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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 | ssel 3091 | . . . . . . 7 | |
2 | vex 2689 | . . . . . . . . 9 | |
3 | vex 2689 | . . . . . . . . 9 | |
4 | 2, 3 | opeldm 4742 | . . . . . . . 8 |
5 | 4 | a1i 9 | . . . . . . 7 |
6 | 1, 5 | jcad 305 | . . . . . 6 |
7 | 6 | adantl 275 | . . . . 5 |
8 | funeu2 5149 | . . . . . . . . . . . 12 | |
9 | 2 | eldm2 4737 | . . . . . . . . . . . . . 14 |
10 | 1 | ancrd 324 | . . . . . . . . . . . . . . 15 |
11 | 10 | eximdv 1852 | . . . . . . . . . . . . . 14 |
12 | 9, 11 | syl5bi 151 | . . . . . . . . . . . . 13 |
13 | 12 | imp 123 | . . . . . . . . . . . 12 |
14 | eupick 2078 | . . . . . . . . . . . 12 | |
15 | 8, 13, 14 | syl2an 287 | . . . . . . . . . . 11 |
16 | 15 | exp43 369 | . . . . . . . . . 10 |
17 | 16 | com23 78 | . . . . . . . . 9 |
18 | 17 | imp 123 | . . . . . . . 8 |
19 | 18 | com34 83 | . . . . . . 7 |
20 | 19 | pm2.43d 50 | . . . . . 6 |
21 | 20 | impd 252 | . . . . 5 |
22 | 7, 21 | impbid 128 | . . . 4 |
23 | 3 | opelres 4824 | . . . 4 |
24 | 22, 23 | syl6rbbr 198 | . . 3 |
25 | 24 | alrimivv 1847 | . 2 |
26 | relres 4847 | . . 3 | |
27 | funrel 5140 | . . . 4 | |
28 | relss 4626 | . . . 4 | |
29 | 27, 28 | mpan9 279 | . . 3 |
30 | eqrel 4628 | . . 3 | |
31 | 26, 29, 30 | sylancr 410 | . 2 |
32 | 25, 31 | mpbird 166 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wal 1329 wceq 1331 wex 1468 wcel 1480 weu 1999 wss 3071 cop 3530 cdm 4539 cres 4541 wrel 4544 wfun 5117 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-res 4551 df-fun 5125 |
This theorem is referenced by: fun2ssres 5166 funcnvres 5196 funssfv 5447 oprssov 5912 |
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