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Mirrors > Home > ILE Home > Th. List > cncnp2m | Unicode version |
Description: A continuous function is continuous at all points. Theorem 7.2(g) of [Munkres] p. 107. (Contributed by Raph Levien, 20-Nov-2006.) (Revised by Jim Kingdon, 30-Mar-2023.) |
Ref | Expression |
---|---|
cncnp.1 | |
cncnp.2 |
Ref | Expression |
---|---|
cncnp2m |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cntop1 12359 | . . . . 5 | |
2 | cncnp.1 | . . . . . 6 | |
3 | 2 | toptopon 12174 | . . . . 5 TopOn |
4 | 1, 3 | sylib 121 | . . . 4 TopOn |
5 | cntop2 12360 | . . . . 5 | |
6 | cncnp.2 | . . . . . 6 | |
7 | 6 | toptopon 12174 | . . . . 5 TopOn |
8 | 5, 7 | sylib 121 | . . . 4 TopOn |
9 | 2, 6 | cnf 12362 | . . . 4 |
10 | 4, 8, 9 | jca31 307 | . . 3 TopOn TopOn |
11 | 10 | adantl 275 | . 2 TopOn TopOn |
12 | 3 | biimpi 119 | . . . . 5 TopOn |
13 | 12 | 3ad2ant1 1002 | . . . 4 TopOn |
14 | 13 | adantr 274 | . . 3 TopOn |
15 | 7 | biimpi 119 | . . . . 5 TopOn |
16 | 15 | 3ad2ant2 1003 | . . . 4 TopOn |
17 | 16 | adantr 274 | . . 3 TopOn |
18 | r19.2m 3444 | . . . . . . 7 | |
19 | 18 | ex 114 | . . . . . 6 |
20 | 19 | 3ad2ant3 1004 | . . . . 5 |
21 | cnpf2 12365 | . . . . . . . 8 TopOn TopOn | |
22 | 21 | 3expia 1183 | . . . . . . 7 TopOn TopOn |
23 | 22 | rexlimdvw 2551 | . . . . . 6 TopOn TopOn |
24 | 13, 16, 23 | syl2anc 408 | . . . . 5 |
25 | 20, 24 | syld 45 | . . . 4 |
26 | 25 | imp 123 | . . 3 |
27 | 14, 17, 26 | jca31 307 | . 2 TopOn TopOn |
28 | cncnp 12388 | . . 3 TopOn TopOn | |
29 | 28 | baibd 908 | . 2 TopOn TopOn |
30 | 11, 27, 29 | pm5.21nd 901 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 962 wceq 1331 wex 1468 wcel 1480 wral 2414 wrex 2415 cuni 3731 wf 5114 cfv 5118 (class class class)co 5767 ctop 12153 TopOnctopon 12166 ccn 12343 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-coll 4038 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-reu 2421 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-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-map 6537 df-topgen 12130 df-top 12154 df-topon 12167 df-cn 12346 df-cnp 12347 |
This theorem is referenced by: (None) |
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