Nearby theorems
|
|
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
| Mirrors > Home > ILE Home > Th. List > txuni | GIF version | ||
| Description: The underlying set of the product of two topologies. (Contributed by Jeff Madsen, 2-Sep-2009.) |
| Ref | Expression |
|---|---|
| txuni.1 | β’ π = βͺ π |
| txuni.2 | β’ π = βͺ π |
| Ref | Expression |
|---|---|
| txuni | β’ ((π β Top β§ π β Top) β (π Γ π) = βͺ (π Γt π)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | txuni.1 | . . . 4 β’ π = βͺ π | |
| 2 | 1 | toptopon 13603 | . . 3 β’ (π β Top β π β (TopOnβπ)) |
| 3 | txuni.2 | . . . 4 β’ π = βͺ π | |
| 4 | 3 | toptopon 13603 | . . 3 β’ (π β Top β π β (TopOnβπ)) |
| 5 | txtopon 13847 | . . 3 β’ ((π β (TopOnβπ) β§ π β (TopOnβπ)) β (π Γt π) β (TopOnβ(π Γ π))) | |
| 6 | 2, 4, 5 | syl2anb 291 | . 2 β’ ((π β Top β§ π β Top) β (π Γt π) β (TopOnβ(π Γ π))) |
| 7 | toponuni 13600 | . 2 β’ ((π Γt π) β (TopOnβ(π Γ π)) β (π Γ π) = βͺ (π Γt π)) | |
| 8 | 6, 7 | syl 14 | 1 β’ ((π β Top β§ π β Top) β (π Γ π) = βͺ (π Γt π)) |
| Colors of variables: wff set class |
| Syntax hints: β wi 4 β§ wa 104 = wceq 1353 β wcel 2148 βͺ cuni 3811 Γ cxp 4626 βcfv 5218 (class class class)co 5877 Topctop 13582 TopOnctopon 13595 Γt ctx 13837 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4120 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 |
| This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-ov 5880 df-oprab 5881 df-mpo 5882 df-1st 6143 df-2nd 6144 df-topgen 12714 df-top 13583 df-topon 13596 df-bases 13628 df-tx 13838 |
| This theorem is referenced by: txunii 13849 neitx 13853 uptx 13859 txcn 13860 txdis 13862 imasnopn 13884 |
| Copyright terms: Public domain | W3C validator |