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Mirrors > Home > ILE Home > Th. List > ssrnres | Unicode version |
Description: Subset of the range of a restriction. (Contributed by NM, 16-Jan-2006.) |
Ref | Expression |
---|---|
ssrnres |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inss2 3158 | . . . . 5 | |
2 | rnss 4564 | . . . . 5 | |
3 | 1, 2 | ax-mp 7 | . . . 4 |
4 | rnxpss 4754 | . . . 4 | |
5 | 3, 4 | sstri 2954 | . . 3 |
6 | eqss 2960 | . . 3 | |
7 | 5, 6 | mpbiran 847 | . 2 |
8 | ssid 2964 | . . . . . . . 8 | |
9 | ssv 2965 | . . . . . . . 8 | |
10 | xpss12 4445 | . . . . . . . 8 | |
11 | 8, 9, 10 | mp2an 402 | . . . . . . 7 |
12 | sslin 3163 | . . . . . . 7 | |
13 | 11, 12 | ax-mp 7 | . . . . . 6 |
14 | df-res 4357 | . . . . . 6 | |
15 | 13, 14 | sseqtr4i 2978 | . . . . 5 |
16 | rnss 4564 | . . . . 5 | |
17 | 15, 16 | ax-mp 7 | . . . 4 |
18 | sstr 2953 | . . . 4 | |
19 | 17, 18 | mpan2 401 | . . 3 |
20 | ssel 2939 | . . . . . . 7 | |
21 | vex 2560 | . . . . . . . 8 | |
22 | 21 | elrn2 4576 | . . . . . . 7 |
23 | 20, 22 | syl6ib 150 | . . . . . 6 |
24 | 23 | ancrd 309 | . . . . 5 |
25 | 21 | elrn2 4576 | . . . . . 6 |
26 | elin 3126 | . . . . . . . 8 | |
27 | opelxp 4374 | . . . . . . . . 9 | |
28 | 27 | anbi2i 430 | . . . . . . . 8 |
29 | 21 | opelres 4617 | . . . . . . . . . 10 |
30 | 29 | anbi1i 431 | . . . . . . . . 9 |
31 | anass 381 | . . . . . . . . 9 | |
32 | 30, 31 | bitr2i 174 | . . . . . . . 8 |
33 | 26, 28, 32 | 3bitri 195 | . . . . . . 7 |
34 | 33 | exbii 1496 | . . . . . 6 |
35 | 19.41v 1782 | . . . . . 6 | |
36 | 25, 34, 35 | 3bitri 195 | . . . . 5 |
37 | 24, 36 | syl6ibr 151 | . . . 4 |
38 | 37 | ssrdv 2951 | . . 3 |
39 | 19, 38 | impbii 117 | . 2 |
40 | 7, 39 | bitr2i 174 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 97 wb 98 wceq 1243 wex 1381 wcel 1393 cvv 2557 cin 2916 wss 2917 cop 3378 cxp 4343 crn 4346 cres 4347 |
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-xp 4351 df-rel 4352 df-cnv 4353 df-dm 4355 df-rn 4356 df-res 4357 |
This theorem is referenced by: rninxp 4764 |
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