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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 2200 | . . . . 5 | |
2 | 1 | anbi2d 459 | . . . 4 |
3 | fveq2 5414 | . . . . 5 | |
4 | 3 | eleq1d 2206 | . . . 4 |
5 | 2, 4 | imbi12d 233 | . . 3 |
6 | funfvop 5525 | . . . . 5 | |
7 | vex 2684 | . . . . . 6 | |
8 | opeq1 3700 | . . . . . . 7 | |
9 | 8 | eleq1d 2206 | . . . . . 6 |
10 | 7, 9 | spcev 2775 | . . . . 5 |
11 | 6, 10 | syl 14 | . . . 4 |
12 | funfvex 5431 | . . . . 5 | |
13 | elrn2g 4724 | . . . . 5 | |
14 | 12, 13 | syl 14 | . . . 4 |
15 | 11, 14 | mpbird 166 | . . 3 |
16 | 5, 15 | vtoclg 2741 | . 2 |
17 | 16 | anabsi7 570 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wex 1468 wcel 1480 cvv 2681 cop 3525 cdm 4534 crn 4535 wfun 5112 cfv 5118 |
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 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-sbc 2905 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-iota 5083 df-fun 5120 df-fn 5121 df-fv 5126 |
This theorem is referenced by: fnfvelrn 5545 eldmrexrn 5554 funfvima 5642 elunirn 5660 frecuzrdgdomlem 10183 frecuzrdgsuctlem 10189 |
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