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Mirrors > Home > MPE Home > Th. List > ustuqtop0 | Structured version Visualization version GIF version |
Description: Lemma for ustuqtop 22855. (Contributed by Thierry Arnoux, 11-Jan-2018.) |
Ref | Expression |
---|---|
utopustuq.1 | ⊢ 𝑁 = (𝑝 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝}))) |
Ref | Expression |
---|---|
ustuqtop0 | ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑁:𝑋⟶𝒫 𝒫 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ustimasn 22837 | . . . . . . . 8 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑣 ∈ 𝑈 ∧ 𝑝 ∈ 𝑋) → (𝑣 “ {𝑝}) ⊆ 𝑋) | |
2 | 1 | 3expa 1114 | . . . . . . 7 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑣 ∈ 𝑈) ∧ 𝑝 ∈ 𝑋) → (𝑣 “ {𝑝}) ⊆ 𝑋) |
3 | 2 | an32s 650 | . . . . . 6 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑣 ∈ 𝑈) → (𝑣 “ {𝑝}) ⊆ 𝑋) |
4 | vex 3497 | . . . . . . . 8 ⊢ 𝑣 ∈ V | |
5 | 4 | imaex 7621 | . . . . . . 7 ⊢ (𝑣 “ {𝑝}) ∈ V |
6 | 5 | elpw 4543 | . . . . . 6 ⊢ ((𝑣 “ {𝑝}) ∈ 𝒫 𝑋 ↔ (𝑣 “ {𝑝}) ⊆ 𝑋) |
7 | 3, 6 | sylibr 236 | . . . . 5 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑣 ∈ 𝑈) → (𝑣 “ {𝑝}) ∈ 𝒫 𝑋) |
8 | 7 | ralrimiva 3182 | . . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → ∀𝑣 ∈ 𝑈 (𝑣 “ {𝑝}) ∈ 𝒫 𝑋) |
9 | eqid 2821 | . . . . 5 ⊢ (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})) = (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})) | |
10 | 9 | rnmptss 6886 | . . . 4 ⊢ (∀𝑣 ∈ 𝑈 (𝑣 “ {𝑝}) ∈ 𝒫 𝑋 → ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})) ⊆ 𝒫 𝑋) |
11 | 8, 10 | syl 17 | . . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})) ⊆ 𝒫 𝑋) |
12 | mptexg 6984 | . . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})) ∈ V) | |
13 | rnexg 7614 | . . . . 5 ⊢ ((𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})) ∈ V → ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})) ∈ V) | |
14 | elpwg 4542 | . . . . 5 ⊢ (ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})) ∈ V → (ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})) ∈ 𝒫 𝒫 𝑋 ↔ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})) ⊆ 𝒫 𝑋)) | |
15 | 12, 13, 14 | 3syl 18 | . . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})) ∈ 𝒫 𝒫 𝑋 ↔ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})) ⊆ 𝒫 𝑋)) |
16 | 15 | adantr 483 | . . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → (ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})) ∈ 𝒫 𝒫 𝑋 ↔ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})) ⊆ 𝒫 𝑋)) |
17 | 11, 16 | mpbird 259 | . 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})) ∈ 𝒫 𝒫 𝑋) |
18 | utopustuq.1 | . 2 ⊢ 𝑁 = (𝑝 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝}))) | |
19 | 17, 18 | fmptd 6878 | 1 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑁:𝑋⟶𝒫 𝒫 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3138 Vcvv 3494 ⊆ wss 3936 𝒫 cpw 4539 {csn 4567 ↦ cmpt 5146 ran crn 5556 “ cima 5558 ⟶wf 6351 ‘cfv 6355 UnifOncust 22808 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ust 22809 |
This theorem is referenced by: ustuqtop 22855 utopsnneiplem 22856 |
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