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Mirrors > Home > ILE Home > Th. List > ecinxp | Unicode version |
Description: Restrict the relation in an equivalence class to a base set. (Contributed by Mario Carneiro, 10-Jul-2015.) |
Ref | Expression |
---|---|
ecinxp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 103 | . . . . . . . 8 | |
2 | 1 | snssd 3509 | . . . . . . 7 |
3 | df-ss 2931 | . . . . . . 7 | |
4 | 2, 3 | sylib 127 | . . . . . 6 |
5 | 4 | imaeq2d 4668 | . . . . 5 |
6 | 5 | ineq1d 3137 | . . . 4 |
7 | imass2 4701 | . . . . . . 7 | |
8 | 2, 7 | syl 14 | . . . . . 6 |
9 | simpl 102 | . . . . . 6 | |
10 | 8, 9 | sstrd 2955 | . . . . 5 |
11 | df-ss 2931 | . . . . 5 | |
12 | 10, 11 | sylib 127 | . . . 4 |
13 | 6, 12 | eqtr2d 2073 | . . 3 |
14 | imainrect 4766 | . . 3 | |
15 | 13, 14 | syl6eqr 2090 | . 2 |
16 | df-ec 6108 | . 2 | |
17 | df-ec 6108 | . 2 | |
18 | 15, 16, 17 | 3eqtr4g 2097 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wceq 1243 wcel 1393 cin 2916 wss 2917 csn 3375 cxp 4343 cima 4348 cec 6104 |
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-eu 1903 df-mo 1904 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-cnv 4353 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-ec 6108 |
This theorem is referenced by: qsinxp 6182 nqnq0pi 6536 |
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