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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 2715 | . . . . 5 | |
2 | 1 | opelres 4872 | . . . 4 |
3 | ssel 3122 | . . . . . . 7 | |
4 | vex 2715 | . . . . . . . . 9 | |
5 | 4, 1 | opeldm 4790 | . . . . . . . 8 |
6 | 5 | a1i 9 | . . . . . . 7 |
7 | 3, 6 | jcad 305 | . . . . . 6 |
8 | 7 | adantl 275 | . . . . 5 |
9 | funeu2 5197 | . . . . . . . . . . . 12 | |
10 | 4 | eldm2 4785 | . . . . . . . . . . . . . 14 |
11 | 3 | ancrd 324 | . . . . . . . . . . . . . . 15 |
12 | 11 | eximdv 1860 | . . . . . . . . . . . . . 14 |
13 | 10, 12 | syl5bi 151 | . . . . . . . . . . . . 13 |
14 | 13 | imp 123 | . . . . . . . . . . . 12 |
15 | eupick 2085 | . . . . . . . . . . . 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 1855 | . 2 |
26 | relres 4895 | . . 3 | |
27 | funrel 5188 | . . . 4 | |
28 | relss 4674 | . . . 4 | |
29 | 27, 28 | mpan9 279 | . . 3 |
30 | eqrel 4676 | . . 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 1333 wceq 1335 wex 1472 weu 2006 wcel 2128 wss 3102 cop 3563 cdm 4587 cres 4589 wrel 4592 wfun 5165 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4083 ax-pow 4136 ax-pr 4170 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-br 3967 df-opab 4027 df-id 4254 df-xp 4593 df-rel 4594 df-cnv 4595 df-co 4596 df-dm 4597 df-res 4599 df-fun 5173 |
This theorem is referenced by: fun2ssres 5214 funcnvres 5244 funssfv 5495 oprssov 5963 |
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