Theorem nelsubclem 49056

Description: Lemma for nelsubc 49057. (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 5246	. . . 4 ⊢ ∅ ∈ V
2		fnconstg 6712	. . . 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 6575	. . 3 ⊢ (𝜑 → (𝐽 Fn (𝑆 × 𝑆) ↔ ((𝑆 × 𝑆) × {∅}) Fn (𝑆 × 𝑆)))
6	3, 5	mpbiri 258	. 2 ⊢ (𝜑 → 𝐽 Fn (𝑆 × 𝑆))
7		nelsubc.s	. . 3 ⊢ (𝜑 → 𝑆 ⊆ 𝐵)
8	4	oveqd 7366	. . . . . 6 ⊢ (𝜑 → (𝑝𝐽𝑞) = (𝑝((𝑆 × 𝑆) × {∅})𝑞))
9	1	ovconst2 7529	. . . . . 6 ⊢ ((𝑝 ∈ 𝑆 ∧ 𝑞 ∈ 𝑆) → (𝑝((𝑆 × 𝑆) × {∅})𝑞) = ∅)
10	8, 9	sylan9eq 2784	. . . . 5 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑆 ∧ 𝑞 ∈ 𝑆)) → (𝑝𝐽𝑞) = ∅)
11		0ss 4351	. . . . 5 ⊢ ∅ ⊆ (𝑝𝐻𝑞)
12	10, 11	eqsstrdi 3980	. . . 4 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑆 ∧ 𝑞 ∈ 𝑆)) → (𝑝𝐽𝑞) ⊆ (𝑝𝐻𝑞))
13	12	ralrimivva 3172	. . 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 17599	. . . . 5 ⊢ 𝐻 Fn (𝐵 × 𝐵)
17	16	a1i 11	. . . 4 ⊢ (𝜑 → 𝐻 Fn (𝐵 × 𝐵))
18	15	fvexi 6836	. . . . 5 ⊢ 𝐵 ∈ V
19	18	a1i 11	. . . 4 ⊢ (𝜑 → 𝐵 ∈ V)
20	6, 17, 19	isssc 17727	. . 3 ⊢ (𝜑 → (𝐽 ⊆cat 𝐻 ↔ (𝑆 ⊆ 𝐵 ∧ ∀𝑝 ∈ 𝑆 ∀𝑞 ∈ 𝑆 (𝑝𝐽𝑞) ⊆ (𝑝𝐻𝑞))))
21	7, 13, 20	mpbir2and 713	. 2 ⊢ (𝜑 → 𝐽 ⊆cat 𝐻)
22		nelsubc.0	. . . . 5 ⊢ (𝜑 → 𝑆 ≠ ∅)
23	4	oveqd 7366	. . . . . . . 8 ⊢ (𝜑 → (𝑥𝐽𝑥) = (𝑥((𝑆 × 𝑆) × {∅})𝑥))
24	1	ovconst2 7529	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆) → (𝑥((𝑆 × 𝑆) × {∅})𝑥) = ∅)
25	24	anidms 566	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑆 → (𝑥((𝑆 × 𝑆) × {∅})𝑥) = ∅)
26	23, 25	sylan9eq 2784	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑥𝐽𝑥) = ∅)
27		nel02 4290	. . . . . . 7 ⊢ ((𝑥𝐽𝑥) = ∅ → ¬ 𝐼 ∈ (𝑥𝐽𝑥))
28	26, 27	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ¬ 𝐼 ∈ (𝑥𝐽𝑥))
29	28	reximdva0 4306	. . . . 5 ⊢ ((𝜑 ∧ 𝑆 ≠ ∅) → ∃𝑥 ∈ 𝑆 ¬ 𝐼 ∈ (𝑥𝐽𝑥))
30	22, 29	mpdan 687	. . . 4 ⊢ (𝜑 → ∃𝑥 ∈ 𝑆 ¬ 𝐼 ∈ (𝑥𝐽𝑥))
31		rexnal 3081	. . . 4 ⊢ (∃𝑥 ∈ 𝑆 ¬ 𝐼 ∈ (𝑥𝐽𝑥) ↔ ¬ ∀𝑥 ∈ 𝑆 𝐼 ∈ (𝑥𝐽𝑥))
32	30, 31	sylib 218	. . 3 ⊢ (𝜑 → ¬ ∀𝑥 ∈ 𝑆 𝐼 ∈ (𝑥𝐽𝑥))
33	4	oveqd 7366	. . . . . . 7 ⊢ (𝜑 → (𝑥𝐽𝑦) = (𝑥((𝑆 × 𝑆) × {∅})𝑦))
34	1	ovconst2 7529	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥((𝑆 × 𝑆) × {∅})𝑦) = ∅)
35	33, 34	sylan9eq 2784	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐽𝑦) = ∅)
36		rzal 4460	. . . . . 6 ⊢ ((𝑥𝐽𝑦) = ∅ → ∀𝑓 ∈ (𝑥𝐽𝑦)𝜓)
37	35, 36	syl 17	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ∀𝑓 ∈ (𝑥𝐽𝑦)𝜓)
38	37	ralrimivw 3125	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)𝜓)
39	38	ralrimivva 3172	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)𝜓)
40	32, 39	jca 511	. 2 ⊢ (𝜑 → (¬ ∀𝑥 ∈ 𝑆 𝐼 ∈ (𝑥𝐽𝑥) ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)𝜓))
41	6, 21, 40	jca32 515	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 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  ∃wrex 3053  Vcvv 3436   ⊆ wss 3903  ∅c0 4284  {csn 4577   class class class wbr 5092   × cxp 5617   Fn wfn 6477  ‘cfv 6482  (class class class)co 7349  Basecbs 17120  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:  nelsubc  49057  nelsubc3  49060