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Mirrors > Home > NFE 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 set.mm contributors, 12-Aug-1994.) |
Ref | Expression |
---|---|
funun |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elun 3220 | . . . . . . . 8 | |
2 | elun 3220 | . . . . . . . 8 | |
3 | 1, 2 | anbi12i 678 | . . . . . . 7 |
4 | anddi 840 | . . . . . . 7 | |
5 | 3, 4 | bitri 240 | . . . . . 6 |
6 | sp 1747 | . . . . . . . . . 10 | |
7 | disj1 3593 | . . . . . . . . . 10 | |
8 | imnan 411 | . . . . . . . . . . 11 | |
9 | 8 | bicomi 193 | . . . . . . . . . 10 |
10 | 6, 7, 9 | 3imtr4i 257 | . . . . . . . . 9 |
11 | opeldm 4910 | . . . . . . . . . 10 | |
12 | opeldm 4910 | . . . . . . . . . 10 | |
13 | 11, 12 | anim12i 549 | . . . . . . . . 9 |
14 | 10, 13 | nsyl 113 | . . . . . . . 8 |
15 | orel2 372 | . . . . . . . 8 | |
16 | 14, 15 | syl 15 | . . . . . . 7 |
17 | sp 1747 | . . . . . . . . . 10 | |
18 | incom 3448 | . . . . . . . . . . . 12 | |
19 | 18 | eqeq1i 2360 | . . . . . . . . . . 11 |
20 | disj1 3593 | . . . . . . . . . . 11 | |
21 | 19, 20 | bitri 240 | . . . . . . . . . 10 |
22 | imnan 411 | . . . . . . . . . . 11 | |
23 | 22 | bicomi 193 | . . . . . . . . . 10 |
24 | 17, 21, 23 | 3imtr4i 257 | . . . . . . . . 9 |
25 | opeldm 4910 | . . . . . . . . . 10 | |
26 | opeldm 4910 | . . . . . . . . . 10 | |
27 | 25, 26 | anim12i 549 | . . . . . . . . 9 |
28 | 24, 27 | nsyl 113 | . . . . . . . 8 |
29 | orel1 371 | . . . . . . . 8 | |
30 | 28, 29 | syl 15 | . . . . . . 7 |
31 | 16, 30 | orim12d 811 | . . . . . 6 |
32 | 5, 31 | syl5bi 208 | . . . . 5 |
33 | dffun4 5121 | . . . . . . . . 9 | |
34 | 33 | biimpi 186 | . . . . . . . 8 |
35 | 34 | 19.21bi 1758 | . . . . . . 7 |
36 | 35 | 19.21bbi 1865 | . . . . . 6 |
37 | dffun4 5121 | . . . . . . . . 9 | |
38 | 37 | biimpi 186 | . . . . . . . 8 |
39 | 38 | 19.21bi 1758 | . . . . . . 7 |
40 | 39 | 19.21bbi 1865 | . . . . . 6 |
41 | 36, 40 | jaao 495 | . . . . 5 |
42 | 32, 41 | sylan9r 639 | . . . 4 |
43 | 42 | alrimiv 1631 | . . 3 |
44 | 43 | alrimivv 1632 | . 2 |
45 | dffun4 5121 | . 2 | |
46 | 44, 45 | sylibr 203 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wo 357 wa 358 wal 1540 wceq 1642 wcel 1710 cun 3207 cin 3208 c0 3550 cop 4561 cdm 4772 wfun 4775 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-13 1712 ax-14 1714 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-xp 4079 ax-cnv 4080 ax-1c 4081 ax-sset 4082 ax-si 4083 ax-ins2 4084 ax-ins3 4085 ax-typlower 4086 ax-sn 4087 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3or 935 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-mo 2209 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-reu 2621 df-rmo 2622 df-rab 2623 df-v 2861 df-sbc 3047 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-pss 3261 df-nul 3551 df-if 3663 df-pw 3724 df-sn 3741 df-pr 3742 df-uni 3892 df-int 3927 df-opk 4058 df-1c 4136 df-pw1 4137 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 df-ssetk 4193 df-imagek 4194 df-idk 4195 df-iota 4339 df-0c 4377 df-addc 4378 df-nnc 4379 df-fin 4380 df-lefin 4440 df-ltfin 4441 df-ncfin 4442 df-tfin 4443 df-evenfin 4444 df-oddfin 4445 df-sfin 4446 df-spfin 4447 df-phi 4565 df-op 4566 df-proj1 4567 df-proj2 4568 df-opab 4623 df-br 4640 df-co 4726 df-ima 4727 df-id 4767 df-cnv 4785 df-rn 4786 df-dm 4787 df-fun 4789 |
This theorem is referenced by: funprg 5149 funprgOLD 5150 fnun 5189 fvun 5378 |
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