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Mirrors > Home > ILE Home > Th. List > funco | Unicode version |
Description: The composition of two functions is a function. Exercise 29 of [TakeutiZaring] p. 25. (Contributed by NM, 26-Jan-1997.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) |
Ref | Expression |
---|---|
funco |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmcoss 4601 | . . . . 5 | |
2 | funmo 4917 | . . . . . . . . . 10 | |
3 | 2 | alrimiv 1754 | . . . . . . . . 9 |
4 | 3 | ralrimivw 2393 | . . . . . . . 8 |
5 | dffun8 4929 | . . . . . . . . 9 | |
6 | 5 | simprbi 260 | . . . . . . . 8 |
7 | 4, 6 | anim12ci 322 | . . . . . . 7 |
8 | r19.26 2441 | . . . . . . 7 | |
9 | 7, 8 | sylibr 137 | . . . . . 6 |
10 | nfv 1421 | . . . . . . . 8 | |
11 | 10 | euexex 1985 | . . . . . . 7 |
12 | 11 | ralimi 2384 | . . . . . 6 |
13 | 9, 12 | syl 14 | . . . . 5 |
14 | ssralv 3004 | . . . . 5 | |
15 | 1, 13, 14 | mpsyl 59 | . . . 4 |
16 | df-br 3765 | . . . . . . 7 | |
17 | df-co 4354 | . . . . . . . 8 | |
18 | 17 | eleq2i 2104 | . . . . . . 7 |
19 | opabid 3994 | . . . . . . 7 | |
20 | 16, 18, 19 | 3bitri 195 | . . . . . 6 |
21 | 20 | mobii 1937 | . . . . 5 |
22 | 21 | ralbii 2330 | . . . 4 |
23 | 15, 22 | sylibr 137 | . . 3 |
24 | relco 4819 | . . 3 | |
25 | 23, 24 | jctil 295 | . 2 |
26 | dffun7 4928 | . 2 | |
27 | 25, 26 | sylibr 137 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wal 1241 wex 1381 wcel 1393 weu 1900 wmo 1901 wral 2306 wss 2917 cop 3378 class class class wbr 3764 copab 3817 cdm 4345 ccom 4349 wrel 4350 wfun 4896 |
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-fun 4904 |
This theorem is referenced by: fnco 5007 f1co 5101 tposfun 5875 |
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