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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 3297 | . . . . 5 | |
2 | rnss 4769 | . . . . 5 | |
3 | 1, 2 | ax-mp 5 | . . . 4 |
4 | rnxpss 4970 | . . . 4 | |
5 | 3, 4 | sstri 3106 | . . 3 |
6 | eqss 3112 | . . 3 | |
7 | 5, 6 | mpbiran 924 | . 2 |
8 | ssid 3117 | . . . . . . . 8 | |
9 | ssv 3119 | . . . . . . . 8 | |
10 | xpss12 4646 | . . . . . . . 8 | |
11 | 8, 9, 10 | mp2an 422 | . . . . . . 7 |
12 | sslin 3302 | . . . . . . 7 | |
13 | 11, 12 | ax-mp 5 | . . . . . 6 |
14 | df-res 4551 | . . . . . 6 | |
15 | 13, 14 | sseqtrri 3132 | . . . . 5 |
16 | rnss 4769 | . . . . 5 | |
17 | 15, 16 | ax-mp 5 | . . . 4 |
18 | sstr 3105 | . . . 4 | |
19 | 17, 18 | mpan2 421 | . . 3 |
20 | ssel 3091 | . . . . . . 7 | |
21 | vex 2689 | . . . . . . . 8 | |
22 | 21 | elrn2 4781 | . . . . . . 7 |
23 | 20, 22 | syl6ib 160 | . . . . . 6 |
24 | 23 | ancrd 324 | . . . . 5 |
25 | 21 | elrn2 4781 | . . . . . 6 |
26 | elin 3259 | . . . . . . . 8 | |
27 | opelxp 4569 | . . . . . . . . 9 | |
28 | 27 | anbi2i 452 | . . . . . . . 8 |
29 | 21 | opelres 4824 | . . . . . . . . . 10 |
30 | 29 | anbi1i 453 | . . . . . . . . 9 |
31 | anass 398 | . . . . . . . . 9 | |
32 | 30, 31 | bitr2i 184 | . . . . . . . 8 |
33 | 26, 28, 32 | 3bitri 205 | . . . . . . 7 |
34 | 33 | exbii 1584 | . . . . . 6 |
35 | 19.41v 1874 | . . . . . 6 | |
36 | 25, 34, 35 | 3bitri 205 | . . . . 5 |
37 | 24, 36 | syl6ibr 161 | . . . 4 |
38 | 37 | ssrdv 3103 | . . 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 1331 wex 1468 wcel 1480 cvv 2686 cin 3070 wss 3071 cop 3530 cxp 4537 crn 4540 cres 4541 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-opab 3990 df-xp 4545 df-rel 4546 df-cnv 4547 df-dm 4549 df-rn 4550 df-res 4551 |
This theorem is referenced by: rninxp 4982 |
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