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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 22923 | . . 3 ⊢ (𝑅 ∈ Top ↔ 𝑅 ∈ (TopOn‘𝑋)) |
| 3 | ptuniconst.2 | . . . 4 ⊢ 𝐽 = (∏t‘(𝐴 × {𝑅})) | |
| 4 | 3 | pttoponconst 23605 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ (TopOn‘𝑋)) → 𝐽 ∈ (TopOn‘(𝑋 ↑m 𝐴))) |
| 5 | 2, 4 | sylan2b 594 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ Top) → 𝐽 ∈ (TopOn‘(𝑋 ↑m 𝐴))) |
| 6 | toponuni 22920 | . 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 1540 ∈ wcel 2108 {csn 4626 ∪ cuni 4907 × cxp 5683 ‘cfv 6561 (class class class)co 7431 ↑m cmap 8866 ∏tcpt 17483 Topctop 22899 TopOnctopon 22916 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1o 8506 df-2o 8507 df-map 8868 df-ixp 8938 df-en 8986 df-fin 8989 df-fi 9451 df-topgen 17488 df-pt 17489 df-top 22900 df-topon 22917 df-bases 22953 |
| This theorem is referenced by: xkopt 23663 xkopjcn 23664 poimirlem29 37656 poimirlem30 37657 poimirlem31 37658 |
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