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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 4938 and dmmptss 4817. (Contributed by Jim Kingdon, 31-Dec-2018.) |
Ref | Expression |
---|---|
dffn5im |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnrel 4997 | . . . 4 | |
2 | dfrel4v 4772 | . . . 4 | |
3 | 1, 2 | sylib 127 | . . 3 |
4 | fnbr 5001 | . . . . . . 7 | |
5 | 4 | ex 108 | . . . . . 6 |
6 | 5 | pm4.71rd 374 | . . . . 5 |
7 | eqcom 2042 | . . . . . . 7 | |
8 | fnbrfvb 5214 | . . . . . . 7 | |
9 | 7, 8 | syl5bb 181 | . . . . . 6 |
10 | 9 | pm5.32da 425 | . . . . 5 |
11 | 6, 10 | bitr4d 180 | . . . 4 |
12 | 11 | opabbidv 3823 | . . 3 |
13 | 3, 12 | eqtrd 2072 | . 2 |
14 | df-mpt 3820 | . 2 | |
15 | 13, 14 | syl6eqr 2090 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wceq 1243 wcel 1393 class class class wbr 3764 copab 3817 cmpt 3818 wrel 4350 wfn 4897 cfv 4902 |
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-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-sbc 2765 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-mpt 3820 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-iota 4867 df-fun 4904 df-fn 4905 df-fv 4910 |
This theorem is referenced by: fnrnfv 5220 feqmptd 5226 dffn5imf 5228 eqfnfv 5265 fndmin 5274 fcompt 5333 resfunexg 5382 eufnfv 5389 fnovim 5609 offveqb 5730 caofinvl 5733 oprabco 5838 df1st2 5840 df2nd2 5841 |
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