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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 3549 | . . . . . . . . 9 | |
4 | 2, 3 | syl 14 | . . . . . . . 8 TopOn TopOn |
5 | simprr 521 | . . . . . . . . . 10 TopOn TopOn | |
6 | simplrr 525 | . . . . . . . . . . 11 TopOn TopOn | |
7 | ffn 5267 | . . . . . . . . . . 11 | |
8 | elpreima 5532 | . . . . . . . . . . 11 | |
9 | 6, 7, 8 | 3syl 17 | . . . . . . . . . 10 TopOn TopOn |
10 | 2, 5, 9 | mpbir2and 928 | . . . . . . . . 9 TopOn TopOn |
11 | 10 | snssd 3660 | . . . . . . . 8 TopOn TopOn |
12 | eleq2 2201 | . . . . . . . . . 10 | |
13 | sseq1 3115 | . . . . . . . . . 10 | |
14 | 12, 13 | anbi12d 464 | . . . . . . . . 9 |
15 | 14 | rspcev 2784 | . . . . . . . 8 |
16 | 1, 4, 11, 15 | syl12anc 1214 | . . . . . . 7 TopOn TopOn |
17 | 16 | expr 372 | . . . . . 6 TopOn TopOn |
18 | 17 | ralrimiva 2503 | . . . . 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 12181 | . . . . 5 TopOn | |
23 | 21, 22 | syl 14 | . . . 4 TopOn TopOn |
24 | simpl1 984 | . . . . 5 TopOn TopOn TopOn | |
25 | toponmax 12181 | . . . . 5 TopOn | |
26 | 24, 25 | syl 14 | . . . 4 TopOn TopOn |
27 | 23, 26 | elmapd 6549 | . . 3 TopOn TopOn |
28 | iscnp3 12361 | . . . 4 TopOn TopOn | |
29 | 28 | adantr 274 | . . 3 TopOn TopOn |
30 | 20, 27, 29 | 3bitr4rd 220 | . 2 TopOn TopOn |
31 | 30 | eqrdv 2135 | 1 TopOn TopOn |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 962 wceq 1331 wcel 1480 wral 2414 wrex 2415 wss 3066 csn 3522 ccnv 4533 cima 4537 wfn 5113 wf 5114 cfv 5118 (class class class)co 5767 cmap 6535 TopOnctopon 12166 ccnp 12344 |
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 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 |
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 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-fv 5126 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-map 6537 df-top 12154 df-topon 12167 df-cnp 12347 |
This theorem is referenced by: (None) |
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