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Mirrors > Home > ILE Home > Th. List > cnpdis | Unicode version |
Description: If is an isolated point in (or equivalently, the singleton is open in ), then every function is continuous at . (Contributed by Mario Carneiro, 9-Sep-2015.) |
Ref | Expression |
---|---|
cnpdis | TopOn TopOn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simplrl 524 | . . . . . . . 8 TopOn TopOn | |
2 | simpll3 1022 | . . . . . . . . 9 TopOn TopOn | |
3 | snidg 3554 | . . . . . . . . 9 | |
4 | 2, 3 | syl 14 | . . . . . . . 8 TopOn TopOn |
5 | simprr 521 | . . . . . . . . . 10 TopOn TopOn | |
6 | simplrr 525 | . . . . . . . . . . 11 TopOn TopOn | |
7 | ffn 5272 | . . . . . . . . . . 11 | |
8 | elpreima 5539 | . . . . . . . . . . 11 | |
9 | 6, 7, 8 | 3syl 17 | . . . . . . . . . 10 TopOn TopOn |
10 | 2, 5, 9 | mpbir2and 928 | . . . . . . . . 9 TopOn TopOn |
11 | 10 | snssd 3665 | . . . . . . . 8 TopOn TopOn |
12 | eleq2 2203 | . . . . . . . . . 10 | |
13 | sseq1 3120 | . . . . . . . . . 10 | |
14 | 12, 13 | anbi12d 464 | . . . . . . . . 9 |
15 | 14 | rspcev 2789 | . . . . . . . 8 |
16 | 1, 4, 11, 15 | syl12anc 1214 | . . . . . . 7 TopOn TopOn |
17 | 16 | expr 372 | . . . . . 6 TopOn TopOn |
18 | 17 | ralrimiva 2505 | . . . . 5 TopOn TopOn |
19 | 18 | expr 372 | . . . 4 TopOn TopOn |
20 | 19 | pm4.71d 390 | . . 3 TopOn TopOn |
21 | simpl2 985 | . . . . 5 TopOn TopOn TopOn | |
22 | toponmax 12192 | . . . . 5 TopOn | |
23 | 21, 22 | syl 14 | . . . 4 TopOn TopOn |
24 | simpl1 984 | . . . . 5 TopOn TopOn TopOn | |
25 | toponmax 12192 | . . . . 5 TopOn | |
26 | 24, 25 | syl 14 | . . . 4 TopOn TopOn |
27 | 23, 26 | elmapd 6556 | . . 3 TopOn TopOn |
28 | iscnp3 12372 | . . . 4 TopOn TopOn | |
29 | 28 | adantr 274 | . . 3 TopOn TopOn |
30 | 20, 27, 29 | 3bitr4rd 220 | . 2 TopOn TopOn |
31 | 30 | eqrdv 2137 | 1 TopOn TopOn |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 962 wceq 1331 wcel 1480 wral 2416 wrex 2417 wss 3071 csn 3527 ccnv 4538 cima 4542 wfn 5118 wf 5119 cfv 5123 (class class class)co 5774 cmap 6542 TopOnctopon 12177 ccnp 12355 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-map 6544 df-top 12165 df-topon 12178 df-cnp 12358 |
This theorem is referenced by: (None) |
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