Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > funun | Unicode version |
Description: The union of functions with disjoint domains is a function. Theorem 4.6 of [Monk1] p. 43. (Contributed by NM, 12-Aug-1994.) |
Ref | Expression |
---|---|
funun |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funrel 5140 | . . . . 5 | |
2 | funrel 5140 | . . . . 5 | |
3 | 1, 2 | anim12i 336 | . . . 4 |
4 | relun 4656 | . . . 4 | |
5 | 3, 4 | sylibr 133 | . . 3 |
6 | 5 | adantr 274 | . 2 |
7 | elun 3217 | . . . . . . . 8 | |
8 | elun 3217 | . . . . . . . 8 | |
9 | 7, 8 | anbi12i 455 | . . . . . . 7 |
10 | anddi 810 | . . . . . . 7 | |
11 | 9, 10 | bitri 183 | . . . . . 6 |
12 | disj1 3413 | . . . . . . . . . . . . 13 | |
13 | 12 | biimpi 119 | . . . . . . . . . . . 12 |
14 | 13 | 19.21bi 1537 | . . . . . . . . . . 11 |
15 | imnan 679 | . . . . . . . . . . 11 | |
16 | 14, 15 | sylib 121 | . . . . . . . . . 10 |
17 | vex 2689 | . . . . . . . . . . . 12 | |
18 | vex 2689 | . . . . . . . . . . . 12 | |
19 | 17, 18 | opeldm 4742 | . . . . . . . . . . 11 |
20 | vex 2689 | . . . . . . . . . . . 12 | |
21 | 17, 20 | opeldm 4742 | . . . . . . . . . . 11 |
22 | 19, 21 | anim12i 336 | . . . . . . . . . 10 |
23 | 16, 22 | nsyl 617 | . . . . . . . . 9 |
24 | orel2 715 | . . . . . . . . 9 | |
25 | 23, 24 | syl 14 | . . . . . . . 8 |
26 | 14 | con2d 613 | . . . . . . . . . . 11 |
27 | imnan 679 | . . . . . . . . . . 11 | |
28 | 26, 27 | sylib 121 | . . . . . . . . . 10 |
29 | 17, 18 | opeldm 4742 | . . . . . . . . . . 11 |
30 | 17, 20 | opeldm 4742 | . . . . . . . . . . 11 |
31 | 29, 30 | anim12i 336 | . . . . . . . . . 10 |
32 | 28, 31 | nsyl 617 | . . . . . . . . 9 |
33 | orel1 714 | . . . . . . . . 9 | |
34 | 32, 33 | syl 14 | . . . . . . . 8 |
35 | 25, 34 | orim12d 775 | . . . . . . 7 |
36 | 35 | adantl 275 | . . . . . 6 |
37 | 11, 36 | syl5bi 151 | . . . . 5 |
38 | dffun4 5134 | . . . . . . . . . 10 | |
39 | 38 | simprbi 273 | . . . . . . . . 9 |
40 | 39 | 19.21bi 1537 | . . . . . . . 8 |
41 | 40 | 19.21bbi 1538 | . . . . . . 7 |
42 | dffun4 5134 | . . . . . . . . . 10 | |
43 | 42 | simprbi 273 | . . . . . . . . 9 |
44 | 43 | 19.21bi 1537 | . . . . . . . 8 |
45 | 44 | 19.21bbi 1538 | . . . . . . 7 |
46 | 41, 45 | jaao 708 | . . . . . 6 |
47 | 46 | adantr 274 | . . . . 5 |
48 | 37, 47 | syld 45 | . . . 4 |
49 | 48 | alrimiv 1846 | . . 3 |
50 | 49 | alrimivv 1847 | . 2 |
51 | dffun4 5134 | . 2 | |
52 | 6, 50, 51 | sylanbrc 413 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wo 697 wal 1329 wceq 1331 wcel 1480 cun 3069 cin 3070 c0 3363 cop 3530 cdm 4539 wrel 4544 wfun 5117 |
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-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-opab 3990 df-id 4215 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-fun 5125 |
This theorem is referenced by: funprg 5173 funtpg 5174 funtp 5176 fnun 5229 fvun1 5487 sbthlem7 6851 sbthlemi8 6852 casefun 6970 caseinj 6974 djufun 6989 djuinj 6991 exmidfodomrlemim 7057 setsfun 11994 setsfun0 11995 strleund 12047 strleun 12048 |
Copyright terms: Public domain | W3C validator |