⊢ (𝜑 → (𝑝𝐽𝑞) = (𝑝((𝑆 × 𝑆) × {∅})𝑞))
9	1	ovconst2 7600	. . . . . 6 ⊢ ((𝑝 ∈ 𝑆 ∧ 𝑞 ∈ 𝑆) → (𝑝((𝑆 × 𝑆) × {∅})𝑞) = ∅)
10	8, 9	sylan9eq 2820	. . . . 5 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑆 ∧ 𝑞 ∈ 𝑆)) → (𝑝𝐽𝑞) = ∅)
11		0ss 4357	. . . . 5 ⊢ ∅ ⊆ (𝑝𝐻𝑞)
12	10, 11	eqsstrdi 3982	. . . 4 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑆 ∧ 𝑞 ∈ 𝑆)) → (𝑝𝐽𝑞) ⊆ (𝑝𝐻𝑞))
13	12	ralrimivva 3210	. . 3 ⊢ (𝜑 → ∀𝑝 ∈ 𝑆 ∀𝑞 ∈ 𝑆 (𝑝𝐽𝑞) ⊆ (𝑝𝐻𝑞))
14		nelsubc.h	. . . . . 6 ⊢ 𝐻 = (Homf ‘𝐶)
15		nelsubc.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐶)
16	14, 15	homffn 17773	. . . . 5 ⊢ 𝐻 Fn (𝐵 × 𝐵)
17	16	a1i 11	. . . 4 ⊢ (𝜑 → 𝐻 Fn (𝐵 × 𝐵))
18	15	fvexi 6899	. . . . 5 ⊢ 𝐵 ∈ V
19	18	a1i 11	. . . 4 ⊢ (𝜑 → 𝐵 ∈ V)
20	6, 17, 19	isssc 17901	. . 3 ⊢ (𝜑 → (𝐽 ⊆cat 𝐻 ↔ (𝑆 ⊆ 𝐵 ∧ ∀𝑝 ∈ 𝑆 ∀𝑞 ∈ 𝑆 (𝑝𝐽𝑞) ⊆ (𝑝𝐻𝑞))))
21	7, 13, 20	mpbir2and 726	. 2 ⊢ (𝜑 → 𝐽 ⊆cat 𝐻)
22		nelsubc.0	. . . . 5 ⊢ (𝜑 → 𝑆 ≠ ∅)
23	4	oveqd 7436	. . . . . . . 8 ⊢ (𝜑 → (𝑥𝐽𝑥) = (𝑥((𝑆 × 𝑆) × {∅})𝑥))
24	1	ovconst2 7600	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆) → (𝑥((𝑆 × 𝑆) × {∅})𝑥) = ∅)
25	24	anidms 577	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑆 → (𝑥((𝑆 × 𝑆) × {∅})𝑥) = ∅)
26	23, 25	sylan9eq 2820	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑥𝐽𝑥) = ∅)
27		nel02 4292	. . . . . . 7 ⊢ ((𝑥𝐽𝑥) = ∅ → ¬ 𝐼 ∈ (𝑥𝐽𝑥))
28	26, 27	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ¬ 𝐼 ∈ (𝑥𝐽𝑥))
29	28	reximdva0 4310	. . . . 5 ⊢ ((𝜑 ∧ 𝑆 ≠ ∅) → ∃𝑥 ∈ 𝑆 ¬ 𝐼 ∈ (𝑥𝐽𝑥))
30	22, 29	mpdan 700	. . . 4 ⊢ (𝜑 → ∃𝑥 ∈ 𝑆 ¬ 𝐼 ∈ (𝑥𝐽𝑥))
31		rexnal 3119	. . . 4 ⊢ (∃𝑥 ∈ 𝑆 ¬ 𝐼 ∈ (𝑥𝐽𝑥) ↔ ¬ ∀𝑥 ∈ 𝑆 𝐼 ∈ (𝑥𝐽𝑥))
32	30, 31	sylib 221	. . 3 ⊢ (𝜑 → ¬ ∀𝑥 ∈ 𝑆 𝐼 ∈ (𝑥𝐽𝑥))
33	4	oveqd 7436	. . . . . . 7 ⊢ (𝜑 → (𝑥𝐽𝑦) = (𝑥((𝑆 × 𝑆) × {∅})𝑦))
34	1	ovconst2 7600	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥((𝑆 × 𝑆) × {∅})𝑦) = ∅)
35	33, 34	sylan9eq 2820	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐽𝑦) = ∅)
36		rzal 4457	. . . . . 6 ⊢ ((𝑥𝐽𝑦) = ∅ → ∀𝑓 ∈ (𝑥𝐽𝑦)𝜓)
37	35, 36	syl 18	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ∀𝑓 ∈ (𝑥𝐽𝑦)𝜓)
38	37	ralrimivw 3163	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)𝜓)
39	38	ralrimivva 3210	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)𝜓)
40	32, 39	jca 521	. 2 ⊢ (𝜑 → (¬ ∀𝑥 ∈ 𝑆 𝐼 ∈ (𝑥𝐽𝑥) ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)𝜓))
41	6, 21, 40	jca32 525	1 ⊢ (𝜑 → (𝐽 Fn (𝑆 × 𝑆) ∧ (𝐽 ⊆cat 𝐻 ∧ (¬ ∀𝑥 ∈ 𝑆 𝐼 ∈ (𝑥𝐽𝑥) ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)𝜓))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146   ≠ wne 2960  ∀wral 3081  ∃wrex 3091  Vcvv 3457   ⊆ wss 3906  ∅c0 4286  {csn 4591   class class class wbr 5111   × cxp 5661   Fn wfn 6535  ‘cfv 6540  (class class class)co 7419  Basecbs 17293  Hom_f chomf 17746   ⊆_cat cssc 17888

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7992  df-2nd 7993  df-ixp 8902  df-homf 17750  df-ssc 17891

This theorem is used by:  nelsubc  49905  nelsubc3  49908