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Mirrors > Home > ILE Home > Th. List > fco | Unicode version |
Description: Composition of two mappings. (Contributed by NM, 29-Aug-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) |
Ref | Expression |
---|---|
fco |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-f 4906 | . . 3 | |
2 | df-f 4906 | . . 3 | |
3 | fnco 5007 | . . . . . . 7 | |
4 | 3 | 3expib 1107 | . . . . . 6 |
5 | 4 | adantr 261 | . . . . 5 |
6 | rncoss 4602 | . . . . . . 7 | |
7 | sstr 2953 | . . . . . . 7 | |
8 | 6, 7 | mpan 400 | . . . . . 6 |
9 | 8 | adantl 262 | . . . . 5 |
10 | 5, 9 | jctird 300 | . . . 4 |
11 | 10 | imp 115 | . . 3 |
12 | 1, 2, 11 | syl2anb 275 | . 2 |
13 | df-f 4906 | . 2 | |
14 | 12, 13 | sylibr 137 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wss 2917 crn 4346 ccom 4349 wfn 4897 wf 4898 |
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-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-fun 4904 df-fn 4905 df-f 4906 |
This theorem is referenced by: fco2 5057 f1co 5101 foco 5116 ialgcvg 9887 |
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