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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 2939 | . . . . . . 7 | |
2 | vex 2560 | . . . . . . . . 9 | |
3 | vex 2560 | . . . . . . . . 9 | |
4 | 2, 3 | opeldm 4538 | . . . . . . . 8 |
5 | 4 | a1i 9 | . . . . . . 7 |
6 | 1, 5 | jcad 291 | . . . . . 6 |
7 | 6 | adantl 262 | . . . . 5 |
8 | funeu2 4927 | . . . . . . . . . . . 12 | |
9 | 2 | eldm2 4533 | . . . . . . . . . . . . . 14 |
10 | 1 | ancrd 309 | . . . . . . . . . . . . . . 15 |
11 | 10 | eximdv 1760 | . . . . . . . . . . . . . 14 |
12 | 9, 11 | syl5bi 141 | . . . . . . . . . . . . 13 |
13 | 12 | imp 115 | . . . . . . . . . . . 12 |
14 | eupick 1979 | . . . . . . . . . . . 12 | |
15 | 8, 13, 14 | syl2an 273 | . . . . . . . . . . 11 |
16 | 15 | exp43 354 | . . . . . . . . . 10 |
17 | 16 | com23 72 | . . . . . . . . 9 |
18 | 17 | imp 115 | . . . . . . . 8 |
19 | 18 | com34 77 | . . . . . . 7 |
20 | 19 | pm2.43d 44 | . . . . . 6 |
21 | 20 | impd 242 | . . . . 5 |
22 | 7, 21 | impbid 120 | . . . 4 |
23 | 3 | opelres 4617 | . . . 4 |
24 | 22, 23 | syl6rbbr 188 | . . 3 |
25 | 24 | alrimivv 1755 | . 2 |
26 | relres 4639 | . . 3 | |
27 | funrel 4919 | . . . 4 | |
28 | relss 4427 | . . . 4 | |
29 | 27, 28 | mpan9 265 | . . 3 |
30 | eqrel 4429 | . . 3 | |
31 | 26, 29, 30 | sylancr 393 | . 2 |
32 | 25, 31 | mpbird 156 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wal 1241 wceq 1243 wex 1381 wcel 1393 weu 1900 wss 2917 cop 3378 cdm 4345 cres 4347 wrel 4350 wfun 4896 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-res 4357 df-fun 4904 |
This theorem is referenced by: fun2ssres 4943 funcnvres 4972 funssfv 5199 oprssov 5642 |
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