Theorem utopsnnei 24074

Description: Images of singletons by entourages 𝑉 are neighborhoods of those singletons. (Contributed by Thierry Arnoux, 13-Jan-2018.)

Hypothesis

Ref	Expression
utoptop.1	⊢ 𝐽 = (unifTop‘𝑈)

Assertion

Ref	Expression
utopsnnei	⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝑃 ∈ 𝑋) → (𝑉 “ {𝑃}) ∈ ((nei‘𝐽)‘{𝑃}))

Proof of Theorem utopsnnei

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2731	. . . 4 ⊢ (𝑉 “ {𝑃}) = (𝑉 “ {𝑃})
2		imaeq1 6054	. . . . 5 ⊢ (𝑣 = 𝑉 → (𝑣 “ {𝑃}) = (𝑉 “ {𝑃}))
3	2	rspceeqv 3633	. . . 4 ⊢ ((𝑉 ∈ 𝑈 ∧ (𝑉 “ {𝑃}) = (𝑉 “ {𝑃})) → ∃𝑣 ∈ 𝑈 (𝑉 “ {𝑃}) = (𝑣 “ {𝑃}))
4	1, 3	mpan2 688	. . 3 ⊢ (𝑉 ∈ 𝑈 → ∃𝑣 ∈ 𝑈 (𝑉 “ {𝑃}) = (𝑣 “ {𝑃}))
5	4	3ad2ant2 1133	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝑃 ∈ 𝑋) → ∃𝑣 ∈ 𝑈 (𝑉 “ {𝑃}) = (𝑣 “ {𝑃}))
6		utoptop.1	. . . . . 6 ⊢ 𝐽 = (unifTop‘𝑈)
7	6	utopsnneip 24073	. . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋) → ((nei‘𝐽)‘{𝑃}) = ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})))
8	7	3adant2 1130	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝑃 ∈ 𝑋) → ((nei‘𝐽)‘{𝑃}) = ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})))
9	8	eleq2d 2818	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝑃 ∈ 𝑋) → ((𝑉 “ {𝑃}) ∈ ((nei‘𝐽)‘{𝑃}) ↔ (𝑉 “ {𝑃}) ∈ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃}))))
10		imaexg 7910	. . . . 5 ⊢ (𝑉 ∈ 𝑈 → (𝑉 “ {𝑃}) ∈ V)
11		eqid 2731	. . . . . 6 ⊢ (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})) = (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃}))
12	11	elrnmpt 5955	. . . . 5 ⊢ ((𝑉 “ {𝑃}) ∈ V → ((𝑉 “ {𝑃}) ∈ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})) ↔ ∃𝑣 ∈ 𝑈 (𝑉 “ {𝑃}) = (𝑣 “ {𝑃})))
13	10, 12	syl 17	. . . 4 ⊢ (𝑉 ∈ 𝑈 → ((𝑉 “ {𝑃}) ∈ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})) ↔ ∃𝑣 ∈ 𝑈 (𝑉 “ {𝑃}) = (𝑣 “ {𝑃})))
14	13	3ad2ant2 1133	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝑃 ∈ 𝑋) → ((𝑉 “ {𝑃}) ∈ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})) ↔ ∃𝑣 ∈ 𝑈 (𝑉 “ {𝑃}) = (𝑣 “ {𝑃})))
15	9, 14	bitrd 279	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝑃 ∈ 𝑋) → ((𝑉 “ {𝑃}) ∈ ((nei‘𝐽)‘{𝑃}) ↔ ∃𝑣 ∈ 𝑈 (𝑉 “ {𝑃}) = (𝑣 “ {𝑃})))
16	5, 15	mpbird 257	1 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝑃 ∈ 𝑋) → (𝑉 “ {𝑃}) ∈ ((nei‘𝐽)‘{𝑃}))

Syntax hints:   → wi 4   ↔ wb 205   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  ∃wrex 3069  Vcvv 3473  {csn 4628   ↦ cmpt 5231  ran crn 5677   “ cima 5679  ‘cfv 6543  neicnei 22921  UnifOncust 24024  unifTopcutop 24055

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-om 7860  df-1o 8472  df-er 8709  df-en 8946  df-fin 8949  df-fi 9412  df-top 22716  df-nei 22922  df-ust 24025  df-utop 24056

This theorem is referenced by:  utop2nei  24075  utop3cls  24076  utopreg  24077