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Mirrors > Home > ILE Home > Th. List > dffn5im | Unicode version |
Description: Representation of a function in terms of its values. The converse holds given the law of the excluded middle; as it is we have most of the converse via funmpt 5250 and dmmptss 5121. (Contributed by Jim Kingdon, 31-Dec-2018.) |
Ref | Expression |
---|---|
dffn5im |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnrel 5310 |
. . . 4
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2 | dfrel4v 5076 |
. . . 4
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3 | 1, 2 | sylib 122 |
. . 3
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4 | fnbr 5314 |
. . . . . . 7
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5 | 4 | ex 115 |
. . . . . 6
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6 | 5 | pm4.71rd 394 |
. . . . 5
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7 | eqcom 2179 |
. . . . . . 7
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8 | fnbrfvb 5552 |
. . . . . . 7
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9 | 7, 8 | bitrid 192 |
. . . . . 6
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10 | 9 | pm5.32da 452 |
. . . . 5
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11 | 6, 10 | bitr4d 191 |
. . . 4
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12 | 11 | opabbidv 4066 |
. . 3
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13 | 3, 12 | eqtrd 2210 |
. 2
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14 | df-mpt 4063 |
. 2
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15 | 13, 14 | eqtr4di 2228 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4206 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-sbc 2963 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4290 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-iota 5174 df-fun 5214 df-fn 5215 df-fv 5220 |
This theorem is referenced by: fnrnfv 5558 feqmptd 5565 dffn5imf 5567 eqfnfv 5609 fndmin 5619 fcompt 5682 resfunexg 5733 eufnfv 5742 fnovim 5977 offveqb 6096 caofinvl 6099 oprabco 6212 df1st2 6214 df2nd2 6215 xpen 6839 cnmpt1st 13448 cnmpt2nd 13449 |
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