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Mirrors > Home > MPE Home > Th. List > funun | Structured version Visualization version 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 5905 | . . . . 5 | |
2 | funrel 5905 | . . . . 5 | |
3 | 1, 2 | anim12i 590 | . . . 4 |
4 | relun 5235 | . . . 4 | |
5 | 3, 4 | sylibr 224 | . . 3 |
6 | 5 | adantr 481 | . 2 |
7 | elun 3753 | . . . . . . . 8 | |
8 | elun 3753 | . . . . . . . 8 | |
9 | 7, 8 | anbi12i 733 | . . . . . . 7 |
10 | anddi 914 | . . . . . . 7 | |
11 | 9, 10 | bitri 264 | . . . . . 6 |
12 | disj1 4019 | . . . . . . . . . . . . 13 | |
13 | 12 | biimpi 206 | . . . . . . . . . . . 12 |
14 | 13 | 19.21bi 2059 | . . . . . . . . . . 11 |
15 | imnan 438 | . . . . . . . . . . 11 | |
16 | 14, 15 | sylib 208 | . . . . . . . . . 10 |
17 | vex 3203 | . . . . . . . . . . . 12 | |
18 | vex 3203 | . . . . . . . . . . . 12 | |
19 | 17, 18 | opeldm 5328 | . . . . . . . . . . 11 |
20 | vex 3203 | . . . . . . . . . . . 12 | |
21 | 17, 20 | opeldm 5328 | . . . . . . . . . . 11 |
22 | 19, 21 | anim12i 590 | . . . . . . . . . 10 |
23 | 16, 22 | nsyl 135 | . . . . . . . . 9 |
24 | orel2 398 | . . . . . . . . 9 | |
25 | 23, 24 | syl 17 | . . . . . . . 8 |
26 | 14 | con2d 129 | . . . . . . . . . . 11 |
27 | imnan 438 | . . . . . . . . . . 11 | |
28 | 26, 27 | sylib 208 | . . . . . . . . . 10 |
29 | 17, 18 | opeldm 5328 | . . . . . . . . . . 11 |
30 | 17, 20 | opeldm 5328 | . . . . . . . . . . 11 |
31 | 29, 30 | anim12i 590 | . . . . . . . . . 10 |
32 | 28, 31 | nsyl 135 | . . . . . . . . 9 |
33 | orel1 397 | . . . . . . . . 9 | |
34 | 32, 33 | syl 17 | . . . . . . . 8 |
35 | 25, 34 | orim12d 883 | . . . . . . 7 |
36 | 35 | adantl 482 | . . . . . 6 |
37 | 11, 36 | syl5bi 232 | . . . . 5 |
38 | dffun4 5900 | . . . . . . . . . 10 | |
39 | 38 | simprbi 480 | . . . . . . . . 9 |
40 | 39 | 19.21bi 2059 | . . . . . . . 8 |
41 | 40 | 19.21bbi 2060 | . . . . . . 7 |
42 | dffun4 5900 | . . . . . . . . . 10 | |
43 | 42 | simprbi 480 | . . . . . . . . 9 |
44 | 43 | 19.21bi 2059 | . . . . . . . 8 |
45 | 44 | 19.21bbi 2060 | . . . . . . 7 |
46 | 41, 45 | jaao 531 | . . . . . 6 |
47 | 46 | adantr 481 | . . . . 5 |
48 | 37, 47 | syld 47 | . . . 4 |
49 | 48 | alrimiv 1855 | . . 3 |
50 | 49 | alrimivv 1856 | . 2 |
51 | dffun4 5900 | . 2 | |
52 | 6, 50, 51 | sylanbrc 698 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wo 383 wa 384 wal 1481 wceq 1483 wcel 1990 cun 3572 cin 3573 c0 3915 cop 4183 cdm 5114 wrel 5119 wfun 5882 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-id 5024 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-fun 5890 |
This theorem is referenced by: funprg 5940 funprgOLD 5941 funtpg 5942 funtpgOLD 5943 funtp 5945 funcnvpr 5950 funcnvtp 5951 funcnvqp 5952 funcnvqpOLD 5953 fnun 5997 fvun 6268 wfrlem13 7427 tfrlem10 7483 sbthlem7 8076 sbthlem8 8077 fodomr 8111 funsnfsupp 8299 axdc3lem4 9275 setsfun 15893 setsfun0 15894 strlemor1OLD 15969 strleun 15972 cnfldfun 19758 bnj1421 31110 noextend 31819 noextendseq 31820 |
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