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Theorem ustneism 24214
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 480 . . 3 ((𝑉 ⊆ (𝑋 × 𝑋) ∧ 𝐴𝑋) → {𝐴} ≠ ∅)
3 xpco 6291 . . 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 6159 . . . . 5 ({𝐴} × (𝑉 “ {𝐴})) = ((𝑉 “ {𝐴}) × {𝐴})
6 ressn 6287 . . . . . . 7 (𝑉 ↾ {𝐴}) = ({𝐴} × (𝑉 “ {𝐴}))
76cnveqi 5872 . . . . . 6 (𝑉 ↾ {𝐴}) = ({𝐴} × (𝑉 “ {𝐴}))
8 resss 6002 . . . . . . 7 (𝑉 ↾ {𝐴}) ⊆ 𝑉
9 cnvss 5870 . . . . . . 7 ((𝑉 ↾ {𝐴}) ⊆ 𝑉(𝑉 ↾ {𝐴}) ⊆ 𝑉)
108, 9ax-mp 5 . . . . . 6 (𝑉 ↾ {𝐴}) ⊆ 𝑉
117, 10eqsstrri 4015 . . . . 5 ({𝐴} × (𝑉 “ {𝐴})) ⊆ 𝑉
125, 11eqsstrri 4015 . . . 4 ((𝑉 “ {𝐴}) × {𝐴}) ⊆ 𝑉
13 coss2 5854 . . . 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 4015 . . . 4 ({𝐴} × (𝑉 “ {𝐴})) ⊆ 𝑉
16 coss1 5853 . . . 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 3990 . 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 394   = wceq 1534  wcel 2099  wne 2930  wss 3947  c0 4323  {csn 4624   × cxp 5671  ccnv 5672  cres 5675  cima 5676  ccom 5677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-11 2147  ax-ext 2697  ax-sep 5295  ax-nul 5302  ax-pr 5424
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3421  df-v 3465  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5145  df-opab 5207  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686
This theorem is referenced by:  neipcfilu  24287
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