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| Mirrors > Home > MPE Home > Th. List > utopsnnei | Structured version Visualization version GIF version | ||
| Description: Images of singletons by entourages 𝑉 are neighborhoods of those singletons. (Contributed by Thierry Arnoux, 13-Jan-2018.) |
| Ref | Expression |
|---|---|
| utoptop.1 | ⊢ 𝐽 = (unifTop‘𝑈) |
| Ref | Expression |
|---|---|
| utopsnnei | ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝑃 ∈ 𝑋) → (𝑉 “ {𝑃}) ∈ ((nei‘𝐽)‘{𝑃})) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqid 2737 | . . . 4 ⊢ (𝑉 “ {𝑃}) = (𝑉 “ {𝑃}) | |
| 2 | imaeq1 6073 | . . . . 5 ⊢ (𝑣 = 𝑉 → (𝑣 “ {𝑃}) = (𝑉 “ {𝑃})) | |
| 3 | 2 | rspceeqv 3645 | . . . 4 ⊢ ((𝑉 ∈ 𝑈 ∧ (𝑉 “ {𝑃}) = (𝑉 “ {𝑃})) → ∃𝑣 ∈ 𝑈 (𝑉 “ {𝑃}) = (𝑣 “ {𝑃})) |
| 4 | 1, 3 | mpan2 691 | . . 3 ⊢ (𝑉 ∈ 𝑈 → ∃𝑣 ∈ 𝑈 (𝑉 “ {𝑃}) = (𝑣 “ {𝑃})) |
| 5 | 4 | 3ad2ant2 1135 | . 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝑃 ∈ 𝑋) → ∃𝑣 ∈ 𝑈 (𝑉 “ {𝑃}) = (𝑣 “ {𝑃})) |
| 6 | utoptop.1 | . . . . . 6 ⊢ 𝐽 = (unifTop‘𝑈) | |
| 7 | 6 | utopsnneip 24257 | . . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋) → ((nei‘𝐽)‘{𝑃}) = ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃}))) |
| 8 | 7 | 3adant2 1132 | . . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝑃 ∈ 𝑋) → ((nei‘𝐽)‘{𝑃}) = ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃}))) |
| 9 | 8 | eleq2d 2827 | . . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝑃 ∈ 𝑋) → ((𝑉 “ {𝑃}) ∈ ((nei‘𝐽)‘{𝑃}) ↔ (𝑉 “ {𝑃}) ∈ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})))) |
| 10 | imaexg 7935 | . . . . 5 ⊢ (𝑉 ∈ 𝑈 → (𝑉 “ {𝑃}) ∈ V) | |
| 11 | eqid 2737 | . . . . . 6 ⊢ (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})) = (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})) | |
| 12 | 11 | elrnmpt 5969 | . . . . 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 1135 | . . 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})) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 Vcvv 3480 {csn 4626 ↦ cmpt 5225 ran crn 5686 “ cima 5688 ‘cfv 6561 neicnei 23105 UnifOncust 24208 unifTopcutop 24239 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-om 7888 df-1o 8506 df-2o 8507 df-en 8986 df-fin 8989 df-fi 9451 df-top 22900 df-nei 23106 df-ust 24209 df-utop 24240 |
| This theorem is referenced by: utop2nei 24259 utop3cls 24260 utopreg 24261 |
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