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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 5110 | . . . . 5 | |
2 | funrel 5110 | . . . . 5 | |
3 | 1, 2 | anim12i 336 | . . . 4 |
4 | relun 4626 | . . . 4 | |
5 | 3, 4 | sylibr 133 | . . 3 |
6 | 5 | adantr 274 | . 2 |
7 | elun 3187 | . . . . . . . 8 | |
8 | elun 3187 | . . . . . . . 8 | |
9 | 7, 8 | anbi12i 455 | . . . . . . 7 |
10 | anddi 795 | . . . . . . 7 | |
11 | 9, 10 | bitri 183 | . . . . . 6 |
12 | disj1 3383 | . . . . . . . . . . . . 13 | |
13 | 12 | biimpi 119 | . . . . . . . . . . . 12 |
14 | 13 | 19.21bi 1522 | . . . . . . . . . . 11 |
15 | imnan 664 | . . . . . . . . . . 11 | |
16 | 14, 15 | sylib 121 | . . . . . . . . . 10 |
17 | vex 2663 | . . . . . . . . . . . 12 | |
18 | vex 2663 | . . . . . . . . . . . 12 | |
19 | 17, 18 | opeldm 4712 | . . . . . . . . . . 11 |
20 | vex 2663 | . . . . . . . . . . . 12 | |
21 | 17, 20 | opeldm 4712 | . . . . . . . . . . 11 |
22 | 19, 21 | anim12i 336 | . . . . . . . . . 10 |
23 | 16, 22 | nsyl 602 | . . . . . . . . 9 |
24 | orel2 700 | . . . . . . . . 9 | |
25 | 23, 24 | syl 14 | . . . . . . . 8 |
26 | 14 | con2d 598 | . . . . . . . . . . 11 |
27 | imnan 664 | . . . . . . . . . . 11 | |
28 | 26, 27 | sylib 121 | . . . . . . . . . 10 |
29 | 17, 18 | opeldm 4712 | . . . . . . . . . . 11 |
30 | 17, 20 | opeldm 4712 | . . . . . . . . . . 11 |
31 | 29, 30 | anim12i 336 | . . . . . . . . . 10 |
32 | 28, 31 | nsyl 602 | . . . . . . . . 9 |
33 | orel1 699 | . . . . . . . . 9 | |
34 | 32, 33 | syl 14 | . . . . . . . 8 |
35 | 25, 34 | orim12d 760 | . . . . . . 7 |
36 | 35 | adantl 275 | . . . . . 6 |
37 | 11, 36 | syl5bi 151 | . . . . 5 |
38 | dffun4 5104 | . . . . . . . . . 10 | |
39 | 38 | simprbi 273 | . . . . . . . . 9 |
40 | 39 | 19.21bi 1522 | . . . . . . . 8 |
41 | 40 | 19.21bbi 1523 | . . . . . . 7 |
42 | dffun4 5104 | . . . . . . . . . 10 | |
43 | 42 | simprbi 273 | . . . . . . . . 9 |
44 | 43 | 19.21bi 1522 | . . . . . . . 8 |
45 | 44 | 19.21bbi 1523 | . . . . . . 7 |
46 | 41, 45 | jaao 693 | . . . . . 6 |
47 | 46 | adantr 274 | . . . . 5 |
48 | 37, 47 | syld 45 | . . . 4 |
49 | 48 | alrimiv 1830 | . . 3 |
50 | 49 | alrimivv 1831 | . 2 |
51 | dffun4 5104 | . 2 | |
52 | 6, 50, 51 | sylanbrc 413 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wo 682 wal 1314 wceq 1316 wcel 1465 cun 3039 cin 3040 c0 3333 cop 3500 cdm 4509 wrel 4514 wfun 5087 |
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 588 ax-in2 589 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-v 2662 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-br 3900 df-opab 3960 df-id 4185 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-fun 5095 |
This theorem is referenced by: funprg 5143 funtpg 5144 funtp 5146 fnun 5199 fvun1 5455 sbthlem7 6819 sbthlemi8 6820 casefun 6938 caseinj 6942 djufun 6957 djuinj 6959 exmidfodomrlemim 7025 setsfun 11905 setsfun0 11906 strleund 11958 strleun 11959 |
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