Theorem nelsubclem 49565

Description: Lemma for nelsubc 49566. (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 5230	. . . 4 ⊢ ∅ ∈ V
2		fnconstg 6716	. . . 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 6579	. . 3 ⊢ (𝜑 → (𝐽 Fn (𝑆 × 𝑆) ↔ ((𝑆 × 𝑆) × {∅}) Fn (𝑆 × 𝑆)))
6	3, 5	mpbiri 259	. 2 ⊢ (𝜑 → 𝐽 Fn (𝑆 × 𝑆))
7		nelsubc.s	. . 3 ⊢ (𝜑 → 𝑆 ⊆ 𝐵)
8	4	oveqd 7374	. . . . . 6 ⊢ (𝜑 → (𝑝𝐽𝑞) = (𝑝((𝑆 × 𝑆) × {∅})𝑞))
9	1	ovconst2 7537	. . . . . 6 ⊢ ((𝑝 ∈ 𝑆 ∧ 𝑞 ∈ 𝑆) → (𝑝((𝑆 × 𝑆) × {∅})𝑞) = ∅)
10	8, 9	sylan9eq 2794	. . . . 5 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑆 ∧ 𝑞 ∈ 𝑆)) → (𝑝𝐽𝑞) = ∅)
11		0ss 4329	. . . . 5 ⊢ ∅ ⊆ (𝑝𝐻𝑞)
12	10, 11	eqsstrdi 3959	. . . 4 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑆 ∧ 𝑞 ∈ 𝑆)) → (𝑝𝐽𝑞) ⊆ (𝑝𝐻𝑞))
13	12	ralrimivva 3182	. . 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 17651	. . . . 5 ⊢ 𝐻 Fn (𝐵 × 𝐵)
17	16	a1i 11	. . . 4 ⊢ (𝜑 → 𝐻 Fn (𝐵 × 𝐵))
18	15	fvexi 6842	. . . . 5 ⊢ 𝐵 ∈ V
19	18	a1i 11	. . . 4 ⊢ (𝜑 → 𝐵 ∈ V)
20	6, 17, 19	isssc 17779	. . 3 ⊢ (𝜑 → (𝐽 ⊆cat 𝐻 ↔ (𝑆 ⊆ 𝐵 ∧ ∀𝑝 ∈ 𝑆 ∀𝑞 ∈ 𝑆 (𝑝𝐽𝑞) ⊆ (𝑝𝐻𝑞))))
21	7, 13, 20	mpbir2and 719	. 2 ⊢ (𝜑 → 𝐽 ⊆cat 𝐻)
22		nelsubc.0	. . . . 5 ⊢ (𝜑 → 𝑆 ≠ ∅)
23	4	oveqd 7374	. . . . . . . 8 ⊢ (𝜑 → (𝑥𝐽𝑥) = (𝑥((𝑆 × 𝑆) × {∅})𝑥))
24	1	ovconst2 7537	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆) → (𝑥((𝑆 × 𝑆) × {∅})𝑥) = ∅)
25	24	anidms 571	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑆 → (𝑥((𝑆 × 𝑆) × {∅})𝑥) = ∅)
26	23, 25	sylan9eq 2794	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑥𝐽𝑥) = ∅)
27		nel02 4268	. . . . . . 7 ⊢ ((𝑥𝐽𝑥) = ∅ → ¬ 𝐼 ∈ (𝑥𝐽𝑥))
28	26, 27	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ¬ 𝐼 ∈ (𝑥𝐽𝑥))
29	28	reximdva0 4284	. . . . 5 ⊢ ((𝜑 ∧ 𝑆 ≠ ∅) → ∃𝑥 ∈ 𝑆 ¬ 𝐼 ∈ (𝑥𝐽𝑥))
30	22, 29	mpdan 693	. . . 4 ⊢ (𝜑 → ∃𝑥 ∈ 𝑆 ¬ 𝐼 ∈ (𝑥𝐽𝑥))
31		rexnal 3091	. . . 4 ⊢ (∃𝑥 ∈ 𝑆 ¬ 𝐼 ∈ (𝑥𝐽𝑥) ↔ ¬ ∀𝑥 ∈ 𝑆 𝐼 ∈ (𝑥𝐽𝑥))
32	30, 31	sylib 219	. . 3 ⊢ (𝜑 → ¬ ∀𝑥 ∈ 𝑆 𝐼 ∈ (𝑥𝐽𝑥))
33	4	oveqd 7374	. . . . . . 7 ⊢ (𝜑 → (𝑥𝐽𝑦) = (𝑥((𝑆 × 𝑆) × {∅})𝑦))
34	1	ovconst2 7537	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥((𝑆 × 𝑆) × {∅})𝑦) = ∅)
35	33, 34	sylan9eq 2794	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐽𝑦) = ∅)
36		rzal 4423	. . . . . 6 ⊢ ((𝑥𝐽𝑦) = ∅ → ∀𝑓 ∈ (𝑥𝐽𝑦)𝜓)
37	35, 36	syl 17	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ∀𝑓 ∈ (𝑥𝐽𝑦)𝜓)
38	37	ralrimivw 3135	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)𝜓)
39	38	ralrimivva 3182	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)𝜓)
40	32, 39	jca 516	. 2 ⊢ (𝜑 → (¬ ∀𝑥 ∈ 𝑆 𝐼 ∈ (𝑥𝐽𝑥) ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)𝜓))
41	6, 21, 40	jca32 520	1 ⊢ (𝜑 → (𝐽 Fn (𝑆 × 𝑆) ∧ (𝐽 ⊆cat 𝐻 ∧ (¬ ∀𝑥 ∈ 𝑆 𝐼 ∈ (𝑥𝐽𝑥) ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)𝜓))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119   ≠ wne 2934  ∀wral 3053  ∃wrex 3063  Vcvv 3431   ⊆ wss 3883  ∅c0 4262  {csn 4556   class class class wbr 5073   × cxp 5617   Fn wfn 6481  ‘cfv 6486  (class class class)co 7357  Basecbs 17171  Hom_f chomf 17624   ⊆_cat cssc 17766

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-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5200  ax-sep 5219  ax-nul 5229  ax-pow 5295  ax-pr 5363  ax-un 7679

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-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4263  df-if 4456  df-pw 4532  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4840  df-iun 4924  df-br 5074  df-opab 5136  df-mpt 5155  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 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7932  df-2nd 7933  df-ixp 8837  df-homf 17628  df-ssc 17769

This theorem is referenced by:  nelsubc  49566  nelsubc3  49569