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Mirrors > Home > MPE Home > Th. List > txuni | Structured version Visualization version GIF version |
Description: The underlying set of the product of two topologies. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Ref | Expression |
---|---|
txuni.1 | ⊢ 𝑋 = ∪ 𝑅 |
txuni.2 | ⊢ 𝑌 = ∪ 𝑆 |
Ref | Expression |
---|---|
txuni | ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑋 × 𝑌) = ∪ (𝑅 ×t 𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | txuni.1 | . . . 4 ⊢ 𝑋 = ∪ 𝑅 | |
2 | 1 | toptopon 21522 | . . 3 ⊢ (𝑅 ∈ Top ↔ 𝑅 ∈ (TopOn‘𝑋)) |
3 | txuni.2 | . . . 4 ⊢ 𝑌 = ∪ 𝑆 | |
4 | 3 | toptopon 21522 | . . 3 ⊢ (𝑆 ∈ Top ↔ 𝑆 ∈ (TopOn‘𝑌)) |
5 | txtopon 22196 | . . 3 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑅 ×t 𝑆) ∈ (TopOn‘(𝑋 × 𝑌))) | |
6 | 2, 4, 5 | syl2anb 600 | . 2 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ (TopOn‘(𝑋 × 𝑌))) |
7 | toponuni 21519 | . 2 ⊢ ((𝑅 ×t 𝑆) ∈ (TopOn‘(𝑋 × 𝑌)) → (𝑋 × 𝑌) = ∪ (𝑅 ×t 𝑆)) | |
8 | 6, 7 | syl 17 | 1 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑋 × 𝑌) = ∪ (𝑅 ×t 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∪ cuni 4800 × cxp 5517 ‘cfv 6324 (class class class)co 7135 Topctop 21498 TopOnctopon 21515 ×t ctx 22165 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-topgen 16709 df-top 21499 df-topon 21516 df-bases 21551 df-tx 22167 |
This theorem is referenced by: txunii 22198 txcld 22208 neitx 22212 uptx 22230 txcn 22231 txdis 22237 txnlly 22242 txcmp 22248 txcmpb 22249 hausdiag 22250 txhaus 22252 tx1stc 22255 txkgen 22257 txconn 22294 imasnopn 22295 imasncld 22296 imasncls 22297 utop2nei 22856 utop3cls 22857 qtophaus 31189 txpconn 32592 |
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