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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 23509 | . . 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 3198 | . 2 β’ ((π΄ β π β§ πΉ:π΄βΆTop) β βπ’ β π΅ βπ£ β π΅ (π’ β© π£) β π΅) |
4 | 1 | ptuni2 23508 | . . . . 5 β’ ((π΄ β π β§ πΉ:π΄βΆTop) β Xπ β π΄ βͺ (πΉβπ) = βͺ π΅) |
5 | ixpexg 8949 | . . . . . 6 β’ (βπ β π΄ βͺ (πΉβπ) β V β Xπ β π΄ βͺ (πΉβπ) β V) | |
6 | fvex 6915 | . . . . . . . 8 β’ (πΉβπ) β V | |
7 | 6 | uniex 7754 | . . . . . . 7 β’ βͺ (πΉβπ) β V |
8 | 7 | a1i 11 | . . . . . 6 β’ (π β π΄ β βͺ (πΉβπ) β V) |
9 | 5, 8 | mprg 3064 | . . . . 5 β’ Xπ β π΄ βͺ (πΉβπ) β V |
10 | 4, 9 | eqeltrrdi 2838 | . . . 4 β’ ((π΄ β π β§ πΉ:π΄βΆTop) β βͺ π΅ β V) |
11 | uniexb 7774 | . . . 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 9458 | . . 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 394 β§ w3a 1084 = wceq 1533 βwex 1773 β wcel 2098 {cab 2705 βwral 3058 βwrex 3067 Vcvv 3473 β cdif 3946 β© cin 3948 βͺ cuni 4912 Fn wfn 6548 βΆwf 6549 βcfv 6553 Xcixp 8924 Fincfn 8972 ficfi 9443 Topctop 22823 |
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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-om 7879 df-1o 8495 df-er 8733 df-ixp 8925 df-en 8973 df-fin 8976 df-fi 9444 df-top 22824 |
This theorem is referenced by: ptbas 23511 ptbasfi 23513 |
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