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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 21395 | . 2 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑋 × 𝑌) = ∪ (𝑅 ×t 𝑆)) |
6 | 1, 2, 5 | mp2an 708 | 1 ⊢ (𝑋 × 𝑌) = ∪ (𝑅 ×t 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 ∈ wcel 1990 ∪ cuni 4436 × cxp 5112 (class class class)co 6650 Topctop 20698 ×t ctx 21363 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-topgen 16104 df-top 20699 df-topon 20716 df-bases 20750 df-tx 21365 |
This theorem is referenced by: txindis 21437 cxpcn3 24489 tpr2rico 29958 raddcn 29975 sxbrsigalem3 30334 dya2iocucvr 30346 sxbrsigalem1 30347 txsconnlem 31222 cvmlift2lem7 31291 cvmlift2lem9 31293 cvmlift2lem10 31294 cvmlift2lem12 31296 cvmlift2lem13 31297 cvmliftphtlem 31299 |
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