Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 2733 | . . . . 5 | |
2 | 1 | opelres 4896 | . . . 4 |
3 | ssel 3141 | . . . . . . 7 | |
4 | vex 2733 | . . . . . . . . 9 | |
5 | 4, 1 | opeldm 4814 | . . . . . . . 8 |
6 | 5 | a1i 9 | . . . . . . 7 |
7 | 3, 6 | jcad 305 | . . . . . 6 |
8 | 7 | adantl 275 | . . . . 5 |
9 | funeu2 5224 | . . . . . . . . . . . 12 | |
10 | 4 | eldm2 4809 | . . . . . . . . . . . . . 14 |
11 | 3 | ancrd 324 | . . . . . . . . . . . . . . 15 |
12 | 11 | eximdv 1873 | . . . . . . . . . . . . . 14 |
13 | 10, 12 | syl5bi 151 | . . . . . . . . . . . . 13 |
14 | 13 | imp 123 | . . . . . . . . . . . 12 |
15 | eupick 2098 | . . . . . . . . . . . 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 1868 | . 2 |
26 | relres 4919 | . . 3 | |
27 | funrel 5215 | . . . 4 | |
28 | relss 4698 | . . . 4 | |
29 | 27, 28 | mpan9 279 | . . 3 |
30 | eqrel 4700 | . . 3 | |
31 | 26, 29, 30 | sylancr 412 | . 2 |
32 | 25, 31 | mpbird 166 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wal 1346 wceq 1348 wex 1485 weu 2019 wcel 2141 wss 3121 cop 3586 cdm 4611 cres 4613 wrel 4616 wfun 5192 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-br 3990 df-opab 4051 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-res 4623 df-fun 5200 |
This theorem is referenced by: fun2ssres 5241 funcnvres 5271 funssfv 5522 oprssov 5994 |
Copyright terms: Public domain | W3C validator |