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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 3592 | . . . . . 6 | |
2 | reseq2 4884 | . . . . . . . 8 | |
3 | 2 | feq1d 5332 | . . . . . . 7 |
4 | feq2 5329 | . . . . . . 7 | |
5 | 3, 4 | bitrd 187 | . . . . . 6 |
6 | 1, 5 | syl 14 | . . . . 5 |
7 | fveq2 5494 | . . . . . 6 | |
8 | 7 | eleq1d 2239 | . . . . 5 |
9 | 6, 8 | bibi12d 234 | . . . 4 |
10 | 9 | imbi2d 229 | . . 3 |
11 | fnressn 5680 | . . . . 5 | |
12 | vsnid 3613 | . . . . . . . . . 10 | |
13 | fvres 5518 | . . . . . . . . . 10 | |
14 | 12, 13 | ax-mp 5 | . . . . . . . . 9 |
15 | 14 | opeq2i 3767 | . . . . . . . 8 |
16 | 15 | sneqi 3593 | . . . . . . 7 |
17 | 16 | eqeq2i 2181 | . . . . . 6 |
18 | vex 2733 | . . . . . . . 8 | |
19 | 18 | fsn2 5668 | . . . . . . 7 |
20 | iba 298 | . . . . . . . 8 | |
21 | 14 | eleq1i 2236 | . . . . . . . 8 |
22 | 20, 21 | bitr3di 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 2796 | . 2 |
28 | 27 | impcom 124 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1348 wcel 2141 csn 3581 cop 3584 cres 4611 wfn 5191 wf 5192 cfv 5196 |
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 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-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 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-ral 2453 df-rex 2454 df-reu 2455 df-v 2732 df-sbc 2956 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 |
This theorem is referenced by: (None) |
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