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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 22194 | . 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 2110 ∪ cuni 4831 × cxp 5547 (class class class)co 7150 Topctop 21495 ×t ctx 22162 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-topgen 16711 df-top 21496 df-topon 21513 df-bases 21548 df-tx 22164 |
This theorem is referenced by: txindis 22236 cxpcn3 25323 tpr2rico 31150 raddcn 31167 sxbrsigalem3 31525 dya2iocucvr 31537 sxbrsigalem1 31538 txsconnlem 32482 cvmlift2lem7 32551 cvmlift2lem9 32553 cvmlift2lem10 32554 cvmlift2lem12 32556 cvmlift2lem13 32557 cvmliftphtlem 32559 |
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