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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 4734 |
. . . . . . 7
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2 | 1 | elin2 3188 |
. . . . . 6
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3 | funres 5055 |
. . . . . . . . 9
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4 | fvelrn 5430 |
. . . . . . . . 9
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5 | 3, 4 | sylan 277 |
. . . . . . . 8
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6 | fvres 5329 |
. . . . . . . . . 10
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7 | 6 | eleq1d 2156 |
. . . . . . . . 9
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8 | df-ima 4451 |
. . . . . . . . . 10
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9 | 8 | eleq2i 2154 |
. . . . . . . . 9
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10 | 7, 9 | syl6rbbr 197 |
. . . . . . . 8
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11 | 5, 10 | syl5ibrcom 155 |
. . . . . . 7
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12 | 11 | ex 113 |
. . . . . 6
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13 | 2, 12 | syl5bir 151 |
. . . . 5
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14 | 13 | expd 254 |
. . . 4
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15 | 14 | com12 30 |
. . 3
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16 | 15 | impd 251 |
. 2
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17 | 16 | pm2.43b 51 |
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 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3957 ax-pow 4009 ax-pr 4036 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ral 2364 df-rex 2365 df-v 2621 df-sbc 2841 df-un 3003 df-in 3005 df-ss 3012 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 df-uni 3654 df-br 3846 df-opab 3900 df-id 4120 df-xp 4444 df-rel 4445 df-cnv 4446 df-co 4447 df-dm 4448 df-rn 4449 df-res 4450 df-ima 4451 df-iota 4980 df-fun 5017 df-fn 5018 df-fv 5023 |
This theorem is referenced by: funfvima2 5527 fiintim 6637 djuun 6758 |
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