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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 4919 | . . . . 5 | |
2 | funrel 4919 | . . . . 5 | |
3 | 1, 2 | anim12i 321 | . . . 4 |
4 | relun 4454 | . . . 4 | |
5 | 3, 4 | sylibr 137 | . . 3 |
6 | 5 | adantr 261 | . 2 |
7 | elun 3084 | . . . . . . . 8 | |
8 | elun 3084 | . . . . . . . 8 | |
9 | 7, 8 | anbi12i 433 | . . . . . . 7 |
10 | anddi 734 | . . . . . . 7 | |
11 | 9, 10 | bitri 173 | . . . . . 6 |
12 | disj1 3270 | . . . . . . . . . . . . 13 | |
13 | 12 | biimpi 113 | . . . . . . . . . . . 12 |
14 | 13 | 19.21bi 1450 | . . . . . . . . . . 11 |
15 | imnan 624 | . . . . . . . . . . 11 | |
16 | 14, 15 | sylib 127 | . . . . . . . . . 10 |
17 | vex 2560 | . . . . . . . . . . . 12 | |
18 | vex 2560 | . . . . . . . . . . . 12 | |
19 | 17, 18 | opeldm 4538 | . . . . . . . . . . 11 |
20 | vex 2560 | . . . . . . . . . . . 12 | |
21 | 17, 20 | opeldm 4538 | . . . . . . . . . . 11 |
22 | 19, 21 | anim12i 321 | . . . . . . . . . 10 |
23 | 16, 22 | nsyl 558 | . . . . . . . . 9 |
24 | orel2 645 | . . . . . . . . 9 | |
25 | 23, 24 | syl 14 | . . . . . . . 8 |
26 | 14 | con2d 554 | . . . . . . . . . . 11 |
27 | imnan 624 | . . . . . . . . . . 11 | |
28 | 26, 27 | sylib 127 | . . . . . . . . . 10 |
29 | 17, 18 | opeldm 4538 | . . . . . . . . . . 11 |
30 | 17, 20 | opeldm 4538 | . . . . . . . . . . 11 |
31 | 29, 30 | anim12i 321 | . . . . . . . . . 10 |
32 | 28, 31 | nsyl 558 | . . . . . . . . 9 |
33 | orel1 644 | . . . . . . . . 9 | |
34 | 32, 33 | syl 14 | . . . . . . . 8 |
35 | 25, 34 | orim12d 700 | . . . . . . 7 |
36 | 35 | adantl 262 | . . . . . 6 |
37 | 11, 36 | syl5bi 141 | . . . . 5 |
38 | dffun4 4913 | . . . . . . . . . 10 | |
39 | 38 | simprbi 260 | . . . . . . . . 9 |
40 | 39 | 19.21bi 1450 | . . . . . . . 8 |
41 | 40 | 19.21bbi 1451 | . . . . . . 7 |
42 | dffun4 4913 | . . . . . . . . . 10 | |
43 | 42 | simprbi 260 | . . . . . . . . 9 |
44 | 43 | 19.21bi 1450 | . . . . . . . 8 |
45 | 44 | 19.21bbi 1451 | . . . . . . 7 |
46 | 41, 45 | jaao 639 | . . . . . 6 |
47 | 46 | adantr 261 | . . . . 5 |
48 | 37, 47 | syld 40 | . . . 4 |
49 | 48 | alrimiv 1754 | . . 3 |
50 | 49 | alrimivv 1755 | . 2 |
51 | dffun4 4913 | . 2 | |
52 | 6, 50, 51 | sylanbrc 394 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 97 wo 629 wal 1241 wceq 1243 wcel 1393 cun 2915 cin 2916 c0 3224 cop 3378 cdm 4345 wrel 4350 wfun 4896 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 df-id 4030 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-fun 4904 |
This theorem is referenced by: funprg 4949 funtpg 4950 funtp 4952 fnun 5005 fvun1 5239 |
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