Theorem nelsubclem 50174

Description: Lemma for nelsubc 50175. (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 5261	. . . 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 7437	. . . . . 6 ⊢ (𝜑 → (𝑝𝐽𝑞) = (𝑝((𝑆 × 𝑆) × {∅})𝑞))
9	1	ovconst2 7601	. . . . . 6 ⊢ ((𝑝 ∈ 𝑆 ∧ 𝑞 ∈ 𝑆) → (𝑝((𝑆 × 𝑆) × {∅})𝑞) = ∅)
10	8, 9	sylan9eq 2816	. . . . 5 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑆 ∧ 𝑞 ∈ 𝑆)) → (𝑝𝐽𝑞) = ∅)
11		0ss 4350	. . . . 5 ⊢ ∅ ⊆ (𝑝𝐻𝑞)
12	10, 11	eqsstrdi 3975	. . . 4 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑆 ∧ 𝑞 ∈ 𝑆)) → (𝑝𝐽𝑞) ⊆ (𝑝𝐻𝑞))
13	12	ralrimivva 3206	. . 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 17867	. . . . 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 17995	. . 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 7437	. . . . . . . 8 ⊢ (𝜑 → (𝑥𝐽𝑥) = (𝑥((𝑆 × 𝑆) × {∅})𝑥))
24	1	ovconst2 7601	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆) → (𝑥((𝑆 × 𝑆) × {∅})𝑥) = ∅)
25	24	anidms 577	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑆 → (𝑥((𝑆 × 𝑆) × {∅})𝑥) = ∅)
26	23, 25	sylan9eq 2816	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑥𝐽𝑥) = ∅)
27		nel02 4285	. . . . . . 7 ⊢ ((𝑥𝐽𝑥) = ∅ → ¬ 𝐼 ∈ (𝑥𝐽𝑥))
28	26, 27	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ¬ 𝐼 ∈ (𝑥𝐽𝑥))
29	28	reximdva0 4303	. . . . 5 ⊢ ((𝜑 ∧ 𝑆 ≠ ∅) → ∃𝑥 ∈ 𝑆 ¬ 𝐼 ∈ (𝑥𝐽𝑥))
30	22, 29	mpdan 700	. . . 4 ⊢ (𝜑 → ∃𝑥 ∈ 𝑆 ¬ 𝐼 ∈ (𝑥𝐽𝑥))
31		rexnal 3115	. . . 4 ⊢ (∃𝑥 ∈ 𝑆 ¬ 𝐼 ∈ (𝑥𝐽𝑥) ↔ ¬ ∀𝑥 ∈ 𝑆 𝐼 ∈ (𝑥𝐽𝑥))
32	30, 31	sylib 221	. . 3 ⊢ (𝜑 → ¬ ∀𝑥 ∈ 𝑆 𝐼 ∈ (𝑥𝐽𝑥))
33	4	oveqd 7437	. . . . . . 7 ⊢ (𝜑 → (𝑥𝐽𝑦) = (𝑥((𝑆 × 𝑆) × {∅})𝑦))
34	1	ovconst2 7601	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥((𝑆 × 𝑆) × {∅})𝑦) = ∅)
35	33, 34	sylan9eq 2816	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐽𝑦) = ∅)
36		rzal 4450	. . . . . 6 ⊢ ((𝑥𝐽𝑦) = ∅ → ∀𝑓 ∈ (𝑥𝐽𝑦)𝜓)
37	35, 36	syl 18	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ∀𝑓 ∈ (𝑥𝐽𝑦)𝜓)
38	37	ralrimivw 3159	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)𝜓)
39	38	ralrimivva 3206	. . 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 2145   ≠ wne 2956  ∀wral 3077  ∃wrex 3087  Vcvv 3451   ⊆ wss 3899  ∅c0 4279  {csn 4584   class class class wbr 5103   × cxp 5649   Fn wfn 6533  ‘cfv 6538  (class class class)co 7420  Basecbs 17387  Hom_f chomf 17840   ⊆_cat cssc 17982

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-ixp 8926  df-homf 17844  df-ssc 17985

This theorem is used by:  nelsubc  50175  nelsubc3  50178