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 5148 | . . . . 5 | |
2 | funrel 5148 | . . . . 5 | |
3 | 1, 2 | anim12i 336 | . . . 4 |
4 | relun 4664 | . . . 4 | |
5 | 3, 4 | sylibr 133 | . . 3 |
6 | 5 | adantr 274 | . 2 |
7 | elun 3222 | . . . . . . . 8 | |
8 | elun 3222 | . . . . . . . 8 | |
9 | 7, 8 | anbi12i 456 | . . . . . . 7 |
10 | anddi 811 | . . . . . . 7 | |
11 | 9, 10 | bitri 183 | . . . . . 6 |
12 | disj1 3418 | . . . . . . . . . . . . 13 | |
13 | 12 | biimpi 119 | . . . . . . . . . . . 12 |
14 | 13 | 19.21bi 1538 | . . . . . . . . . . 11 |
15 | imnan 680 | . . . . . . . . . . 11 | |
16 | 14, 15 | sylib 121 | . . . . . . . . . 10 |
17 | vex 2692 | . . . . . . . . . . . 12 | |
18 | vex 2692 | . . . . . . . . . . . 12 | |
19 | 17, 18 | opeldm 4750 | . . . . . . . . . . 11 |
20 | vex 2692 | . . . . . . . . . . . 12 | |
21 | 17, 20 | opeldm 4750 | . . . . . . . . . . 11 |
22 | 19, 21 | anim12i 336 | . . . . . . . . . 10 |
23 | 16, 22 | nsyl 618 | . . . . . . . . 9 |
24 | orel2 716 | . . . . . . . . 9 | |
25 | 23, 24 | syl 14 | . . . . . . . 8 |
26 | 14 | con2d 614 | . . . . . . . . . . 11 |
27 | imnan 680 | . . . . . . . . . . 11 | |
28 | 26, 27 | sylib 121 | . . . . . . . . . 10 |
29 | 17, 18 | opeldm 4750 | . . . . . . . . . . 11 |
30 | 17, 20 | opeldm 4750 | . . . . . . . . . . 11 |
31 | 29, 30 | anim12i 336 | . . . . . . . . . 10 |
32 | 28, 31 | nsyl 618 | . . . . . . . . 9 |
33 | orel1 715 | . . . . . . . . 9 | |
34 | 32, 33 | syl 14 | . . . . . . . 8 |
35 | 25, 34 | orim12d 776 | . . . . . . 7 |
36 | 35 | adantl 275 | . . . . . 6 |
37 | 11, 36 | syl5bi 151 | . . . . 5 |
38 | dffun4 5142 | . . . . . . . . . 10 | |
39 | 38 | simprbi 273 | . . . . . . . . 9 |
40 | 39 | 19.21bi 1538 | . . . . . . . 8 |
41 | 40 | 19.21bbi 1539 | . . . . . . 7 |
42 | dffun4 5142 | . . . . . . . . . 10 | |
43 | 42 | simprbi 273 | . . . . . . . . 9 |
44 | 43 | 19.21bi 1538 | . . . . . . . 8 |
45 | 44 | 19.21bbi 1539 | . . . . . . 7 |
46 | 41, 45 | jaao 709 | . . . . . 6 |
47 | 46 | adantr 274 | . . . . 5 |
48 | 37, 47 | syld 45 | . . . 4 |
49 | 48 | alrimiv 1847 | . . 3 |
50 | 49 | alrimivv 1848 | . 2 |
51 | dffun4 5142 | . 2 | |
52 | 6, 50, 51 | sylanbrc 414 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wo 698 wal 1330 wceq 1332 wcel 1481 cun 3074 cin 3075 c0 3368 cop 3535 cdm 4547 wrel 4552 wfun 5125 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-br 3938 df-opab 3998 df-id 4223 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-fun 5133 |
This theorem is referenced by: funprg 5181 funtpg 5182 funtp 5184 fnun 5237 fvun1 5495 sbthlem7 6859 sbthlemi8 6860 casefun 6978 caseinj 6982 djufun 6997 djuinj 6999 exmidfodomrlemim 7074 setsfun 12033 setsfun0 12034 strleund 12086 strleun 12087 |
Copyright terms: Public domain | W3C validator |