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Theorem ustneism 22766
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 4709 . . . 4 (𝐴𝑋 → {𝐴} ≠ ∅)
21adantl 482 . . 3 ((𝑉 ⊆ (𝑋 × 𝑋) ∧ 𝐴𝑋) → {𝐴} ≠ ∅)
3 xpco 6139 . . 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 6013 . . . . 5 ({𝐴} × (𝑉 “ {𝐴})) = ((𝑉 “ {𝐴}) × {𝐴})
6 ressn 6135 . . . . . . 7 (𝑉 ↾ {𝐴}) = ({𝐴} × (𝑉 “ {𝐴}))
76cnveqi 5744 . . . . . 6 (𝑉 ↾ {𝐴}) = ({𝐴} × (𝑉 “ {𝐴}))
8 resss 5877 . . . . . . 7 (𝑉 ↾ {𝐴}) ⊆ 𝑉
9 cnvss 5742 . . . . . . 7 ((𝑉 ↾ {𝐴}) ⊆ 𝑉(𝑉 ↾ {𝐴}) ⊆ 𝑉)
108, 9ax-mp 5 . . . . . 6 (𝑉 ↾ {𝐴}) ⊆ 𝑉
117, 10eqsstrri 4006 . . . . 5 ({𝐴} × (𝑉 “ {𝐴})) ⊆ 𝑉
125, 11eqsstrri 4006 . . . 4 ((𝑉 “ {𝐴}) × {𝐴}) ⊆ 𝑉
13 coss2 5726 . . . 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 4006 . . . 4 ({𝐴} × (𝑉 “ {𝐴})) ⊆ 𝑉
16 coss1 5725 . . . 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 3981 . 2 ((𝑉 ⊆ (𝑋 × 𝑋) ∧ 𝐴𝑋) → (({𝐴} × (𝑉 “ {𝐴})) ∘ ((𝑉 “ {𝐴}) × {𝐴})) ⊆ (𝑉𝑉))
194, 18eqsstrrd 4010 1 ((𝑉 ⊆ (𝑋 × 𝑋) ∧ 𝐴𝑋) → ((𝑉 “ {𝐴}) × (𝑉 “ {𝐴})) ⊆ (𝑉𝑉))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wcel 2107  wne 3021  wss 3940  c0 4295  {csn 4564   × cxp 5552  ccnv 5553  cres 5556  cima 5557  ccom 5558
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-br 5064  df-opab 5126  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567
This theorem is referenced by:  neipcfilu  22839
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