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| Mirrors > Home > NFE Home > Th. List > dmpprod | Unicode version | ||
| Description: The domain of a parallel product. (Contributed by SF, 24-Feb-2015.) |
| Ref | Expression |
|---|---|
| dmpprod |
  PProd      
 |
are mutually distinct and distinct from all other
variables.| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | vex 2862 |
. . . . . . 7
  | |
| 2 | vex 2862 |
. . . . . . 7
| |
| 3 | 1, 2 | opex 4588 |
. . . . . 6
  |
| 4 | 3 | isseti 2865 |
. . . . 5
|
| 5 | 19.41v 1901 |
. . . . 5
![](wedge.gif "/\"){kind=small} 
| |
| 6 | 4, 5 | mpbiran 884 |
. . . 4
|
| 7 | 6 | 2exbii 1583 |
. . 3
|
| 8 | df-br 4640 |
. . . 4
PProd
PProd | |
| 9 | eldm 4898 |
. . . 4
PProd
PProd | |
| 10 | brpprod 5839 |
. . . . . . 7
PProd
| |
| 11 | 19.42vv 1907 |
. . . . . . . . 9
| |
| 12 | 3anass 938 |
. . . . . . . . . . 11
| |
| 13 | eqcom 2355 |
. . . . . . . . . . . . 13
| |
| 14 | opth 4602 |
. . . . . . . . . . . . 13
| |
| 15 | 13, 14 | bitri 240 |
. . . . . . . . . . . 12
|
| 16 | 15 | anbi1i 676 |
. . . . . . . . . . 11
|
| 17 | 12, 16 | bitri 240 |
. . . . . . . . . 10
|
| 18 | 17 | 2exbii 1583 |
. . . . . . . . 9
|
| 19 | df-3an 936 |
. . . . . . . . 9
| |
| 20 | 11, 18, 19 | 3bitr4i 268 |
. . . . . . . 8
|
| 21 | 20 | 2exbii 1583 |
. . . . . . 7
|
| 22 | vex 2862 |
. . . . . . . 8
| |
| 23 | vex 2862 |
. . . . . . . 8
| |
| 24 | breq1 4642 |
. . . . . . . . . . 11
 | |
| 25 | 24 | anbi1d 685 |
. . . . . . . . . 10
|
| 26 | 25 | anbi2d 684 |
. . . . . . . . 9
|
| 27 | 26 | 2exbidv 1628 |
. . . . . . . 8
|
| 28 | breq1 4642 |
. . . . . . . . . . 11
| |
| 29 | 28 | anbi2d 684 |
. . . . . . . . . 10
|
| 30 | 29 | anbi2d 684 |
. . . . . . . . 9
|
| 31 | 30 | 2exbidv 1628 |
. . . . . . . 8
|
| 32 | 22, 23, 27, 31 | ceqsex2v 2896 |
. . . . . . 7
|
| 33 | 10, 21, 32 | 3bitri 262 |
. . . . . 6
PProd
|
| 34 | 33 | exbii 1582 |
. . . . 5
PProd
|
| 35 | exrot3 1744 |
. . . . 5
| |
| 36 | 34, 35 | bitri 240 |
. . . 4
PProd
|
| 37 | 8, 9, 36 | 3bitri 262 |
. . 3
PProd
|
| 38 | eldm 4898 |
. . . . 5
| |
| 39 | eldm 4898 |
. . . . 5
| |
| 40 | 38, 39 | anbi12i 678 |
. . . 4
|
| 41 | brxp 4812 |
. . . 4
| |
| 42 | eeanv 1913 |
. . . 4
| |
| 43 | 40, 41, 42 | 3bitr4i 268 |
. . 3
|
| 44 | 7, 37, 43 | 3bitr4i 268 |
. 2
PProd
|
| 45 | 44 | eqbrriv 4851 |
1
PProd
|
| Colors of variables: wff setvar class |
| Syntax hints: wa 358
w3a 934
wex 1541
wceq 1642
wcel 1710
cop 4561
class class class wbr 4639 cxp 4770 cdm 4772 PProd cpprod 5737 |
| This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 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-1st 4723 df-co 4726 df-ima 4727 df-xp 4784 df-cnv 4785 df-rn 4786 df-dm 4787 df-2nd 4797 df-txp 5736 df-pprod 5738 |
| This theorem is referenced by: rnpprod 5842 fnpprod 5843 |
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