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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 3713 | . . . . . . 7 |
3 | df-ss 3125 | . . . . . . 7 | |
4 | 2, 3 | sylib 121 | . . . . . 6 |
5 | 4 | imaeq2d 4941 | . . . . 5 |
6 | 5 | ineq1d 3318 | . . . 4 |
7 | imass2 4975 | . . . . . . 7 | |
8 | 2, 7 | syl 14 | . . . . . 6 |
9 | simpl 108 | . . . . . 6 | |
10 | 8, 9 | sstrd 3148 | . . . . 5 |
11 | df-ss 3125 | . . . . 5 | |
12 | 10, 11 | sylib 121 | . . . 4 |
13 | 6, 12 | eqtr2d 2198 | . . 3 |
14 | imainrect 5044 | . . 3 | |
15 | 13, 14 | eqtr4di 2215 | . 2 |
16 | df-ec 6495 | . 2 | |
17 | df-ec 6495 | . 2 | |
18 | 15, 16, 17 | 3eqtr4g 2222 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1342 wcel 2135 cin 3111 wss 3112 csn 3571 cxp 4597 cima 4602 cec 6491 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4095 ax-pow 4148 ax-pr 4182 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2724 df-un 3116 df-in 3118 df-ss 3125 df-pw 3556 df-sn 3577 df-pr 3578 df-op 3580 df-br 3978 df-opab 4039 df-xp 4605 df-rel 4606 df-cnv 4607 df-dm 4609 df-rn 4610 df-res 4611 df-ima 4612 df-ec 6495 |
This theorem is referenced by: qsinxp 6569 nqnq0pi 7371 |
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