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Theorem ustneism 21837
 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) ∧ 𝐴𝑋) → ((𝑉 “ {𝐴}) × (𝑉 “ {𝐴})) ⊆ (𝑉𝑉))

Proof of Theorem ustneism
StepHypRef Expression
1 snnzg 4251 . . . 4 (𝐴𝑋 → {𝐴} ≠ ∅)
21adantl 481 . . 3 ((𝑉 ⊆ (𝑋 × 𝑋) ∧ 𝐴𝑋) → {𝐴} ≠ ∅)
3 xpco 5592 . . 3 ({𝐴} ≠ ∅ → (({𝐴} × (𝑉 “ {𝐴})) ∘ ((𝑉 “ {𝐴}) × {𝐴})) = ((𝑉 “ {𝐴}) × (𝑉 “ {𝐴})))
42, 3syl 17 . 2 ((𝑉 ⊆ (𝑋 × 𝑋) ∧ 𝐴𝑋) → (({𝐴} × (𝑉 “ {𝐴})) ∘ ((𝑉 “ {𝐴}) × {𝐴})) = ((𝑉 “ {𝐴}) × (𝑉 “ {𝐴})))
5 cnvxp 5470 . . . . 5 ({𝐴} × (𝑉 “ {𝐴})) = ((𝑉 “ {𝐴}) × {𝐴})
6 ressn 5588 . . . . . . 7 (𝑉 ↾ {𝐴}) = ({𝐴} × (𝑉 “ {𝐴}))
76cnveqi 5219 . . . . . 6 (𝑉 ↾ {𝐴}) = ({𝐴} × (𝑉 “ {𝐴}))
8 resss 5342 . . . . . . 7 (𝑉 ↾ {𝐴}) ⊆ 𝑉
9 cnvss 5216 . . . . . . 7 ((𝑉 ↾ {𝐴}) ⊆ 𝑉(𝑉 ↾ {𝐴}) ⊆ 𝑉)
108, 9ax-mp 5 . . . . . 6 (𝑉 ↾ {𝐴}) ⊆ 𝑉
117, 10eqsstr3i 3599 . . . . 5 ({𝐴} × (𝑉 “ {𝐴})) ⊆ 𝑉
125, 11eqsstr3i 3599 . . . 4 ((𝑉 “ {𝐴}) × {𝐴}) ⊆ 𝑉
13 coss2 5200 . . . 4 (((𝑉 “ {𝐴}) × {𝐴}) ⊆ 𝑉 → (({𝐴} × (𝑉 “ {𝐴})) ∘ ((𝑉 “ {𝐴}) × {𝐴})) ⊆ (({𝐴} × (𝑉 “ {𝐴})) ∘ 𝑉))
1412, 13mp1i 13 . . 3 ((𝑉 ⊆ (𝑋 × 𝑋) ∧ 𝐴𝑋) → (({𝐴} × (𝑉 “ {𝐴})) ∘ ((𝑉 “ {𝐴}) × {𝐴})) ⊆ (({𝐴} × (𝑉 “ {𝐴})) ∘ 𝑉))
156, 8eqsstr3i 3599 . . . 4 ({𝐴} × (𝑉 “ {𝐴})) ⊆ 𝑉
16 coss1 5199 . . . 4 (({𝐴} × (𝑉 “ {𝐴})) ⊆ 𝑉 → (({𝐴} × (𝑉 “ {𝐴})) ∘ 𝑉) ⊆ (𝑉𝑉))
1715, 16mp1i 13 . . 3 ((𝑉 ⊆ (𝑋 × 𝑋) ∧ 𝐴𝑋) → (({𝐴} × (𝑉 “ {𝐴})) ∘ 𝑉) ⊆ (𝑉𝑉))
1814, 17sstrd 3578 . 2 ((𝑉 ⊆ (𝑋 × 𝑋) ∧ 𝐴𝑋) → (({𝐴} × (𝑉 “ {𝐴})) ∘ ((𝑉 “ {𝐴}) × {𝐴})) ⊆ (𝑉𝑉))
194, 18eqsstr3d 3603 1 ((𝑉 ⊆ (𝑋 × 𝑋) ∧ 𝐴𝑋) → ((𝑉 “ {𝐴}) × (𝑉 “ {𝐴})) ⊆ (𝑉𝑉))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780   ⊆ wss 3540  ∅c0 3874  {csn 4125   × cxp 5036  ◡ccnv 5037   ↾ cres 5040   “ cima 5041   ∘ ccom 5042 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051 This theorem is referenced by:  neipcfilu  21910
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