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Mirrors > Home > ILE Home > Th. List > 2elresin | Unicode version |
Description: Membership in two functions restricted by each other's domain. (Contributed by NM, 8-Aug-1994.) |
Ref | Expression |
---|---|
2elresin |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnop 5002 | . . . . . . . 8 | |
2 | fnop 5002 | . . . . . . . 8 | |
3 | 1, 2 | anim12i 321 | . . . . . . 7 |
4 | 3 | an4s 522 | . . . . . 6 |
5 | elin 3126 | . . . . . 6 | |
6 | 4, 5 | sylibr 137 | . . . . 5 |
7 | vex 2560 | . . . . . . . 8 | |
8 | 7 | opres 4621 | . . . . . . 7 |
9 | vex 2560 | . . . . . . . 8 | |
10 | 9 | opres 4621 | . . . . . . 7 |
11 | 8, 10 | anbi12d 442 | . . . . . 6 |
12 | 11 | biimprd 147 | . . . . 5 |
13 | 6, 12 | syl 14 | . . . 4 |
14 | 13 | ex 108 | . . 3 |
15 | 14 | pm2.43d 44 | . 2 |
16 | resss 4635 | . . . 4 | |
17 | 16 | sseli 2941 | . . 3 |
18 | resss 4635 | . . . 4 | |
19 | 18 | sseli 2941 | . . 3 |
20 | 17, 19 | anim12i 321 | . 2 |
21 | 15, 20 | impbid1 130 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wcel 1393 cin 2916 cop 3378 cres 4347 wfn 4897 |
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-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-dm 4355 df-res 4357 df-fun 4904 df-fn 4905 |
This theorem is referenced by: (None) |
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