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Mirrors > Home > ILE Home > Th. List > elcncf | Unicode version |
Description: Membership in the set of continuous complex functions from to . (Contributed by Paul Chapman, 11-Oct-2007.) (Revised by Mario Carneiro, 9-Nov-2013.) |
Ref | Expression |
---|---|
elcncf |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cncfval 13029 | . . . 4 | |
2 | 1 | eleq2d 2227 | . . 3 |
3 | fveq1 5469 | . . . . . . . . . 10 | |
4 | fveq1 5469 | . . . . . . . . . 10 | |
5 | 3, 4 | oveq12d 5844 | . . . . . . . . 9 |
6 | 5 | fveq2d 5474 | . . . . . . . 8 |
7 | 6 | breq1d 3977 | . . . . . . 7 |
8 | 7 | imbi2d 229 | . . . . . 6 |
9 | 8 | rexralbidv 2483 | . . . . 5 |
10 | 9 | 2ralbidv 2481 | . . . 4 |
11 | 10 | elrab 2868 | . . 3 |
12 | 2, 11 | bitrdi 195 | . 2 |
13 | cnex 7858 | . . . . 5 | |
14 | 13 | ssex 4103 | . . . 4 |
15 | 13 | ssex 4103 | . . . 4 |
16 | elmapg 6608 | . . . 4 | |
17 | 14, 15, 16 | syl2anr 288 | . . 3 |
18 | 17 | anbi1d 461 | . 2 |
19 | 12, 18 | bitrd 187 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1335 wcel 2128 wral 2435 wrex 2436 crab 2439 cvv 2712 wss 3102 class class class wbr 3967 wf 5168 cfv 5172 (class class class)co 5826 cmap 6595 cc 7732 clt 7914 cmin 8050 crp 9566 cabs 10908 ccncf 13027 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4084 ax-pow 4137 ax-pr 4171 ax-un 4395 ax-setind 4498 ax-cnex 7825 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-br 3968 df-opab 4028 df-id 4255 df-xp 4594 df-rel 4595 df-cnv 4596 df-co 4597 df-dm 4598 df-rn 4599 df-iota 5137 df-fun 5174 df-fn 5175 df-f 5176 df-fv 5180 df-ov 5829 df-oprab 5830 df-mpo 5831 df-map 6597 df-cncf 13028 |
This theorem is referenced by: elcncf2 13031 cncff 13034 elcncf1di 13036 rescncf 13038 cncfmet 13049 |
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