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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 4855 | . . . . 5 | |
2 | funmo 5185 | . . . . . . . . . 10 | |
3 | 2 | alrimiv 1854 | . . . . . . . . 9 |
4 | 3 | ralrimivw 2531 | . . . . . . . 8 |
5 | dffun8 5198 | . . . . . . . . 9 | |
6 | 5 | simprbi 273 | . . . . . . . 8 |
7 | 4, 6 | anim12ci 337 | . . . . . . 7 |
8 | r19.26 2583 | . . . . . . 7 | |
9 | 7, 8 | sylibr 133 | . . . . . 6 |
10 | nfv 1508 | . . . . . . . 8 | |
11 | 10 | euexex 2091 | . . . . . . 7 |
12 | 11 | ralimi 2520 | . . . . . 6 |
13 | 9, 12 | syl 14 | . . . . 5 |
14 | ssralv 3192 | . . . . 5 | |
15 | 1, 13, 14 | mpsyl 65 | . . . 4 |
16 | df-br 3966 | . . . . . . 7 | |
17 | df-co 4595 | . . . . . . . 8 | |
18 | 17 | eleq2i 2224 | . . . . . . 7 |
19 | opabid 4217 | . . . . . . 7 | |
20 | 16, 18, 19 | 3bitri 205 | . . . . . 6 |
21 | 20 | mobii 2043 | . . . . 5 |
22 | 21 | ralbii 2463 | . . . 4 |
23 | 15, 22 | sylibr 133 | . . 3 |
24 | relco 5084 | . . 3 | |
25 | 23, 24 | jctil 310 | . 2 |
26 | dffun7 5197 | . 2 | |
27 | 25, 26 | sylibr 133 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wal 1333 wex 1472 weu 2006 wmo 2007 wcel 2128 wral 2435 wss 3102 cop 3563 class class class wbr 3965 copab 4024 cdm 4586 ccom 4590 wrel 4591 wfun 5164 |
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 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-br 3966 df-opab 4026 df-id 4253 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-fun 5172 |
This theorem is referenced by: fnco 5278 f1co 5387 tposfun 6207 casefun 7029 caseinj 7033 caseinl 7035 caseinr 7036 djufun 7048 djuinj 7050 ctssdccl 7055 |
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