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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 13353 | . . . 4 | |
2 | 1 | eleq2d 2240 | . . 3 |
3 | fveq1 5495 | . . . . . . . . . 10 | |
4 | fveq1 5495 | . . . . . . . . . 10 | |
5 | 3, 4 | oveq12d 5871 | . . . . . . . . 9 |
6 | 5 | fveq2d 5500 | . . . . . . . 8 |
7 | 6 | breq1d 3999 | . . . . . . 7 |
8 | 7 | imbi2d 229 | . . . . . 6 |
9 | 8 | rexralbidv 2496 | . . . . 5 |
10 | 9 | 2ralbidv 2494 | . . . 4 |
11 | 10 | elrab 2886 | . . 3 |
12 | 2, 11 | bitrdi 195 | . 2 |
13 | cnex 7898 | . . . . 5 | |
14 | 13 | ssex 4126 | . . . 4 |
15 | 13 | ssex 4126 | . . . 4 |
16 | elmapg 6639 | . . . 4 | |
17 | 14, 15, 16 | syl2anr 288 | . . 3 |
18 | 17 | anbi1d 462 | . 2 |
19 | 12, 18 | bitrd 187 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1348 wcel 2141 wral 2448 wrex 2449 crab 2452 cvv 2730 wss 3121 class class class wbr 3989 wf 5194 cfv 5198 (class class class)co 5853 cmap 6626 cc 7772 clt 7954 cmin 8090 crp 9610 cabs 10961 ccncf 13351 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-cnex 7865 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-fv 5206 df-ov 5856 df-oprab 5857 df-mpo 5858 df-map 6628 df-cncf 13352 |
This theorem is referenced by: elcncf2 13355 cncff 13358 elcncf1di 13360 rescncf 13362 cncfmet 13373 |
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