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Mirrors > Home > MPE Home > Th. List > ptbasin2 | Structured version Visualization version GIF version |
Description: The basis for a product topology is closed under intersections. (Contributed by Mario Carneiro, 19-Mar-2015.) |
Ref | Expression |
---|---|
ptbas.1 | β’ π΅ = {π₯ β£ βπ((π Fn π΄ β§ βπ¦ β π΄ (πβπ¦) β (πΉβπ¦) β§ βπ§ β Fin βπ¦ β (π΄ β π§)(πβπ¦) = βͺ (πΉβπ¦)) β§ π₯ = Xπ¦ β π΄ (πβπ¦))} |
Ref | Expression |
---|---|
ptbasin2 | β’ ((π΄ β π β§ πΉ:π΄βΆTop) β (fiβπ΅) = π΅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ptbas.1 | . . . 4 β’ π΅ = {π₯ β£ βπ((π Fn π΄ β§ βπ¦ β π΄ (πβπ¦) β (πΉβπ¦) β§ βπ§ β Fin βπ¦ β (π΄ β π§)(πβπ¦) = βͺ (πΉβπ¦)) β§ π₯ = Xπ¦ β π΄ (πβπ¦))} | |
2 | 1 | ptbasin 23436 | . . 3 β’ (((π΄ β π β§ πΉ:π΄βΆTop) β§ (π’ β π΅ β§ π£ β π΅)) β (π’ β© π£) β π΅) |
3 | 2 | ralrimivva 3194 | . 2 β’ ((π΄ β π β§ πΉ:π΄βΆTop) β βπ’ β π΅ βπ£ β π΅ (π’ β© π£) β π΅) |
4 | 1 | ptuni2 23435 | . . . . 5 β’ ((π΄ β π β§ πΉ:π΄βΆTop) β Xπ β π΄ βͺ (πΉβπ) = βͺ π΅) |
5 | ixpexg 8918 | . . . . . 6 β’ (βπ β π΄ βͺ (πΉβπ) β V β Xπ β π΄ βͺ (πΉβπ) β V) | |
6 | fvex 6898 | . . . . . . . 8 β’ (πΉβπ) β V | |
7 | 6 | uniex 7728 | . . . . . . 7 β’ βͺ (πΉβπ) β V |
8 | 7 | a1i 11 | . . . . . 6 β’ (π β π΄ β βͺ (πΉβπ) β V) |
9 | 5, 8 | mprg 3061 | . . . . 5 β’ Xπ β π΄ βͺ (πΉβπ) β V |
10 | 4, 9 | eqeltrrdi 2836 | . . . 4 β’ ((π΄ β π β§ πΉ:π΄βΆTop) β βͺ π΅ β V) |
11 | uniexb 7748 | . . . 4 β’ (π΅ β V β βͺ π΅ β V) | |
12 | 10, 11 | sylibr 233 | . . 3 β’ ((π΄ β π β§ πΉ:π΄βΆTop) β π΅ β V) |
13 | inficl 9422 | . . 3 β’ (π΅ β V β (βπ’ β π΅ βπ£ β π΅ (π’ β© π£) β π΅ β (fiβπ΅) = π΅)) | |
14 | 12, 13 | syl 17 | . 2 β’ ((π΄ β π β§ πΉ:π΄βΆTop) β (βπ’ β π΅ βπ£ β π΅ (π’ β© π£) β π΅ β (fiβπ΅) = π΅)) |
15 | 3, 14 | mpbid 231 | 1 β’ ((π΄ β π β§ πΉ:π΄βΆTop) β (fiβπ΅) = π΅) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 β§ w3a 1084 = wceq 1533 βwex 1773 β wcel 2098 {cab 2703 βwral 3055 βwrex 3064 Vcvv 3468 β cdif 3940 β© cin 3942 βͺ cuni 4902 Fn wfn 6532 βΆwf 6533 βcfv 6537 Xcixp 8893 Fincfn 8941 ficfi 9407 Topctop 22750 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-om 7853 df-1o 8467 df-er 8705 df-ixp 8894 df-en 8942 df-fin 8945 df-fi 9408 df-top 22751 |
This theorem is referenced by: ptbas 23438 ptbasfi 23440 |
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