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Mirrors > Home > ILE Home > Th. List > fnco | Unicode version |
Description: Composition of two functions. (Contributed by NM, 22-May-2006.) |
Ref | Expression |
---|---|
fnco |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnfun 5190 | . . . 4 | |
2 | fnfun 5190 | . . . 4 | |
3 | funco 5133 | . . . 4 | |
4 | 1, 2, 3 | syl2an 287 | . . 3 |
5 | 4 | 3adant3 986 | . 2 |
6 | fndm 5192 | . . . . . . 7 | |
7 | 6 | sseq2d 3097 | . . . . . 6 |
8 | 7 | biimpar 295 | . . . . 5 |
9 | dmcosseq 4780 | . . . . 5 | |
10 | 8, 9 | syl 14 | . . . 4 |
11 | 10 | 3adant2 985 | . . 3 |
12 | fndm 5192 | . . . 4 | |
13 | 12 | 3ad2ant2 988 | . . 3 |
14 | 11, 13 | eqtrd 2150 | . 2 |
15 | df-fn 5096 | . 2 | |
16 | 5, 14, 15 | sylanbrc 413 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 947 wceq 1316 wss 3041 cdm 4509 crn 4510 ccom 4513 wfun 5087 wfn 5088 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-br 3900 df-opab 3960 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-fun 5095 df-fn 5096 |
This theorem is referenced by: fco 5258 fnfco 5267 updjudhcoinlf 6933 updjudhcoinrg 6934 upxp 12352 uptx 12354 |
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