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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 21528 | . . 3 ⊢ (𝑅 ∈ Top ↔ 𝑅 ∈ (TopOn‘𝑋)) |
3 | txuni.2 | . . . 4 ⊢ 𝑌 = ∪ 𝑆 | |
4 | 3 | toptopon 21528 | . . 3 ⊢ (𝑆 ∈ Top ↔ 𝑆 ∈ (TopOn‘𝑌)) |
5 | txtopon 22202 | . . 3 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑅 ×t 𝑆) ∈ (TopOn‘(𝑋 × 𝑌))) | |
6 | 2, 4, 5 | syl2anb 599 | . 2 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ (TopOn‘(𝑋 × 𝑌))) |
7 | toponuni 21525 | . 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 398 = wceq 1536 ∈ wcel 2113 ∪ cuni 4841 × cxp 5556 ‘cfv 6358 (class class class)co 7159 Topctop 21504 TopOnctopon 21521 ×t ctx 22171 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-fv 6366 df-ov 7162 df-oprab 7163 df-mpo 7164 df-1st 7692 df-2nd 7693 df-topgen 16720 df-top 21505 df-topon 21522 df-bases 21557 df-tx 22173 |
This theorem is referenced by: txunii 22204 txcld 22214 neitx 22218 uptx 22236 txcn 22237 txdis 22243 txnlly 22248 txcmp 22254 txcmpb 22255 hausdiag 22256 txhaus 22258 tx1stc 22261 txkgen 22263 txconn 22300 imasnopn 22301 imasncld 22302 imasncls 22303 utop2nei 22862 utop3cls 22863 qtophaus 31104 txpconn 32483 |
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