Theorem ustneism 23719

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 4777	. . . 4 ⊢ (𝐴 ∈ 𝑋 → {𝐴} ≠ ∅)
2	1	adantl 482	. . 3 ⊢ ((𝑉 ⊆ (𝑋 × 𝑋) ∧ 𝐴 ∈ 𝑋) → {𝐴} ≠ ∅)
3		xpco 6285	. . 3 ⊢ ({𝐴} ≠ ∅ → (({𝐴} × (𝑉 “ {𝐴})) ∘ ((𝑉 “ {𝐴}) × {𝐴})) = ((𝑉 “ {𝐴}) × (𝑉 “ {𝐴})))
4	2, 3	syl 17	. 2 ⊢ ((𝑉 ⊆ (𝑋 × 𝑋) ∧ 𝐴 ∈ 𝑋) → (({𝐴} × (𝑉 “ {𝐴})) ∘ ((𝑉 “ {𝐴}) × {𝐴})) = ((𝑉 “ {𝐴}) × (𝑉 “ {𝐴})))
5		cnvxp 6153	. . . . 5 ⊢ ◡({𝐴} × (𝑉 “ {𝐴})) = ((𝑉 “ {𝐴}) × {𝐴})
6		ressn 6281	. . . . . . 7 ⊢ (𝑉 ↾ {𝐴}) = ({𝐴} × (𝑉 “ {𝐴}))
7	6	cnveqi 5872	. . . . . 6 ⊢ ◡(𝑉 ↾ {𝐴}) = ◡({𝐴} × (𝑉 “ {𝐴}))
8		resss 6004	. . . . . . 7 ⊢ (𝑉 ↾ {𝐴}) ⊆ 𝑉
9		cnvss 5870	. . . . . . 7 ⊢ ((𝑉 ↾ {𝐴}) ⊆ 𝑉 → ◡(𝑉 ↾ {𝐴}) ⊆ ◡𝑉)
10	8, 9	ax-mp 5	. . . . . 6 ⊢ ◡(𝑉 ↾ {𝐴}) ⊆ ◡𝑉
11	7, 10	eqsstrri 4016	. . . . 5 ⊢ ◡({𝐴} × (𝑉 “ {𝐴})) ⊆ ◡𝑉
12	5, 11	eqsstrri 4016	. . . 4 ⊢ ((𝑉 “ {𝐴}) × {𝐴}) ⊆ ◡𝑉
13		coss2 5854	. . . 4 ⊢ (((𝑉 “ {𝐴}) × {𝐴}) ⊆ ◡𝑉 → (({𝐴} × (𝑉 “ {𝐴})) ∘ ((𝑉 “ {𝐴}) × {𝐴})) ⊆ (({𝐴} × (𝑉 “ {𝐴})) ∘ ◡𝑉))
14	12, 13	mp1i 13	. . 3 ⊢ ((𝑉 ⊆ (𝑋 × 𝑋) ∧ 𝐴 ∈ 𝑋) → (({𝐴} × (𝑉 “ {𝐴})) ∘ ((𝑉 “ {𝐴}) × {𝐴})) ⊆ (({𝐴} × (𝑉 “ {𝐴})) ∘ ◡𝑉))
15	6, 8	eqsstrri 4016	. . . 4 ⊢ ({𝐴} × (𝑉 “ {𝐴})) ⊆ 𝑉
16		coss1 5853	. . . 4 ⊢ (({𝐴} × (𝑉 “ {𝐴})) ⊆ 𝑉 → (({𝐴} × (𝑉 “ {𝐴})) ∘ ◡𝑉) ⊆ (𝑉 ∘ ◡𝑉))
17	15, 16	mp1i 13	. . 3 ⊢ ((𝑉 ⊆ (𝑋 × 𝑋) ∧ 𝐴 ∈ 𝑋) → (({𝐴} × (𝑉 “ {𝐴})) ∘ ◡𝑉) ⊆ (𝑉 ∘ ◡𝑉))
18	14, 17	sstrd 3991	. 2 ⊢ ((𝑉 ⊆ (𝑋 × 𝑋) ∧ 𝐴 ∈ 𝑋) → (({𝐴} × (𝑉 “ {𝐴})) ∘ ((𝑉 “ {𝐴}) × {𝐴})) ⊆ (𝑉 ∘ ◡𝑉))
19	4, 18	eqsstrrd 4020	1 ⊢ ((𝑉 ⊆ (𝑋 × 𝑋) ∧ 𝐴 ∈ 𝑋) → ((𝑉 “ {𝐴}) × (𝑉 “ {𝐴})) ⊆ (𝑉 ∘ ◡𝑉))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106   ≠ wne 2940   ⊆ wss 3947  ∅c0 4321  {csn 4627   × cxp 5673  ^◡ccnv 5674   ↾ cres 5677   “ cima 5678   ∘ ccom 5679

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688

This theorem is referenced by:  neipcfilu  23792