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Mirrors > Home > ILE Home > Th. List > funfvima | Unicode version |
Description: A function's value in a preimage belongs to the image. (Contributed by NM, 23-Sep-2003.) |
Ref | Expression |
---|---|
funfvima |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmres 4796 |
. . . . . . 7
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2 | 1 | elin2 3228 |
. . . . . 6
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3 | funres 5120 |
. . . . . . . . 9
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4 | fvelrn 5503 |
. . . . . . . . 9
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5 | 3, 4 | sylan 279 |
. . . . . . . 8
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6 | fvres 5397 |
. . . . . . . . . 10
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7 | 6 | eleq1d 2181 |
. . . . . . . . 9
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8 | df-ima 4510 |
. . . . . . . . . 10
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9 | 8 | eleq2i 2179 |
. . . . . . . . 9
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10 | 7, 9 | syl6rbbr 198 |
. . . . . . . 8
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11 | 5, 10 | syl5ibrcom 156 |
. . . . . . 7
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12 | 11 | ex 114 |
. . . . . 6
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13 | 2, 12 | syl5bir 152 |
. . . . 5
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14 | 13 | expd 256 |
. . . 4
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15 | 14 | com12 30 |
. . 3
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16 | 15 | impd 252 |
. 2
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17 | 16 | pm2.43b 52 |
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-eu 1976 df-mo 1977 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ral 2393 df-rex 2394 df-v 2657 df-sbc 2877 df-un 3039 df-in 3041 df-ss 3048 df-pw 3476 df-sn 3497 df-pr 3498 df-op 3500 df-uni 3701 df-br 3894 df-opab 3948 df-id 4173 df-xp 4503 df-rel 4504 df-cnv 4505 df-co 4506 df-dm 4507 df-rn 4508 df-res 4509 df-ima 4510 df-iota 5044 df-fun 5081 df-fn 5082 df-fv 5087 |
This theorem is referenced by: funfvima2 5602 fiintim 6767 caseinl 6925 caseinr 6926 ctssdccl 6945 ennnfonelemex 11765 ctinfomlemom 11778 txcnp 12275 |
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