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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 3409 | . . . . . 6 | |
2 | reseq2 4625 | . . . . . . . 8 | |
3 | 2 | feq1d 5054 | . . . . . . 7 |
4 | feq2 5051 | . . . . . . 7 | |
5 | 3, 4 | bitrd 186 | . . . . . 6 |
6 | 1, 5 | syl 14 | . . . . 5 |
7 | fveq2 5198 | . . . . . 6 | |
8 | 7 | eleq1d 2147 | . . . . 5 |
9 | 6, 8 | bibi12d 233 | . . . 4 |
10 | 9 | imbi2d 228 | . . 3 |
11 | fnressn 5370 | . . . . 5 | |
12 | vsnid 3426 | . . . . . . . . . 10 | |
13 | fvres 5219 | . . . . . . . . . 10 | |
14 | 12, 13 | ax-mp 7 | . . . . . . . . 9 |
15 | 14 | opeq2i 3574 | . . . . . . . 8 |
16 | 15 | sneqi 3410 | . . . . . . 7 |
17 | 16 | eqeq2i 2091 | . . . . . 6 |
18 | vex 2604 | . . . . . . . 8 | |
19 | 18 | fsn2 5358 | . . . . . . 7 |
20 | 14 | eleq1i 2144 | . . . . . . . 8 |
21 | iba 294 | . . . . . . . 8 | |
22 | 20, 21 | syl5rbbr 193 | . . . . . . 7 |
23 | 19, 22 | syl5bb 190 | . . . . . 6 |
24 | 17, 23 | sylbir 133 | . . . . 5 |
25 | 11, 24 | syl 14 | . . . 4 |
26 | 25 | expcom 114 | . . 3 |
27 | 10, 26 | vtoclga 2664 | . 2 |
28 | 27 | impcom 123 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 wceq 1284 wcel 1433 csn 3398 cop 3401 cres 4365 wfn 4917 wf 4918 cfv 4922 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-reu 2355 df-v 2603 df-sbc 2816 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 |
This theorem is referenced by: (None) |
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