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Mirrors > Home > MPE Home > Th. List > ustuqtop0 | Structured version Visualization version GIF version |
Description: Lemma for ustuqtop 22377. (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 22359 | . . . . . . . 8 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑣 ∈ 𝑈 ∧ 𝑝 ∈ 𝑋) → (𝑣 “ {𝑝}) ⊆ 𝑋) | |
2 | 1 | 3expa 1148 | . . . . . . 7 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑣 ∈ 𝑈) ∧ 𝑝 ∈ 𝑋) → (𝑣 “ {𝑝}) ⊆ 𝑋) |
3 | 2 | an32s 643 | . . . . . 6 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑣 ∈ 𝑈) → (𝑣 “ {𝑝}) ⊆ 𝑋) |
4 | vex 3389 | . . . . . . . 8 ⊢ 𝑣 ∈ V | |
5 | 4 | imaex 7340 | . . . . . . 7 ⊢ (𝑣 “ {𝑝}) ∈ V |
6 | 5 | elpw 4356 | . . . . . 6 ⊢ ((𝑣 “ {𝑝}) ∈ 𝒫 𝑋 ↔ (𝑣 “ {𝑝}) ⊆ 𝑋) |
7 | 3, 6 | sylibr 226 | . . . . 5 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑣 ∈ 𝑈) → (𝑣 “ {𝑝}) ∈ 𝒫 𝑋) |
8 | 7 | ralrimiva 3148 | . . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → ∀𝑣 ∈ 𝑈 (𝑣 “ {𝑝}) ∈ 𝒫 𝑋) |
9 | eqid 2800 | . . . . 5 ⊢ (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})) = (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})) | |
10 | 9 | rnmptss 6619 | . . . 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 6714 | . . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})) ∈ V) | |
13 | rnexg 7333 | . . . . 5 ⊢ ((𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})) ∈ V → ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})) ∈ V) | |
14 | elpwg 4358 | . . . . 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 473 | . . 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 249 | . 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 6611 | 1 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑁:𝑋⟶𝒫 𝒫 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 385 = wceq 1653 ∈ wcel 2157 ∀wral 3090 Vcvv 3386 ⊆ wss 3770 𝒫 cpw 4350 {csn 4369 ↦ cmpt 4923 ran crn 5314 “ cima 5316 ⟶wf 6098 ‘cfv 6102 UnifOncust 22330 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2378 ax-ext 2778 ax-rep 4965 ax-sep 4976 ax-nul 4984 ax-pow 5036 ax-pr 5098 ax-un 7184 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2592 df-eu 2610 df-clab 2787 df-cleq 2793 df-clel 2796 df-nfc 2931 df-ne 2973 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3388 df-sbc 3635 df-csb 3730 df-dif 3773 df-un 3775 df-in 3777 df-ss 3784 df-nul 4117 df-if 4279 df-pw 4352 df-sn 4370 df-pr 4372 df-op 4376 df-uni 4630 df-iun 4713 df-br 4845 df-opab 4907 df-mpt 4924 df-id 5221 df-xp 5319 df-rel 5320 df-cnv 5321 df-co 5322 df-dm 5323 df-rn 5324 df-res 5325 df-ima 5326 df-iota 6065 df-fun 6104 df-fn 6105 df-f 6106 df-f1 6107 df-fo 6108 df-f1o 6109 df-fv 6110 df-ust 22331 |
This theorem is referenced by: ustuqtop 22377 utopsnneiplem 22378 |
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