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Mirrors > Home > MPE Home > Th. List > ptuniconst | Structured version Visualization version GIF version |
Description: The base set for a product topology when all factors are the same. (Contributed by Mario Carneiro, 3-Feb-2015.) |
Ref | Expression |
---|---|
ptuniconst.2 | ⊢ 𝐽 = (∏t‘(𝐴 × {𝑅})) |
ptuniconst.1 | ⊢ 𝑋 = ∪ 𝑅 |
Ref | Expression |
---|---|
ptuniconst | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ Top) → (𝑋 ↑𝑚 𝐴) = ∪ 𝐽) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ptuniconst.1 | . . . 4 ⊢ 𝑋 = ∪ 𝑅 | |
2 | 1 | toptopon 21229 | . . 3 ⊢ (𝑅 ∈ Top ↔ 𝑅 ∈ (TopOn‘𝑋)) |
3 | ptuniconst.2 | . . . 4 ⊢ 𝐽 = (∏t‘(𝐴 × {𝑅})) | |
4 | 3 | pttoponconst 21909 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ (TopOn‘𝑋)) → 𝐽 ∈ (TopOn‘(𝑋 ↑𝑚 𝐴))) |
5 | 2, 4 | sylan2b 584 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ Top) → 𝐽 ∈ (TopOn‘(𝑋 ↑𝑚 𝐴))) |
6 | toponuni 21226 | . 2 ⊢ (𝐽 ∈ (TopOn‘(𝑋 ↑𝑚 𝐴)) → (𝑋 ↑𝑚 𝐴) = ∪ 𝐽) | |
7 | 5, 6 | syl 17 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ Top) → (𝑋 ↑𝑚 𝐴) = ∪ 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1507 ∈ wcel 2050 {csn 4441 ∪ cuni 4712 × cxp 5405 ‘cfv 6188 (class class class)co 6976 ↑𝑚 cmap 8206 ∏tcpt 16568 Topctop 21205 TopOnctopon 21222 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2751 ax-rep 5049 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2760 df-cleq 2772 df-clel 2847 df-nfc 2919 df-ne 2969 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3418 df-sbc 3683 df-csb 3788 df-dif 3833 df-un 3835 df-in 3837 df-ss 3844 df-pss 3846 df-nul 4180 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-int 4750 df-iun 4794 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-ov 6979 df-oprab 6980 df-mpo 6981 df-om 7397 df-wrecs 7750 df-recs 7812 df-rdg 7850 df-1o 7905 df-oadd 7909 df-er 8089 df-map 8208 df-ixp 8260 df-en 8307 df-fin 8310 df-fi 8670 df-topgen 16573 df-pt 16574 df-top 21206 df-topon 21223 df-bases 21258 |
This theorem is referenced by: xkopt 21967 xkopjcn 21968 poimirlem29 34359 poimirlem30 34360 poimirlem31 34361 |
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