| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| 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 23600 | . 2 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑋 × 𝑌) = ∪ (𝑅 ×t 𝑆)) |
| 6 | 1, 2, 5 | mp2an 692 | 1 ⊢ (𝑋 × 𝑌) = ∪ (𝑅 ×t 𝑆) |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∈ wcel 2108 ∪ cuni 4907 × cxp 5683 (class class class)co 7431 Topctop 22899 ×t ctx 23568 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-topgen 17488 df-top 22900 df-topon 22917 df-bases 22953 df-tx 23570 |
| This theorem is referenced by: txindis 23642 cxpcn3 26791 tpr2rico 33911 raddcn 33928 sxbrsigalem3 34274 dya2iocucvr 34286 sxbrsigalem1 34287 txsconnlem 35245 cvmlift2lem7 35314 cvmlift2lem9 35316 cvmlift2lem10 35317 cvmlift2lem12 35319 cvmlift2lem13 35320 cvmliftphtlem 35322 |
| Copyright terms: Public domain | W3C validator |