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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 21453 | . . 3 ⊢ (𝑅 ∈ Top ↔ 𝑅 ∈ (TopOn‘𝑋)) |
3 | ptuniconst.2 | . . . 4 ⊢ 𝐽 = (∏t‘(𝐴 × {𝑅})) | |
4 | 3 | pttoponconst 22133 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ (TopOn‘𝑋)) → 𝐽 ∈ (TopOn‘(𝑋 ↑m 𝐴))) |
5 | 2, 4 | sylan2b 593 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ Top) → 𝐽 ∈ (TopOn‘(𝑋 ↑m 𝐴))) |
6 | toponuni 21450 | . 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 396 = wceq 1528 ∈ wcel 2105 {csn 4557 ∪ cuni 4830 × cxp 5546 ‘cfv 6348 (class class class)co 7145 ↑m cmap 8395 ∏tcpt 16700 Topctop 21429 TopOnctopon 21446 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-map 8397 df-ixp 8450 df-en 8498 df-fin 8501 df-fi 8863 df-topgen 16705 df-pt 16706 df-top 21430 df-topon 21447 df-bases 21482 |
This theorem is referenced by: xkopt 22191 xkopjcn 22192 poimirlem29 34802 poimirlem30 34803 poimirlem31 34804 |
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