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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 5073 |
. . . . . . . 8
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2 | fnop 5073 |
. . . . . . . 8
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3 | 1, 2 | anim12i 331 |
. . . . . . 7
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4 | 3 | an4s 553 |
. . . . . 6
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5 | elin 3169 |
. . . . . 6
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6 | 4, 5 | sylibr 132 |
. . . . 5
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7 | vex 2617 |
. . . . . . . 8
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8 | 7 | opres 4683 |
. . . . . . 7
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9 | vex 2617 |
. . . . . . . 8
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10 | 9 | opres 4683 |
. . . . . . 7
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11 | 8, 10 | anbi12d 457 |
. . . . . 6
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12 | 11 | biimprd 156 |
. . . . 5
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13 | 6, 12 | syl 14 |
. . . 4
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14 | 13 | ex 113 |
. . 3
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15 | 14 | pm2.43d 49 |
. 2
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16 | resss 4697 |
. . . 4
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17 | 16 | sseli 3008 |
. . 3
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18 | resss 4697 |
. . . 4
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19 | 18 | sseli 3008 |
. . 3
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20 | 17, 19 | anim12i 331 |
. 2
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21 | 15, 20 | impbid1 140 |
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 104 ax-ia2 105 ax-ia3 106 ax-io 663 ax-5 1379 ax-7 1380 ax-gen 1381 ax-ie1 1425 ax-ie2 1426 ax-8 1438 ax-10 1439 ax-11 1440 ax-i12 1441 ax-bndl 1442 ax-4 1443 ax-14 1448 ax-17 1462 ax-i9 1466 ax-ial 1470 ax-i5r 1471 ax-ext 2067 ax-sep 3925 ax-pow 3977 ax-pr 4003 |
This theorem depends on definitions: df-bi 115 df-3an 924 df-tru 1290 df-nf 1393 df-sb 1690 df-clab 2072 df-cleq 2078 df-clel 2081 df-nfc 2214 df-ral 2360 df-rex 2361 df-v 2616 df-un 2990 df-in 2992 df-ss 2999 df-pw 3411 df-sn 3431 df-pr 3432 df-op 3434 df-br 3815 df-opab 3869 df-xp 4410 df-rel 4411 df-dm 4414 df-res 4416 df-fun 4974 df-fn 4975 |
This theorem is referenced by: (None) |
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