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Mirrors > Home > ILE Home > Th. List > fvelrn | Unicode version |
Description: A function's value belongs to its range. (Contributed by NM, 14-Oct-1996.) |
Ref | Expression |
---|---|
fvelrn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2180 | . . . . 5 | |
2 | 1 | anbi2d 459 | . . . 4 |
3 | fveq2 5389 | . . . . 5 | |
4 | 3 | eleq1d 2186 | . . . 4 |
5 | 2, 4 | imbi12d 233 | . . 3 |
6 | funfvop 5500 | . . . . 5 | |
7 | vex 2663 | . . . . . 6 | |
8 | opeq1 3675 | . . . . . . 7 | |
9 | 8 | eleq1d 2186 | . . . . . 6 |
10 | 7, 9 | spcev 2754 | . . . . 5 |
11 | 6, 10 | syl 14 | . . . 4 |
12 | funfvex 5406 | . . . . 5 | |
13 | elrn2g 4699 | . . . . 5 | |
14 | 12, 13 | syl 14 | . . . 4 |
15 | 11, 14 | mpbird 166 | . . 3 |
16 | 5, 15 | vtoclg 2720 | . 2 |
17 | 16 | anabsi7 555 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1316 wex 1453 wcel 1465 cvv 2660 cop 3500 cdm 4509 crn 4510 wfun 5087 cfv 5093 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-sbc 2883 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-iota 5058 df-fun 5095 df-fn 5096 df-fv 5101 |
This theorem is referenced by: fnfvelrn 5520 eldmrexrn 5529 funfvima 5617 elunirn 5635 frecuzrdgdomlem 10145 frecuzrdgsuctlem 10151 |
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