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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 3348 | . . . . 5 | |
2 | rnss 4841 | . . . . 5 | |
3 | 1, 2 | ax-mp 5 | . . . 4 |
4 | rnxpss 5042 | . . . 4 | |
5 | 3, 4 | sstri 3156 | . . 3 |
6 | eqss 3162 | . . 3 | |
7 | 5, 6 | mpbiran 935 | . 2 |
8 | ssid 3167 | . . . . . . . 8 | |
9 | ssv 3169 | . . . . . . . 8 | |
10 | xpss12 4718 | . . . . . . . 8 | |
11 | 8, 9, 10 | mp2an 424 | . . . . . . 7 |
12 | sslin 3353 | . . . . . . 7 | |
13 | 11, 12 | ax-mp 5 | . . . . . 6 |
14 | df-res 4623 | . . . . . 6 | |
15 | 13, 14 | sseqtrri 3182 | . . . . 5 |
16 | rnss 4841 | . . . . 5 | |
17 | 15, 16 | ax-mp 5 | . . . 4 |
18 | sstr 3155 | . . . 4 | |
19 | 17, 18 | mpan2 423 | . . 3 |
20 | ssel 3141 | . . . . . . 7 | |
21 | vex 2733 | . . . . . . . 8 | |
22 | 21 | elrn2 4853 | . . . . . . 7 |
23 | 20, 22 | syl6ib 160 | . . . . . 6 |
24 | 23 | ancrd 324 | . . . . 5 |
25 | 21 | elrn2 4853 | . . . . . 6 |
26 | elin 3310 | . . . . . . . 8 | |
27 | opelxp 4641 | . . . . . . . . 9 | |
28 | 27 | anbi2i 454 | . . . . . . . 8 |
29 | 21 | opelres 4896 | . . . . . . . . . 10 |
30 | 29 | anbi1i 455 | . . . . . . . . 9 |
31 | anass 399 | . . . . . . . . 9 | |
32 | 30, 31 | bitr2i 184 | . . . . . . . 8 |
33 | 26, 28, 32 | 3bitri 205 | . . . . . . 7 |
34 | 33 | exbii 1598 | . . . . . 6 |
35 | 19.41v 1895 | . . . . . 6 | |
36 | 25, 34, 35 | 3bitri 205 | . . . . 5 |
37 | 24, 36 | syl6ibr 161 | . . . 4 |
38 | 37 | ssrdv 3153 | . . 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 1348 wex 1485 wcel 2141 cvv 2730 cin 3120 wss 3121 cop 3586 cxp 4609 crn 4612 cres 4613 |
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-xp 4617 df-rel 4618 df-cnv 4619 df-dm 4621 df-rn 4622 df-res 4623 |
This theorem is referenced by: rninxp 5054 |
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