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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 3328 | . . . . 5 | |
2 | rnss 4817 | . . . . 5 | |
3 | 1, 2 | ax-mp 5 | . . . 4 |
4 | rnxpss 5018 | . . . 4 | |
5 | 3, 4 | sstri 3137 | . . 3 |
6 | eqss 3143 | . . 3 | |
7 | 5, 6 | mpbiran 925 | . 2 |
8 | ssid 3148 | . . . . . . . 8 | |
9 | ssv 3150 | . . . . . . . 8 | |
10 | xpss12 4694 | . . . . . . . 8 | |
11 | 8, 9, 10 | mp2an 423 | . . . . . . 7 |
12 | sslin 3333 | . . . . . . 7 | |
13 | 11, 12 | ax-mp 5 | . . . . . 6 |
14 | df-res 4599 | . . . . . 6 | |
15 | 13, 14 | sseqtrri 3163 | . . . . 5 |
16 | rnss 4817 | . . . . 5 | |
17 | 15, 16 | ax-mp 5 | . . . 4 |
18 | sstr 3136 | . . . 4 | |
19 | 17, 18 | mpan2 422 | . . 3 |
20 | ssel 3122 | . . . . . . 7 | |
21 | vex 2715 | . . . . . . . 8 | |
22 | 21 | elrn2 4829 | . . . . . . 7 |
23 | 20, 22 | syl6ib 160 | . . . . . 6 |
24 | 23 | ancrd 324 | . . . . 5 |
25 | 21 | elrn2 4829 | . . . . . 6 |
26 | elin 3290 | . . . . . . . 8 | |
27 | opelxp 4617 | . . . . . . . . 9 | |
28 | 27 | anbi2i 453 | . . . . . . . 8 |
29 | 21 | opelres 4872 | . . . . . . . . . 10 |
30 | 29 | anbi1i 454 | . . . . . . . . 9 |
31 | anass 399 | . . . . . . . . 9 | |
32 | 30, 31 | bitr2i 184 | . . . . . . . 8 |
33 | 26, 28, 32 | 3bitri 205 | . . . . . . 7 |
34 | 33 | exbii 1585 | . . . . . 6 |
35 | 19.41v 1882 | . . . . . 6 | |
36 | 25, 34, 35 | 3bitri 205 | . . . . 5 |
37 | 24, 36 | syl6ibr 161 | . . . 4 |
38 | 37 | ssrdv 3134 | . . 3 |
39 | 19, 38 | impbii 125 | . 2 |
40 | 7, 39 | bitr2i 184 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wceq 1335 wex 1472 wcel 2128 cvv 2712 cin 3101 wss 3102 cop 3563 cxp 4585 crn 4588 cres 4589 |
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-xp 4593 df-rel 4594 df-cnv 4595 df-dm 4597 df-rn 4598 df-res 4599 |
This theorem is referenced by: rninxp 5030 |
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