Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 3187 | . . . . 5 | |
2 | rnss 4582 | . . . . 5 | |
3 | 1, 2 | ax-mp 7 | . . . 4 |
4 | rnxpss 4774 | . . . 4 | |
5 | 3, 4 | sstri 3008 | . . 3 |
6 | eqss 3014 | . . 3 | |
7 | 5, 6 | mpbiran 881 | . 2 |
8 | ssid 3018 | . . . . . . . 8 | |
9 | ssv 3019 | . . . . . . . 8 | |
10 | xpss12 4463 | . . . . . . . 8 | |
11 | 8, 9, 10 | mp2an 416 | . . . . . . 7 |
12 | sslin 3192 | . . . . . . 7 | |
13 | 11, 12 | ax-mp 7 | . . . . . 6 |
14 | df-res 4375 | . . . . . 6 | |
15 | 13, 14 | sseqtr4i 3032 | . . . . 5 |
16 | rnss 4582 | . . . . 5 | |
17 | 15, 16 | ax-mp 7 | . . . 4 |
18 | sstr 3007 | . . . 4 | |
19 | 17, 18 | mpan2 415 | . . 3 |
20 | ssel 2993 | . . . . . . 7 | |
21 | vex 2604 | . . . . . . . 8 | |
22 | 21 | elrn2 4594 | . . . . . . 7 |
23 | 20, 22 | syl6ib 159 | . . . . . 6 |
24 | 23 | ancrd 319 | . . . . 5 |
25 | 21 | elrn2 4594 | . . . . . 6 |
26 | elin 3155 | . . . . . . . 8 | |
27 | opelxp 4392 | . . . . . . . . 9 | |
28 | 27 | anbi2i 444 | . . . . . . . 8 |
29 | 21 | opelres 4635 | . . . . . . . . . 10 |
30 | 29 | anbi1i 445 | . . . . . . . . 9 |
31 | anass 393 | . . . . . . . . 9 | |
32 | 30, 31 | bitr2i 183 | . . . . . . . 8 |
33 | 26, 28, 32 | 3bitri 204 | . . . . . . 7 |
34 | 33 | exbii 1536 | . . . . . 6 |
35 | 19.41v 1823 | . . . . . 6 | |
36 | 25, 34, 35 | 3bitri 204 | . . . . 5 |
37 | 24, 36 | syl6ibr 160 | . . . 4 |
38 | 37 | ssrdv 3005 | . . 3 |
39 | 19, 38 | impbii 124 | . 2 |
40 | 7, 39 | bitr2i 183 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 102 wb 103 wceq 1284 wex 1421 wcel 1433 cvv 2601 cin 2972 wss 2973 cop 3401 cxp 4361 crn 4364 cres 4365 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-br 3786 df-opab 3840 df-xp 4369 df-rel 4370 df-cnv 4371 df-dm 4373 df-rn 4374 df-res 4375 |
This theorem is referenced by: rninxp 4784 |
Copyright terms: Public domain | W3C validator |