Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 4880 | . . . . 5 | |
2 | funmo 5213 | . . . . . . . . . 10 | |
3 | 2 | alrimiv 1867 | . . . . . . . . 9 |
4 | 3 | ralrimivw 2544 | . . . . . . . 8 |
5 | dffun8 5226 | . . . . . . . . 9 | |
6 | 5 | simprbi 273 | . . . . . . . 8 |
7 | 4, 6 | anim12ci 337 | . . . . . . 7 |
8 | r19.26 2596 | . . . . . . 7 | |
9 | 7, 8 | sylibr 133 | . . . . . 6 |
10 | nfv 1521 | . . . . . . . 8 | |
11 | 10 | euexex 2104 | . . . . . . 7 |
12 | 11 | ralimi 2533 | . . . . . 6 |
13 | 9, 12 | syl 14 | . . . . 5 |
14 | ssralv 3211 | . . . . 5 | |
15 | 1, 13, 14 | mpsyl 65 | . . . 4 |
16 | df-br 3990 | . . . . . . 7 | |
17 | df-co 4620 | . . . . . . . 8 | |
18 | 17 | eleq2i 2237 | . . . . . . 7 |
19 | opabid 4242 | . . . . . . 7 | |
20 | 16, 18, 19 | 3bitri 205 | . . . . . 6 |
21 | 20 | mobii 2056 | . . . . 5 |
22 | 21 | ralbii 2476 | . . . 4 |
23 | 15, 22 | sylibr 133 | . . 3 |
24 | relco 5109 | . . 3 | |
25 | 23, 24 | jctil 310 | . 2 |
26 | dffun7 5225 | . 2 | |
27 | 25, 26 | sylibr 133 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wal 1346 wex 1485 weu 2019 wmo 2020 wcel 2141 wral 2448 wss 3121 cop 3586 class class class wbr 3989 copab 4049 cdm 4611 ccom 4615 wrel 4616 wfun 5192 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-br 3990 df-opab 4051 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-fun 5200 |
This theorem is referenced by: fnco 5306 f1co 5415 tposfun 6239 casefun 7062 caseinj 7066 caseinl 7068 caseinr 7069 djufun 7081 djuinj 7083 ctssdccl 7088 |
Copyright terms: Public domain | W3C validator |