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Mirrors > Home > ILE Home > Th. List > releldm2 | Unicode version |
Description: Two ways of expressing membership in the domain of a relation. (Contributed by NM, 22-Sep-2013.) |
Ref | Expression |
---|---|
releldm2 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2700 |
. . 3
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2 | 1 | anim2i 340 |
. 2
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3 | id 19 |
. . . . 5
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4 | vex 2692 |
. . . . . 6
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5 | 1stexg 6073 |
. . . . . 6
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6 | 4, 5 | ax-mp 5 |
. . . . 5
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7 | 3, 6 | eqeltrrdi 2232 |
. . . 4
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8 | 7 | rexlimivw 2548 |
. . 3
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9 | 8 | anim2i 340 |
. 2
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10 | eldm2g 4743 |
. . . 4
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11 | 10 | adantl 275 |
. . 3
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12 | df-rel 4554 |
. . . . . . . . 9
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13 | ssel 3096 |
. . . . . . . . 9
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14 | 12, 13 | sylbi 120 |
. . . . . . . 8
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15 | 14 | imp 123 |
. . . . . . 7
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16 | op1steq 6085 |
. . . . . . 7
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17 | 15, 16 | syl 14 |
. . . . . 6
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18 | 17 | rexbidva 2435 |
. . . . 5
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19 | 18 | adantr 274 |
. . . 4
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20 | rexcom4 2712 |
. . . . 5
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21 | risset 2466 |
. . . . . 6
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22 | 21 | exbii 1585 |
. . . . 5
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23 | 20, 22 | bitr4i 186 |
. . . 4
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24 | 19, 23 | syl6bb 195 |
. . 3
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25 | 11, 24 | bitr4d 190 |
. 2
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26 | 2, 9, 25 | pm5.21nd 902 |
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 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-sbc 2914 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-fo 5137 df-fv 5139 df-1st 6046 df-2nd 6047 |
This theorem is referenced by: reldm 6092 |
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