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Mirrors > Home > ILE Home > Th. List > relelfvdm | Unicode version |
Description: If a function value has a member, the argument belongs to the domain. (Contributed by Jim Kingdon, 22-Jan-2019.) |
Ref | Expression |
---|---|
relelfvdm |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfv 5371 |
. . . . . 6
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2 | exsimpr 1578 |
. . . . . 6
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3 | 1, 2 | sylbi 120 |
. . . . 5
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4 | equsb1 1739 |
. . . . . . . 8
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5 | spsbbi 1796 |
. . . . . . . 8
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6 | 4, 5 | mpbiri 167 |
. . . . . . 7
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7 | nfv 1489 |
. . . . . . . 8
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8 | breq2 3897 |
. . . . . . . 8
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9 | 7, 8 | sbie 1745 |
. . . . . . 7
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10 | 6, 9 | sylib 121 |
. . . . . 6
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11 | 10 | eximi 1560 |
. . . . 5
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12 | 3, 11 | syl 14 |
. . . 4
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13 | 12 | anim2i 337 |
. . 3
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14 | 19.42v 1858 |
. . 3
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15 | 13, 14 | sylibr 133 |
. 2
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16 | releldm 4732 |
. . 3
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17 | 16 | exlimiv 1558 |
. 2
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18 | 15, 17 | syl 14 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-14 1473 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-sep 4004 ax-pow 4056 ax-pr 4089 |
This theorem depends on definitions: df-bi 116 df-3an 945 df-tru 1315 df-nf 1418 df-sb 1717 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ral 2393 df-rex 2394 df-v 2657 df-un 3039 df-in 3041 df-ss 3048 df-pw 3476 df-sn 3497 df-pr 3498 df-op 3500 df-uni 3701 df-br 3894 df-opab 3948 df-xp 4503 df-rel 4504 df-dm 4507 df-iota 5044 df-fv 5087 |
This theorem is referenced by: mptrcl 5455 elfvmptrab1 5467 elmpocl 5920 oprssdmm 6020 mpoxopn0yelv 6087 eluzel2 9226 hashinfom 10410 istopon 12016 istps 12035 topontopn 12040 eltg4i 12060 eltg3 12062 tg1 12064 tg2 12065 tgclb 12070 cldrcl 12107 neiss2 12147 lmrcl 12196 cnprcl2k 12210 metflem 12331 xmetf 12332 ismet2 12336 xmeteq0 12341 xmettri2 12343 xmetpsmet 12351 xmetres2 12361 blfvalps 12367 blex 12369 blvalps 12370 blval 12371 blfps 12391 blf 12392 mopnval 12424 isxms2 12434 comet 12481 |
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