![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > cnmptkc | Structured version Visualization version GIF version |
Description: The curried first projection function is continuous. (Contributed by Mario Carneiro, 23-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.) |
Ref | Expression |
---|---|
cnmptk1.j | β’ (π β π½ β (TopOnβπ)) |
cnmptk1.k | β’ (π β πΎ β (TopOnβπ)) |
Ref | Expression |
---|---|
cnmptkc | β’ (π β (π₯ β π β¦ (π¦ β π β¦ π₯)) β (π½ Cn (π½ βko πΎ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fconstmpt 5742 | . . 3 β’ (π Γ {π₯}) = (π¦ β π β¦ π₯) | |
2 | 1 | mpteq2i 5255 | . 2 β’ (π₯ β π β¦ (π Γ {π₯})) = (π₯ β π β¦ (π¦ β π β¦ π₯)) |
3 | cnmptk1.k | . . 3 β’ (π β πΎ β (TopOnβπ)) | |
4 | cnmptk1.j | . . 3 β’ (π β π½ β (TopOnβπ)) | |
5 | xkoccn 23541 | . . 3 β’ ((πΎ β (TopOnβπ) β§ π½ β (TopOnβπ)) β (π₯ β π β¦ (π Γ {π₯})) β (π½ Cn (π½ βko πΎ))) | |
6 | 3, 4, 5 | syl2anc 582 | . 2 β’ (π β (π₯ β π β¦ (π Γ {π₯})) β (π½ Cn (π½ βko πΎ))) |
7 | 2, 6 | eqeltrrid 2833 | 1 β’ (π β (π₯ β π β¦ (π¦ β π β¦ π₯)) β (π½ Cn (π½ βko πΎ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wcel 2098 {csn 4630 β¦ cmpt 5233 Γ cxp 5678 βcfv 6551 (class class class)co 7424 TopOnctopon 22830 Cn ccn 23146 βko cxko 23483 |
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 2698 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 |
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 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-int 4952 df-iun 5000 df-iin 5001 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7875 df-1st 7997 df-2nd 7998 df-1o 8491 df-er 8729 df-map 8851 df-en 8969 df-dom 8970 df-fin 8972 df-fi 9440 df-rest 17409 df-topgen 17430 df-top 22814 df-topon 22831 df-bases 22867 df-cn 23149 df-cnp 23150 df-cmp 23309 df-xko 23485 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |