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Mirrors > Home > ILE Home > Th. List > cncnpi | Unicode version |
Description: A continuous function is continuous at all points. One direction of Theorem 7.2(g) of [Munkres] p. 107. (Contributed by Raph Levien, 20-Nov-2006.) (Proof shortened by Mario Carneiro, 21-Aug-2015.) |
Ref | Expression |
---|---|
cnsscnp.1 |
Ref | Expression |
---|---|
cncnpi |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnsscnp.1 | . . . 4 | |
2 | eqid 2164 | . . . 4 | |
3 | 1, 2 | cnf 12751 | . . 3 |
4 | 3 | adantr 274 | . 2 |
5 | cnima 12767 | . . . . . 6 | |
6 | 5 | ad2ant2r 501 | . . . . 5 |
7 | simpr 109 | . . . . . . 7 | |
8 | 7 | adantr 274 | . . . . . 6 |
9 | simprr 522 | . . . . . 6 | |
10 | 3 | ad2antrr 480 | . . . . . . 7 |
11 | ffn 5331 | . . . . . . 7 | |
12 | elpreima 5598 | . . . . . . 7 | |
13 | 10, 11, 12 | 3syl 17 | . . . . . 6 |
14 | 8, 9, 13 | mpbir2and 933 | . . . . 5 |
15 | eqimss 3191 | . . . . . . . 8 | |
16 | 15 | biantrud 302 | . . . . . . 7 |
17 | eleq2 2228 | . . . . . . 7 | |
18 | 16, 17 | bitr3d 189 | . . . . . 6 |
19 | 18 | rspcev 2825 | . . . . 5 |
20 | 6, 14, 19 | syl2anc 409 | . . . 4 |
21 | 20 | expr 373 | . . 3 |
22 | 21 | ralrimiva 2537 | . 2 |
23 | cntop1 12748 | . . . . 5 | |
24 | 23 | adantr 274 | . . . 4 |
25 | 1 | toptopon 12563 | . . . 4 TopOn |
26 | 24, 25 | sylib 121 | . . 3 TopOn |
27 | cntop2 12749 | . . . . 5 | |
28 | 27 | adantr 274 | . . . 4 |
29 | 2 | toptopon 12563 | . . . 4 TopOn |
30 | 28, 29 | sylib 121 | . . 3 TopOn |
31 | iscnp3 12750 | . . 3 TopOn TopOn | |
32 | 26, 30, 7, 31 | syl3anc 1227 | . 2 |
33 | 4, 22, 32 | mpbir2and 933 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1342 wcel 2135 wral 2442 wrex 2443 wss 3111 cuni 3783 ccnv 4597 cima 4601 wfn 5177 wf 5178 cfv 5182 (class class class)co 5836 ctop 12542 TopOnctopon 12555 ccn 12732 ccnp 12733 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-map 6607 df-top 12543 df-topon 12556 df-cn 12735 df-cnp 12736 |
This theorem is referenced by: cnsscnp 12776 cncnp 12777 lmcn 12798 dvcnp2cntop 13210 dvaddxxbr 13212 dvmulxxbr 13213 dvcoapbr 13218 dvcjbr 13219 |
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