Theorem ustneism 24232

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 4774	. . . 4 ⊢ (𝐴 ∈ 𝑋 → {𝐴} ≠ ∅)
2	1	adantl 481	. . 3 ⊢ ((𝑉 ⊆ (𝑋 × 𝑋) ∧ 𝐴 ∈ 𝑋) → {𝐴} ≠ ∅)
3		xpco 6309	. . 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 6177	. . . . 5 ⊢ ◡({𝐴} × (𝑉 “ {𝐴})) = ((𝑉 “ {𝐴}) × {𝐴})
6		ressn 6305	. . . . . . 7 ⊢ (𝑉 ↾ {𝐴}) = ({𝐴} × (𝑉 “ {𝐴}))
7	6	cnveqi 5885	. . . . . 6 ⊢ ◡(𝑉 ↾ {𝐴}) = ◡({𝐴} × (𝑉 “ {𝐴}))
8		resss 6019	. . . . . . 7 ⊢ (𝑉 ↾ {𝐴}) ⊆ 𝑉
9		cnvss 5883	. . . . . . 7 ⊢ ((𝑉 ↾ {𝐴}) ⊆ 𝑉 → ◡(𝑉 ↾ {𝐴}) ⊆ ◡𝑉)
10	8, 9	ax-mp 5	. . . . . 6 ⊢ ◡(𝑉 ↾ {𝐴}) ⊆ ◡𝑉
11	7, 10	eqsstrri 4031	. . . . 5 ⊢ ◡({𝐴} × (𝑉 “ {𝐴})) ⊆ ◡𝑉
12	5, 11	eqsstrri 4031	. . . 4 ⊢ ((𝑉 “ {𝐴}) × {𝐴}) ⊆ ◡𝑉
13		coss2 5867	. . . 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 4031	. . . 4 ⊢ ({𝐴} × (𝑉 “ {𝐴})) ⊆ 𝑉
16		coss1 5866	. . . 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 3994	. 2 ⊢ ((𝑉 ⊆ (𝑋 × 𝑋) ∧ 𝐴 ∈ 𝑋) → (({𝐴} × (𝑉 “ {𝐴})) ∘ ((𝑉 “ {𝐴}) × {𝐴})) ⊆ (𝑉 ∘ ◡𝑉))
19	4, 18	eqsstrrd 4019	1 ⊢ ((𝑉 ⊆ (𝑋 × 𝑋) ∧ 𝐴 ∈ 𝑋) → ((𝑉 “ {𝐴}) × (𝑉 “ {𝐴})) ⊆ (𝑉 ∘ ◡𝑉))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108   ≠ wne 2940   ⊆ wss 3951  ∅c0 4333  {csn 4626   × cxp 5683  ^◡ccnv 5684   ↾ cres 5687   “ cima 5688   ∘ ccom 5689

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 2007  ax-8 2110  ax-9 2118  ax-11 2157  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698

This theorem is referenced by:  neipcfilu  24305