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Mirrors > Home > ILE Home > Th. List > rescncf | Unicode version |
Description: A continuous complex function restricted to a subset is continuous. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 25-Aug-2014.) |
Ref | Expression |
---|---|
rescncf |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 109 | . . . . . 6 | |
2 | cncfrss 13129 | . . . . . . . 8 | |
3 | 2 | adantl 275 | . . . . . . 7 |
4 | cncfrss2 13130 | . . . . . . . 8 | |
5 | 4 | adantl 275 | . . . . . . 7 |
6 | elcncf 13127 | . . . . . . 7 | |
7 | 3, 5, 6 | syl2anc 409 | . . . . . 6 |
8 | 1, 7 | mpbid 146 | . . . . 5 |
9 | 8 | simpld 111 | . . . 4 |
10 | simpl 108 | . . . 4 | |
11 | 9, 10 | fssresd 5359 | . . 3 |
12 | 8 | simprd 113 | . . . 4 |
13 | ssralv 3202 | . . . . 5 | |
14 | ssralv 3202 | . . . . . . . . 9 | |
15 | fvres 5505 | . . . . . . . . . . . . . . 15 | |
16 | fvres 5505 | . . . . . . . . . . . . . . 15 | |
17 | 15, 16 | oveqan12d 5856 | . . . . . . . . . . . . . 14 |
18 | 17 | fveq2d 5485 | . . . . . . . . . . . . 13 |
19 | 18 | breq1d 3987 | . . . . . . . . . . . 12 |
20 | 19 | imbi2d 229 | . . . . . . . . . . 11 |
21 | 20 | biimprd 157 | . . . . . . . . . 10 |
22 | 21 | ralimdva 2531 | . . . . . . . . 9 |
23 | 14, 22 | sylan9 407 | . . . . . . . 8 |
24 | 23 | reximdv 2565 | . . . . . . 7 |
25 | 24 | ralimdv 2532 | . . . . . 6 |
26 | 25 | ralimdva 2531 | . . . . 5 |
27 | 13, 26 | syld 45 | . . . 4 |
28 | 10, 12, 27 | sylc 62 | . . 3 |
29 | 10, 3 | sstrd 3148 | . . . 4 |
30 | elcncf 13127 | . . . 4 | |
31 | 29, 5, 30 | syl2anc 409 | . . 3 |
32 | 11, 28, 31 | mpbir2and 933 | . 2 |
33 | 32 | ex 114 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wcel 2135 wral 2442 wrex 2443 wss 3112 class class class wbr 3977 cres 4601 wf 5179 cfv 5183 (class class class)co 5837 cc 7743 clt 7925 cmin 8061 crp 9581 cabs 10929 ccncf 13124 |
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 4095 ax-pow 4148 ax-pr 4182 ax-un 4406 ax-setind 4509 ax-cnex 7836 |
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 2724 df-sbc 2948 df-dif 3114 df-un 3116 df-in 3118 df-ss 3125 df-pw 3556 df-sn 3577 df-pr 3578 df-op 3580 df-uni 3785 df-br 3978 df-opab 4039 df-id 4266 df-xp 4605 df-rel 4606 df-cnv 4607 df-co 4608 df-dm 4609 df-rn 4610 df-res 4611 df-iota 5148 df-fun 5185 df-fn 5186 df-f 5187 df-fv 5191 df-ov 5840 df-oprab 5841 df-mpo 5842 df-map 6608 df-cncf 13125 |
This theorem is referenced by: (None) |
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