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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 109 | . . . . . . . 8 | |
2 | 1 | snssd 3665 | . . . . . . 7 |
3 | df-ss 3084 | . . . . . . 7 | |
4 | 2, 3 | sylib 121 | . . . . . 6 |
5 | 4 | imaeq2d 4881 | . . . . 5 |
6 | 5 | ineq1d 3276 | . . . 4 |
7 | imass2 4915 | . . . . . . 7 | |
8 | 2, 7 | syl 14 | . . . . . 6 |
9 | simpl 108 | . . . . . 6 | |
10 | 8, 9 | sstrd 3107 | . . . . 5 |
11 | df-ss 3084 | . . . . 5 | |
12 | 10, 11 | sylib 121 | . . . 4 |
13 | 6, 12 | eqtr2d 2173 | . . 3 |
14 | imainrect 4984 | . . 3 | |
15 | 13, 14 | syl6eqr 2190 | . 2 |
16 | df-ec 6431 | . 2 | |
17 | df-ec 6431 | . 2 | |
18 | 15, 16, 17 | 3eqtr4g 2197 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 cin 3070 wss 3071 csn 3527 cxp 4537 cima 4542 cec 6427 |
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 df-ima 4552 df-ec 6431 |
This theorem is referenced by: qsinxp 6505 nqnq0pi 7246 |
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