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Mirrors > Home > MPE Home > Th. List > resdmres | Structured version Visualization version Unicode version |
Description: Restriction to the domain of a restriction. (Contributed by NM, 8-Apr-2007.) |
Ref | Expression |
---|---|
resdmres |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | in12 3634 |
. . . 4
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2 | df-res 4851 |
. . . . . 6
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3 | resdm2 5332 |
. . . . . 6
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4 | 2, 3 | eqtr3i 2495 |
. . . . 5
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5 | 4 | ineq2i 3622 |
. . . 4
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6 | incom 3616 |
. . . 4
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7 | 1, 5, 6 | 3eqtri 2497 |
. . 3
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8 | df-res 4851 |
. . . 4
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9 | dmres 5131 |
. . . . . . 7
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10 | 9 | xpeq1i 4859 |
. . . . . 6
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11 | xpindir 4974 |
. . . . . 6
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12 | 10, 11 | eqtri 2493 |
. . . . 5
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13 | 12 | ineq2i 3622 |
. . . 4
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14 | 8, 13 | eqtri 2493 |
. . 3
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15 | df-res 4851 |
. . 3
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16 | 7, 14, 15 | 3eqtr4i 2503 |
. 2
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17 | rescnvcnv 5305 |
. 2
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18 | 16, 17 | eqtri 2493 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1677 ax-4 1690 ax-5 1766 ax-6 1813 ax-7 1859 ax-9 1913 ax-10 1932 ax-11 1937 ax-12 1950 ax-13 2104 ax-ext 2451 ax-sep 4518 ax-nul 4527 ax-pr 4639 |
This theorem depends on definitions: df-bi 190 df-or 377 df-an 378 df-3an 1009 df-tru 1455 df-ex 1672 df-nf 1676 df-sb 1806 df-eu 2323 df-mo 2324 df-clab 2458 df-cleq 2464 df-clel 2467 df-nfc 2601 df-ne 2643 df-ral 2761 df-rex 2762 df-rab 2765 df-v 3033 df-dif 3393 df-un 3395 df-in 3397 df-ss 3404 df-nul 3723 df-if 3873 df-sn 3960 df-pr 3962 df-op 3966 df-br 4396 df-opab 4455 df-xp 4845 df-rel 4846 df-cnv 4847 df-dm 4849 df-rn 4850 df-res 4851 |
This theorem is referenced by: imadmres 5334 lindfres 19458 imacmp 20489 metreslem 21455 volres 22560 uhgrares 25114 umgrares 25130 usgrares 25175 resresdm 39154 uhgres 40199 |
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