Theorem ustneism 24432

Description: For a point 𝐴 in 𝑋, (𝑉 “ {𝐴}) is small enough in (𝑉 ∘ ^◡𝑉). This proposition actually does not require any axiom of the definition of uniform structures. (Contributed by Thierry Arnoux, 18-Nov-2017.)

Assertion

Ref	Expression
ustneism	⊢ ((𝑉 ⊆ (𝑋 × 𝑋) ∧ 𝐴 ∈ 𝑋) → ((𝑉 “ {𝐴}) × (𝑉 “ {𝐴})) ⊆ (𝑉 ∘ ◡𝑉))

Proof of Theorem ustneism

Step	Hyp	Ref	Expression
1		snnzg 4742	. . . 4 ⊢ (𝐴 ∈ 𝑋 → {𝐴} ≠ ∅)
2	1	adantl 487	. . 3 ⊢ ((𝑉 ⊆ (𝑋 × 𝑋) ∧ 𝐴 ∈ 𝑋) → {𝐴} ≠ ∅)
3		xpco 6294	. . 3 ⊢ ({𝐴} ≠ ∅ → (({𝐴} × (𝑉 “ {𝐴})) ∘ ((𝑉 “ {𝐴}) × {𝐴})) = ((𝑉 “ {𝐴}) × (𝑉 “ {𝐴})))
4	2, 3	syl 18	. 2 ⊢ ((𝑉 ⊆ (𝑋 × 𝑋) ∧ 𝐴 ∈ 𝑋) → (({𝐴} × (𝑉 “ {𝐴})) ∘ ((𝑉 “ {𝐴}) × {𝐴})) = ((𝑉 “ {𝐴}) × (𝑉 “ {𝐴})))
5		cnvxp 6156	. . . . 5 ⊢ ◡({𝐴} × (𝑉 “ {𝐴})) = ((𝑉 “ {𝐴}) × {𝐴})
6		ressn 6290	. . . . . . 7 ⊢ (𝑉 ↾ {𝐴}) = ({𝐴} × (𝑉 “ {𝐴}))
7	6	cnveqi 5862	. . . . . 6 ⊢ ◡(𝑉 ↾ {𝐴}) = ◡({𝐴} × (𝑉 “ {𝐴}))
8		resss 6002	. . . . . . 7 ⊢ (𝑉 ↾ {𝐴}) ⊆ 𝑉
9		cnvss 5860	. . . . . . 7 ⊢ ((𝑉 ↾ {𝐴}) ⊆ 𝑉 → ◡(𝑉 ↾ {𝐴}) ⊆ ◡𝑉)
10	8, 9	ax-mp 5	. . . . . 6 ⊢ ◡(𝑉 ↾ {𝐴}) ⊆ ◡𝑉
11	7, 10	eqsstrri 3985	. . . . 5 ⊢ ◡({𝐴} × (𝑉 “ {𝐴})) ⊆ ◡𝑉
12	5, 11	eqsstrri 3985	. . . 4 ⊢ ((𝑉 “ {𝐴}) × {𝐴}) ⊆ ◡𝑉
13		coss2 5844	. . . 4 ⊢ (((𝑉 “ {𝐴}) × {𝐴}) ⊆ ◡𝑉 → (({𝐴} × (𝑉 “ {𝐴})) ∘ ((𝑉 “ {𝐴}) × {𝐴})) ⊆ (({𝐴} × (𝑉 “ {𝐴})) ∘ ◡𝑉))
14	12, 13	mp1i 14	. . 3 ⊢ ((𝑉 ⊆ (𝑋 × 𝑋) ∧ 𝐴 ∈ 𝑋) → (({𝐴} × (𝑉 “ {𝐴})) ∘ ((𝑉 “ {𝐴}) × {𝐴})) ⊆ (({𝐴} × (𝑉 “ {𝐴})) ∘ ◡𝑉))
15	6, 8	eqsstrri 3985	. . . 4 ⊢ ({𝐴} × (𝑉 “ {𝐴})) ⊆ 𝑉
16		coss1 5843	. . . 4 ⊢ (({𝐴} × (𝑉 “ {𝐴})) ⊆ 𝑉 → (({𝐴} × (𝑉 “ {𝐴})) ∘ ◡𝑉) ⊆ (𝑉 ∘ ◡𝑉))
17	15, 16	mp1i 14	. . 3 ⊢ ((𝑉 ⊆ (𝑋 × 𝑋) ∧ 𝐴 ∈ 𝑋) → (({𝐴} × (𝑉 “ {𝐴})) ∘ ◡𝑉) ⊆ (𝑉 ∘ ◡𝑉))
18	14, 17	sstrd 3948	. 2 ⊢ ((𝑉 ⊆ (𝑋 × 𝑋) ∧ 𝐴 ∈ 𝑋) → (({𝐴} × (𝑉 “ {𝐴})) ∘ ((𝑉 “ {𝐴}) × {𝐴})) ⊆ (𝑉 ∘ ◡𝑉))
19	4, 18	eqsstrrd 3973	1 ⊢ ((𝑉 ⊆ (𝑋 × 𝑋) ∧ 𝐴 ∈ 𝑋) → ((𝑉 “ {𝐴}) × (𝑉 “ {𝐴})) ⊆ (𝑉 ∘ ◡𝑉))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146   ≠ wne 2960   ⊆ wss 3906  ∅c0 4286  {csn 4591   × cxp 5661  ^◡ccnv 5662   ↾ cres 5665   “ cima 5666   ∘ ccom 5667

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-11 2195  ax-ext 2737  ax-sep 5259  ax-pr 5406

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-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676

This theorem is used by:  neipcfilu  24503