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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 3538 | . . . . . 6 | |
2 | reseq2 4814 | . . . . . . . 8 | |
3 | 2 | feq1d 5259 | . . . . . . 7 |
4 | feq2 5256 | . . . . . . 7 | |
5 | 3, 4 | bitrd 187 | . . . . . 6 |
6 | 1, 5 | syl 14 | . . . . 5 |
7 | fveq2 5421 | . . . . . 6 | |
8 | 7 | eleq1d 2208 | . . . . 5 |
9 | 6, 8 | bibi12d 234 | . . . 4 |
10 | 9 | imbi2d 229 | . . 3 |
11 | fnressn 5606 | . . . . 5 | |
12 | vsnid 3557 | . . . . . . . . . 10 | |
13 | fvres 5445 | . . . . . . . . . 10 | |
14 | 12, 13 | ax-mp 5 | . . . . . . . . 9 |
15 | 14 | opeq2i 3709 | . . . . . . . 8 |
16 | 15 | sneqi 3539 | . . . . . . 7 |
17 | 16 | eqeq2i 2150 | . . . . . 6 |
18 | vex 2689 | . . . . . . . 8 | |
19 | 18 | fsn2 5594 | . . . . . . 7 |
20 | 14 | eleq1i 2205 | . . . . . . . 8 |
21 | iba 298 | . . . . . . . 8 | |
22 | 20, 21 | syl5rbbr 194 | . . . . . . 7 |
23 | 19, 22 | syl5bb 191 | . . . . . 6 |
24 | 17, 23 | sylbir 134 | . . . . 5 |
25 | 11, 24 | syl 14 | . . . 4 |
26 | 25 | expcom 115 | . . 3 |
27 | 10, 26 | vtoclga 2752 | . 2 |
28 | 27 | impcom 124 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wcel 1480 csn 3527 cop 3530 cres 4541 wfn 5118 wf 5119 cfv 5123 |
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-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-reu 2423 df-v 2688 df-sbc 2910 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 |
This theorem is referenced by: (None) |
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