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Mirrors > Home > ILE Home > Th. List > funssres | 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 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2729 | . . . . 5 | |
2 | 1 | opelres 4889 | . . . 4 |
3 | ssel 3136 | . . . . . . 7 | |
4 | vex 2729 | . . . . . . . . 9 | |
5 | 4, 1 | opeldm 4807 | . . . . . . . 8 |
6 | 5 | a1i 9 | . . . . . . 7 |
7 | 3, 6 | jcad 305 | . . . . . 6 |
8 | 7 | adantl 275 | . . . . 5 |
9 | funeu2 5214 | . . . . . . . . . . . 12 | |
10 | 4 | eldm2 4802 | . . . . . . . . . . . . . 14 |
11 | 3 | ancrd 324 | . . . . . . . . . . . . . . 15 |
12 | 11 | eximdv 1868 | . . . . . . . . . . . . . 14 |
13 | 10, 12 | syl5bi 151 | . . . . . . . . . . . . 13 |
14 | 13 | imp 123 | . . . . . . . . . . . 12 |
15 | eupick 2093 | . . . . . . . . . . . 12 | |
16 | 9, 14, 15 | syl2an 287 | . . . . . . . . . . 11 |
17 | 16 | exp43 370 | . . . . . . . . . 10 |
18 | 17 | com23 78 | . . . . . . . . 9 |
19 | 18 | imp 123 | . . . . . . . 8 |
20 | 19 | com34 83 | . . . . . . 7 |
21 | 20 | pm2.43d 50 | . . . . . 6 |
22 | 21 | impd 252 | . . . . 5 |
23 | 8, 22 | impbid 128 | . . . 4 |
24 | 2, 23 | bitr4id 198 | . . 3 |
25 | 24 | alrimivv 1863 | . 2 |
26 | relres 4912 | . . 3 | |
27 | funrel 5205 | . . . 4 | |
28 | relss 4691 | . . . 4 | |
29 | 27, 28 | mpan9 279 | . . 3 |
30 | eqrel 4693 | . . 3 | |
31 | 26, 29, 30 | sylancr 411 | . 2 |
32 | 25, 31 | mpbird 166 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wal 1341 wceq 1343 wex 1480 weu 2014 wcel 2136 wss 3116 cop 3579 cdm 4604 cres 4606 wrel 4609 wfun 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 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-14 2139 ax-ext 2147 ax-sep 4100 ax-pow 4153 ax-pr 4187 |
This theorem depends on definitions: df-bi 116 df-3an 970 df-tru 1346 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ral 2449 df-rex 2450 df-v 2728 df-un 3120 df-in 3122 df-ss 3129 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-br 3983 df-opab 4044 df-id 4271 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-res 4616 df-fun 5190 |
This theorem is referenced by: fun2ssres 5231 funcnvres 5261 funssfv 5512 oprssov 5983 |
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