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Mirrors > Home > MPE Home > Th. List > funssres | Structured version Visualization version Unicode version |
Description: The restriction of a function to the domain of a subclass equals the subclass. (Contributed by NM, 15-Aug-1994.) |
Ref | Expression |
---|---|
funssres |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssel 3426 |
. . . . . . 7
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2 | vex 3048 |
. . . . . . . . 9
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3 | vex 3048 |
. . . . . . . . 9
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4 | 2, 3 | opeldm 5038 |
. . . . . . . 8
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5 | 4 | a1i 11 |
. . . . . . 7
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6 | 1, 5 | jcad 536 |
. . . . . 6
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7 | 6 | adantl 468 |
. . . . 5
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8 | funeu2 5607 |
. . . . . . . . . . . 12
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9 | 2 | eldm2 5033 |
. . . . . . . . . . . . . 14
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10 | 1 | ancrd 557 |
. . . . . . . . . . . . . . 15
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11 | 10 | eximdv 1764 |
. . . . . . . . . . . . . 14
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12 | 9, 11 | syl5bi 221 |
. . . . . . . . . . . . 13
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13 | 12 | imp 431 |
. . . . . . . . . . . 12
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14 | eupick 2365 |
. . . . . . . . . . . 12
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15 | 8, 13, 14 | syl2an 480 |
. . . . . . . . . . 11
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16 | 15 | exp43 617 |
. . . . . . . . . 10
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17 | 16 | com23 81 |
. . . . . . . . 9
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18 | 17 | imp 431 |
. . . . . . . 8
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19 | 18 | com34 86 |
. . . . . . 7
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20 | 19 | pm2.43d 50 |
. . . . . 6
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21 | 20 | impd 433 |
. . . . 5
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22 | 7, 21 | impbid 194 |
. . . 4
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23 | 3 | opelres 5110 |
. . . 4
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24 | 22, 23 | syl6rbbr 268 |
. . 3
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25 | 24 | alrimivv 1774 |
. 2
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26 | relres 5132 |
. . 3
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27 | funrel 5599 |
. . . 4
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28 | relss 4922 |
. . . 4
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29 | 27, 28 | mpan9 472 |
. . 3
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30 | eqrel 4924 |
. . 3
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31 | 26, 29, 30 | sylancr 669 |
. 2
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32 | 25, 31 | mpbird 236 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1669 ax-4 1682 ax-5 1758 ax-6 1805 ax-7 1851 ax-9 1896 ax-10 1915 ax-11 1920 ax-12 1933 ax-13 2091 ax-ext 2431 ax-sep 4525 ax-nul 4534 ax-pr 4639 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3an 987 df-tru 1447 df-ex 1664 df-nf 1668 df-sb 1798 df-eu 2303 df-mo 2304 df-clab 2438 df-cleq 2444 df-clel 2447 df-nfc 2581 df-ne 2624 df-ral 2742 df-rex 2743 df-rab 2746 df-v 3047 df-dif 3407 df-un 3409 df-in 3411 df-ss 3418 df-nul 3732 df-if 3882 df-sn 3969 df-pr 3971 df-op 3975 df-br 4403 df-opab 4462 df-id 4749 df-xp 4840 df-rel 4841 df-cnv 4842 df-co 4843 df-dm 4844 df-res 4846 df-fun 5584 |
This theorem is referenced by: fun2ssres 5623 funcnvres 5652 funssfv 5880 oprssov 6438 isngp2 21611 dvres3 22868 dvres3a 22869 dchrelbas2 24165 funpsstri 30406 funsseq 30409 f1ssf1 39021 issubgr2 39344 uhgrissubgr 39347 |
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