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Mirrors > Home > ILE Home > Th. List > fn0 | Unicode version |
Description: A function with empty domain is empty. (Contributed by NM, 15-Apr-1998.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) |
Ref | Expression |
---|---|
fn0 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnrel 4940 |
. . 3
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2 | fndm 4941 |
. . 3
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3 | reldm0 4496 |
. . . 4
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4 | 3 | biimpar 281 |
. . 3
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5 | 1, 2, 4 | syl2anc 391 |
. 2
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6 | fun0 4900 |
. . . 4
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7 | dm0 4492 |
. . . 4
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8 | df-fn 4848 |
. . . 4
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9 | 6, 7, 8 | mpbir2an 848 |
. . 3
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10 | fneq1 4930 |
. . 3
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11 | 9, 10 | mpbiri 157 |
. 2
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12 | 5, 11 | impbii 117 |
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-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-nul 3874 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-fal 1248 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-nul 3219 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-br 3756 df-opab 3810 df-id 4021 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-fun 4847 df-fn 4848 |
This theorem is referenced by: mpt0 4969 f0 5023 f00 5024 f1o00 5104 fo00 5105 tpos0 5830 0fz1 8679 |
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