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Mirrors > Home > ILE Home > Th. List > fressnfv | Unicode version |
Description: The value of a function restricted to a singleton. (Contributed by NM, 9-Oct-2004.) |
Ref | Expression |
---|---|
fressnfv |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sneq 3386 | . . . . . 6 | |
2 | reseq2 4607 | . . . . . . . 8 | |
3 | 2 | feq1d 5034 | . . . . . . 7 |
4 | feq2 5031 | . . . . . . 7 | |
5 | 3, 4 | bitrd 177 | . . . . . 6 |
6 | 1, 5 | syl 14 | . . . . 5 |
7 | fveq2 5178 | . . . . . 6 | |
8 | 7 | eleq1d 2106 | . . . . 5 |
9 | 6, 8 | bibi12d 224 | . . . 4 |
10 | 9 | imbi2d 219 | . . 3 |
11 | fnressn 5349 | . . . . 5 | |
12 | vsnid 3403 | . . . . . . . . . 10 | |
13 | fvres 5198 | . . . . . . . . . 10 | |
14 | 12, 13 | ax-mp 7 | . . . . . . . . 9 |
15 | 14 | opeq2i 3553 | . . . . . . . 8 |
16 | 15 | sneqi 3387 | . . . . . . 7 |
17 | 16 | eqeq2i 2050 | . . . . . 6 |
18 | vex 2560 | . . . . . . . 8 | |
19 | 18 | fsn2 5337 | . . . . . . 7 |
20 | 14 | eleq1i 2103 | . . . . . . . 8 |
21 | iba 284 | . . . . . . . 8 | |
22 | 20, 21 | syl5rbbr 184 | . . . . . . 7 |
23 | 19, 22 | syl5bb 181 | . . . . . 6 |
24 | 17, 23 | sylbir 125 | . . . . 5 |
25 | 11, 24 | syl 14 | . . . 4 |
26 | 25 | expcom 109 | . . 3 |
27 | 10, 26 | vtoclga 2619 | . 2 |
28 | 27 | impcom 116 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wceq 1243 wcel 1393 csn 3375 cop 3378 cres 4347 wfn 4897 wf 4898 cfv 4902 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-reu 2313 df-v 2559 df-sbc 2765 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 |
This theorem is referenced by: (None) |
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