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Mirrors > Home > ILE Home > Th. List > elreldm | Unicode version |
Description: The first member of an ordered pair in a relation belongs to the domain of the relation. (Contributed by NM, 28-Jul-2004.) |
Ref | Expression |
---|---|
elreldm |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rel 4352 |
. . . . 5
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2 | ssel 2939 |
. . . . 5
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3 | 1, 2 | sylbi 114 |
. . . 4
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4 | elvv 4402 |
. . . 4
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5 | 3, 4 | syl6ib 150 |
. . 3
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6 | eleq1 2100 |
. . . . . 6
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7 | vex 2560 |
. . . . . . 7
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8 | vex 2560 |
. . . . . . 7
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9 | 7, 8 | opeldm 4538 |
. . . . . 6
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10 | 6, 9 | syl6bi 152 |
. . . . 5
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11 | inteq 3618 |
. . . . . . . 8
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12 | 11 | inteqd 3620 |
. . . . . . 7
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13 | 7, 8 | op1stb 4209 |
. . . . . . 7
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14 | 12, 13 | syl6eq 2088 |
. . . . . 6
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15 | 14 | eleq1d 2106 |
. . . . 5
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16 | 10, 15 | sylibrd 158 |
. . . 4
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17 | 16 | exlimivv 1776 |
. . 3
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18 | 5, 17 | syli 33 |
. 2
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19 | 18 | imp 115 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-int 3616 df-br 3765 df-opab 3819 df-xp 4351 df-rel 4352 df-dm 4355 |
This theorem is referenced by: 1stdm 5808 fundmen 6286 |
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