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Theorem ustneism 22320
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 4464 . . . 4 (𝐴𝑋 → {𝐴} ≠ ∅)
21adantl 473 . . 3 ((𝑉 ⊆ (𝑋 × 𝑋) ∧ 𝐴𝑋) → {𝐴} ≠ ∅)
3 xpco 5863 . . 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 5736 . . . . 5 ({𝐴} × (𝑉 “ {𝐴})) = ((𝑉 “ {𝐴}) × {𝐴})
6 ressn 5859 . . . . . . 7 (𝑉 ↾ {𝐴}) = ({𝐴} × (𝑉 “ {𝐴}))
76cnveqi 5467 . . . . . 6 (𝑉 ↾ {𝐴}) = ({𝐴} × (𝑉 “ {𝐴}))
8 resss 5599 . . . . . . 7 (𝑉 ↾ {𝐴}) ⊆ 𝑉
9 cnvss 5465 . . . . . . 7 ((𝑉 ↾ {𝐴}) ⊆ 𝑉(𝑉 ↾ {𝐴}) ⊆ 𝑉)
108, 9ax-mp 5 . . . . . 6 (𝑉 ↾ {𝐴}) ⊆ 𝑉
117, 10eqsstr3i 3798 . . . . 5 ({𝐴} × (𝑉 “ {𝐴})) ⊆ 𝑉
125, 11eqsstr3i 3798 . . . 4 ((𝑉 “ {𝐴}) × {𝐴}) ⊆ 𝑉
13 coss2 5449 . . . 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 3798 . . . 4 ({𝐴} × (𝑉 “ {𝐴})) ⊆ 𝑉
16 coss1 5448 . . . 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 3773 . 2 ((𝑉 ⊆ (𝑋 × 𝑋) ∧ 𝐴𝑋) → (({𝐴} × (𝑉 “ {𝐴})) ∘ ((𝑉 “ {𝐴}) × {𝐴})) ⊆ (𝑉𝑉))
194, 18eqsstr3d 3802 1 ((𝑉 ⊆ (𝑋 × 𝑋) ∧ 𝐴𝑋) → ((𝑉 “ {𝐴}) × (𝑉 “ {𝐴})) ⊆ (𝑉𝑉))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1652  wcel 2155  wne 2937  wss 3734  c0 4081  {csn 4336   × cxp 5277  ccnv 5278  cres 5281  cima 5282  ccom 5283
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pr 5064
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3599  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-sn 4337  df-pr 4339  df-op 4343  df-br 4812  df-opab 4874  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292
This theorem is referenced by:  neipcfilu  22393
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