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Mirrors > Home > ILE Home > Th. List > ffvresb | Unicode version |
Description: A necessary and sufficient condition for a restricted function. (Contributed by Mario Carneiro, 14-Nov-2013.) |
Ref | Expression |
---|---|
ffvresb |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fdm 5286 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
2 | dmres 4848 |
. . . . . . 7
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3 | inss2 3302 |
. . . . . . 7
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4 | 2, 3 | eqsstri 3134 |
. . . . . 6
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5 | 1, 4 | eqsstrrdi 3155 |
. . . . 5
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6 | 5 | sselda 3102 |
. . . 4
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7 | fvres 5453 |
. . . . . 6
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8 | 7 | adantl 275 |
. . . . 5
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9 | ffvelrn 5561 |
. . . . 5
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10 | 8, 9 | eqeltrrd 2218 |
. . . 4
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11 | 6, 10 | jca 304 |
. . 3
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12 | 11 | ralrimiva 2508 |
. 2
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13 | simpl 108 |
. . . . . . 7
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14 | 13 | ralimi 2498 |
. . . . . 6
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15 | dfss3 3092 |
. . . . . 6
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16 | 14, 15 | sylibr 133 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
17 | funfn 5161 |
. . . . . 6
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18 | fnssres 5244 |
. . . . . 6
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19 | 17, 18 | sylanb 282 |
. . . . 5
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20 | 16, 19 | sylan2 284 |
. . . 4
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21 | simpr 109 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
22 | 7 | eleq1d 2209 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
23 | 21, 22 | syl5ibr 155 |
. . . . . . 7
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24 | 23 | ralimia 2496 |
. . . . . 6
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25 | 24 | adantl 275 |
. . . . 5
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26 | fnfvrnss 5588 |
. . . . 5
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27 | 20, 25, 26 | syl2anc 409 |
. . . 4
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28 | df-f 5135 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
29 | 20, 27, 28 | sylanbrc 414 |
. . 3
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30 | 29 | ex 114 |
. 2
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31 | 12, 30 | impbid2 142 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-sbc 2914 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-fv 5139 |
This theorem is referenced by: resflem 5592 tfrcl 6269 frecfcllem 6309 lmbr2 12422 lmff 12457 |
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