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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 2748 |
. . 3
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2 | 1 | anim2i 342 |
. 2
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3 | id 19 |
. . . . 5
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4 | vex 2740 |
. . . . . 6
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5 | 1stexg 6167 |
. . . . . 6
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6 | 4, 5 | ax-mp 5 |
. . . . 5
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7 | 3, 6 | eqeltrrdi 2269 |
. . . 4
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8 | 7 | rexlimivw 2590 |
. . 3
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9 | 8 | anim2i 342 |
. 2
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10 | eldm2g 4823 |
. . . 4
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11 | 10 | adantl 277 |
. . 3
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12 | df-rel 4633 |
. . . . . . . . 9
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13 | ssel 3149 |
. . . . . . . . 9
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14 | 12, 13 | sylbi 121 |
. . . . . . . 8
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15 | 14 | imp 124 |
. . . . . . 7
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16 | op1steq 6179 |
. . . . . . 7
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17 | 15, 16 | syl 14 |
. . . . . 6
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18 | 17 | rexbidva 2474 |
. . . . 5
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19 | 18 | adantr 276 |
. . . 4
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20 | rexcom4 2760 |
. . . . 5
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21 | risset 2505 |
. . . . . 6
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22 | 21 | exbii 1605 |
. . . . 5
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23 | 20, 22 | bitr4i 187 |
. . . 4
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24 | 19, 23 | bitrdi 196 |
. . 3
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25 | 11, 24 | bitr4d 191 |
. 2
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26 | 2, 9, 25 | pm5.21nd 916 |
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-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 |
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-sbc 2963 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-fo 5222 df-fv 5224 df-1st 6140 df-2nd 6141 |
This theorem is referenced by: reldm 6186 |
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