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Theorem ustneism 24232
Description: For a point 𝐴 in 𝑋, (𝑉 “ {𝐴}) is small enough in (𝑉𝑉). 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 4774 . . . 4 (𝐴𝑋 → {𝐴} ≠ ∅)
21adantl 481 . . 3 ((𝑉 ⊆ (𝑋 × 𝑋) ∧ 𝐴𝑋) → {𝐴} ≠ ∅)
3 xpco 6309 . . 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 6177 . . . . 5 ({𝐴} × (𝑉 “ {𝐴})) = ((𝑉 “ {𝐴}) × {𝐴})
6 ressn 6305 . . . . . . 7 (𝑉 ↾ {𝐴}) = ({𝐴} × (𝑉 “ {𝐴}))
76cnveqi 5885 . . . . . 6 (𝑉 ↾ {𝐴}) = ({𝐴} × (𝑉 “ {𝐴}))
8 resss 6019 . . . . . . 7 (𝑉 ↾ {𝐴}) ⊆ 𝑉
9 cnvss 5883 . . . . . . 7 ((𝑉 ↾ {𝐴}) ⊆ 𝑉(𝑉 ↾ {𝐴}) ⊆ 𝑉)
108, 9ax-mp 5 . . . . . 6 (𝑉 ↾ {𝐴}) ⊆ 𝑉
117, 10eqsstrri 4031 . . . . 5 ({𝐴} × (𝑉 “ {𝐴})) ⊆ 𝑉
125, 11eqsstrri 4031 . . . 4 ((𝑉 “ {𝐴}) × {𝐴}) ⊆ 𝑉
13 coss2 5867 . . . 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, 8eqsstrri 4031 . . . 4 ({𝐴} × (𝑉 “ {𝐴})) ⊆ 𝑉
16 coss1 5866 . . . 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 3994 . 2 ((𝑉 ⊆ (𝑋 × 𝑋) ∧ 𝐴𝑋) → (({𝐴} × (𝑉 “ {𝐴})) ∘ ((𝑉 “ {𝐴}) × {𝐴})) ⊆ (𝑉𝑉))
194, 18eqsstrrd 4019 1 ((𝑉 ⊆ (𝑋 × 𝑋) ∧ 𝐴𝑋) → ((𝑉 “ {𝐴}) × (𝑉 “ {𝐴})) ⊆ (𝑉𝑉))
Colors of variables: wff setvar class
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
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