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Mirrors > Home > ILE Home > Th. List > 2elresin | Unicode version |
Description: Membership in two functions restricted by each other's domain. (Contributed by NM, 8-Aug-1994.) |
Ref | Expression |
---|---|
2elresin |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnop 5182 |
. . . . . . . 8
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2 | fnop 5182 |
. . . . . . . 8
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3 | 1, 2 | anim12i 334 |
. . . . . . 7
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4 | 3 | an4s 560 |
. . . . . 6
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5 | elin 3223 |
. . . . . 6
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6 | 4, 5 | sylibr 133 |
. . . . 5
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7 | vex 2658 |
. . . . . . . 8
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8 | 7 | opres 4784 |
. . . . . . 7
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9 | vex 2658 |
. . . . . . . 8
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10 | 9 | opres 4784 |
. . . . . . 7
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11 | 8, 10 | anbi12d 462 |
. . . . . 6
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12 | 11 | biimprd 157 |
. . . . 5
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13 | 6, 12 | syl 14 |
. . . 4
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14 | 13 | ex 114 |
. . 3
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15 | 14 | pm2.43d 50 |
. 2
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16 | resss 4799 |
. . . 4
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17 | 16 | sseli 3057 |
. . 3
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18 | resss 4799 |
. . . 4
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19 | 18 | sseli 3057 |
. . 3
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20 | 17, 19 | anim12i 334 |
. 2
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21 | 15, 20 | impbid1 141 |
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 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-14 1473 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-sep 4004 ax-pow 4056 ax-pr 4089 |
This theorem depends on definitions: df-bi 116 df-3an 945 df-tru 1315 df-nf 1418 df-sb 1717 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ral 2393 df-rex 2394 df-v 2657 df-un 3039 df-in 3041 df-ss 3048 df-pw 3476 df-sn 3497 df-pr 3498 df-op 3500 df-br 3894 df-opab 3948 df-xp 4503 df-rel 4504 df-dm 4507 df-res 4509 df-fun 5081 df-fn 5082 |
This theorem is referenced by: (None) |
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