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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 22806 | . . 3 ⊢ (𝑅 ∈ Top ↔ 𝑅 ∈ (TopOn‘𝑋)) |
3 | ptuniconst.2 | . . . 4 ⊢ 𝐽 = (∏t‘(𝐴 × {𝑅})) | |
4 | 3 | pttoponconst 23488 | . . 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 22803 | . 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 395 = wceq 1534 ∈ wcel 2099 {csn 4624 ∪ cuni 4903 × cxp 5670 ‘cfv 6542 (class class class)co 7414 ↑m cmap 8836 ∏tcpt 17411 Topctop 22782 TopOnctopon 22799 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1o 8480 df-er 8718 df-map 8838 df-ixp 8908 df-en 8956 df-fin 8959 df-fi 9426 df-topgen 17416 df-pt 17417 df-top 22783 df-topon 22800 df-bases 22836 |
This theorem is referenced by: xkopt 23546 xkopjcn 23547 poimirlem29 37057 poimirlem30 37058 poimirlem31 37059 |
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