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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 110 |
. . . . . . . 8
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2 | 1 | snssd 3737 |
. . . . . . 7
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3 | df-ss 3142 |
. . . . . . 7
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4 | 2, 3 | sylib 122 |
. . . . . 6
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5 | 4 | imaeq2d 4970 |
. . . . 5
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6 | 5 | ineq1d 3335 |
. . . 4
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7 | imass2 5004 |
. . . . . . 7
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8 | 2, 7 | syl 14 |
. . . . . 6
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9 | simpl 109 |
. . . . . 6
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10 | 8, 9 | sstrd 3165 |
. . . . 5
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11 | df-ss 3142 |
. . . . 5
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12 | 10, 11 | sylib 122 |
. . . 4
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13 | 6, 12 | eqtr2d 2211 |
. . 3
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14 | imainrect 5074 |
. . 3
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15 | 13, 14 | eqtr4di 2228 |
. 2
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16 | df-ec 6536 |
. 2
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17 | df-ec 6536 |
. 2
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18 | 15, 16, 17 | 3eqtr4g 2235 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-br 4004 df-opab 4065 df-xp 4632 df-rel 4633 df-cnv 4634 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-ec 6536 |
This theorem is referenced by: qsinxp 6610 nqnq0pi 7436 |
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