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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 21724 | . 2 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑋 × 𝑌) = ∪ (𝑅 ×t 𝑆)) |
6 | 1, 2, 5 | mp2an 684 | 1 ⊢ (𝑋 × 𝑌) = ∪ (𝑅 ×t 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1653 ∈ wcel 2157 ∪ cuni 4628 × cxp 5310 (class class class)co 6878 Topctop 21026 ×t ctx 21692 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-id 5220 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-fv 6109 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-1st 7401 df-2nd 7402 df-topgen 16419 df-top 21027 df-topon 21044 df-bases 21079 df-tx 21694 |
This theorem is referenced by: txindis 21766 cxpcn3 24833 tpr2rico 30474 raddcn 30491 sxbrsigalem3 30850 dya2iocucvr 30862 sxbrsigalem1 30863 txsconnlem 31739 cvmlift2lem7 31808 cvmlift2lem9 31810 cvmlift2lem10 31811 cvmlift2lem12 31813 cvmlift2lem13 31814 cvmliftphtlem 31816 |
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