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Mirrors > Home > ILE Home > Th. List > icnpimaex | Unicode version |
Description: Property of a function continuous at a point. (Contributed by FL, 31-Dec-2006.) (Revised by Jim Kingdon, 28-Mar-2023.) |
Ref | Expression |
---|---|
icnpimaex | TopOn TopOn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr3 989 | . 2 TopOn TopOn | |
2 | eleq2 2203 | . . . 4 | |
3 | sseq2 3121 | . . . . . 6 | |
4 | 3 | anbi2d 459 | . . . . 5 |
5 | 4 | rexbidv 2438 | . . . 4 |
6 | 2, 5 | imbi12d 233 | . . 3 |
7 | simpr1 987 | . . . . 5 TopOn TopOn | |
8 | iscnp 12371 | . . . . . 6 TopOn TopOn | |
9 | 8 | adantr 274 | . . . . 5 TopOn TopOn |
10 | 7, 9 | mpbid 146 | . . . 4 TopOn TopOn |
11 | 10 | simprd 113 | . . 3 TopOn TopOn |
12 | simpr2 988 | . . 3 TopOn TopOn | |
13 | 6, 11, 12 | rspcdva 2794 | . 2 TopOn TopOn |
14 | 1, 13 | mpd 13 | 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 cima 4542 wf 5119 cfv 5123 (class class class)co 5774 TopOnctopon 12180 ccnp 12358 |
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 12168 df-topon 12181 df-cnp 12361 |
This theorem is referenced by: iscnp4 12390 cnpnei 12391 cnptopco 12394 cncnp 12402 cnptopresti 12410 lmtopcnp 12422 txcnp 12443 |
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