Theorem utopsnnei 24561

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 2761	. . . 4 ⊢ (𝑉 “ {𝑃}) = (𝑉 “ {𝑃})
2		imaeq1 6047	. . . . 5 ⊢ (𝑣 = 𝑉 → (𝑣 “ {𝑃}) = (𝑉 “ {𝑃}))
3	2	rspceeqv 3599	. . . 4 ⊢ ((𝑉 ∈ 𝑈 ∧ (𝑉 “ {𝑃}) = (𝑉 “ {𝑃})) → ∃𝑣 ∈ 𝑈 (𝑉 “ {𝑃}) = (𝑣 “ {𝑃}))
4	1, 3	mpan2 704	. . 3 ⊢ (𝑉 ∈ 𝑈 → ∃𝑣 ∈ 𝑈 (𝑉 “ {𝑃}) = (𝑣 “ {𝑃}))
5	4	3ad2ant2 1152	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝑃 ∈ 𝑋) → ∃𝑣 ∈ 𝑈 (𝑉 “ {𝑃}) = (𝑣 “ {𝑃}))
6		utoptop.1	. . . . . 6 ⊢ 𝐽 = (unifTop‘𝑈)
7	6	utopsnneip 24560	. . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋) → ((nei‘𝐽)‘{𝑃}) = ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})))
8	7	3adant2 1149	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝑃 ∈ 𝑋) → ((nei‘𝐽)‘{𝑃}) = ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})))
9	8	eleq2d 2847	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝑃 ∈ 𝑋) → ((𝑉 “ {𝑃}) ∈ ((nei‘𝐽)‘{𝑃}) ↔ (𝑉 “ {𝑃}) ∈ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃}))))
10		imaexg 7923	. . . . 5 ⊢ (𝑉 ∈ 𝑈 → (𝑉 “ {𝑃}) ∈ V)
11		eqid 2761	. . . . . 6 ⊢ (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})) = (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃}))
12	11	elrnmpt 5940	. . . . 5 ⊢ ((𝑉 “ {𝑃}) ∈ V → ((𝑉 “ {𝑃}) ∈ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})) ↔ ∃𝑣 ∈ 𝑈 (𝑉 “ {𝑃}) = (𝑣 “ {𝑃})))
13	10, 12	syl 18	. . . 4 ⊢ (𝑉 ∈ 𝑈 → ((𝑉 “ {𝑃}) ∈ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})) ↔ ∃𝑣 ∈ 𝑈 (𝑉 “ {𝑃}) = (𝑣 “ {𝑃})))
14	13	3ad2ant2 1152	. . 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 282	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝑃 ∈ 𝑋) → ((𝑉 “ {𝑃}) ∈ ((nei‘𝐽)‘{𝑃}) ↔ ∃𝑣 ∈ 𝑈 (𝑉 “ {𝑃}) = (𝑣 “ {𝑃})))
16	5, 15	mpbird 260	1 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝑃 ∈ 𝑋) → (𝑉 “ {𝑃}) ∈ ((nei‘𝐽)‘{𝑃}))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∃wrex 3087  Vcvv 3451  {csn 4584   ↦ cmpt 5186  ran crn 5652   “ cima 5654  ‘cfv 6537  neicnei 23408  UnifOncust 24512  unifTopcutop 24542

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-om 7876  df-1o 8469  df-2o 8470  df-en 8967  df-fin 8970  df-fi 9396  df-top 23205  df-nei 23409  df-ust 24513  df-utop 24543

This theorem is used by:  utop2nei  24562  utop3cls  24563  utopreg  24564