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Mirrors > Home > MPE Home > Th. List > txunii | Structured version Visualization version GIF version |
Description: The underlying set of the product of two topologies. (Contributed by Jeff Madsen, 15-Jun-2010.) |
Ref | Expression |
---|---|
txunii.1 | ⊢ 𝑅 ∈ Top |
txunii.2 | ⊢ 𝑆 ∈ Top |
txunii.3 | ⊢ 𝑋 = ∪ 𝑅 |
txunii.4 | ⊢ 𝑌 = ∪ 𝑆 |
Ref | Expression |
---|---|
txunii | ⊢ (𝑋 × 𝑌) = ∪ (𝑅 ×t 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | txunii.1 | . 2 ⊢ 𝑅 ∈ Top | |
2 | txunii.2 | . 2 ⊢ 𝑆 ∈ Top | |
3 | txunii.3 | . . 3 ⊢ 𝑋 = ∪ 𝑅 | |
4 | txunii.4 | . . 3 ⊢ 𝑌 = ∪ 𝑆 | |
5 | 3, 4 | txuni 23540 | . 2 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑋 × 𝑌) = ∪ (𝑅 ×t 𝑆)) |
6 | 1, 2, 5 | mp2an 690 | 1 ⊢ (𝑋 × 𝑌) = ∪ (𝑅 ×t 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2098 ∪ cuni 4909 × cxp 5676 (class class class)co 7419 Topctop 22839 ×t ctx 23508 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-fv 6557 df-ov 7422 df-oprab 7423 df-mpo 7424 df-1st 7994 df-2nd 7995 df-topgen 17428 df-top 22840 df-topon 22857 df-bases 22893 df-tx 23510 |
This theorem is referenced by: txindis 23582 cxpcn3 26728 tpr2rico 33644 raddcn 33661 sxbrsigalem3 34023 dya2iocucvr 34035 sxbrsigalem1 34036 txsconnlem 34981 cvmlift2lem7 35050 cvmlift2lem9 35052 cvmlift2lem10 35053 cvmlift2lem12 35055 cvmlift2lem13 35056 cvmliftphtlem 35058 |
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