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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 2227 | . . . . 5 | |
2 | 1 | anbi2d 460 | . . . 4 |
3 | fveq2 5480 | . . . . 5 | |
4 | 3 | eleq1d 2233 | . . . 4 |
5 | 2, 4 | imbi12d 233 | . . 3 |
6 | funfvop 5591 | . . . . 5 | |
7 | vex 2724 | . . . . . 6 | |
8 | opeq1 3752 | . . . . . . 7 | |
9 | 8 | eleq1d 2233 | . . . . . 6 |
10 | 7, 9 | spcev 2816 | . . . . 5 |
11 | 6, 10 | syl 14 | . . . 4 |
12 | funfvex 5497 | . . . . 5 | |
13 | elrn2g 4788 | . . . . 5 | |
14 | 12, 13 | syl 14 | . . . 4 |
15 | 11, 14 | mpbird 166 | . . 3 |
16 | 5, 15 | vtoclg 2781 | . 2 |
17 | 16 | anabsi7 571 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1342 wex 1479 wcel 2135 cvv 2721 cop 3573 cdm 4598 crn 4599 wfun 5176 cfv 5182 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-sbc 2947 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-iota 5147 df-fun 5184 df-fn 5185 df-fv 5190 |
This theorem is referenced by: fnfvelrn 5611 eldmrexrn 5620 funfvima 5710 elunirn 5728 frecuzrdgdomlem 10342 frecuzrdgsuctlem 10348 |
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