Theorem dmpprod 5841

Description: The domain of a parallel product. (Contributed by SF, 24-Feb-2015.)

Assertion

Ref	Expression
dmpprod	|- dom PProd (A, B) = (dom A X. dom B)

Proof of Theorem dmpprod

Dummy variables a b c d x t u are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2863	. . . . . . 7 |- c e. _V
2		vex 2863	. . . . . . 7 |- d e. _V
3	1, 2	opex 4589	. . . . . 6 |- <.c, d>. e. _V
4	3	isseti 2866	. . . . 5 |- E.x x = <.c, d>.
5		19.41v 1901	. . . . 5 |- (E.x(x = <.c, d>. /\ (aAc /\ bBd)) <-> (E.x x = <.c, d>. /\ (aAc /\ bBd)))
6	4, 5	mpbiran 884	. . . 4 |- (E.x(x = <.c, d>. /\ (aAc /\ bBd)) <-> (aAc /\ bBd))
7	6	2exbii 1583	. . 3 |- (E.cE.dE.x(x = <.c, d>. /\ (aAc /\ bBd)) <-> E.cE.d(aAc /\ bBd))
8		df-br 4641	. . . 4 |- (adom PProd (A, B)b <-> <.a, b>. e. dom PProd (A, B))
9		eldm 4899	. . . 4 |- (<.a, b>. e. dom PProd (A, B) <-> E.x<.a, b>. PProd (A, B)x)
10		brpprod 5840	. . . . . . 7 |- (<.a, b>. PProd (A, B)x <-> E.tE.uE.cE.d(<.a, b>. = <.t, u>. /\ x = <.c, d>. /\ (tAc /\ uBd)))
11		19.42vv 1907	. . . . . . . . 9 |- (E.cE.d((t = a /\ u = b) /\ (x = <.c, d>. /\ (tAc /\ uBd))) <-> ((t = a /\ u = b) /\ E.cE.d(x = <.c, d>. /\ (tAc /\ uBd))))
12		3anass 938	. . . . . . . . . . 11 |- ((<.a, b>. = <.t, u>. /\ x = <.c, d>. /\ (tAc /\ uBd)) <-> (<.a, b>. = <.t, u>. /\ (x = <.c, d>. /\ (tAc /\ uBd))))
13		eqcom 2355	. . . . . . . . . . . . 13 |- (<.a, b>. = <.t, u>. <-> <.t, u>. = <.a, b>.)
14		opth 4603	. . . . . . . . . . . . 13 |- (<.t, u>. = <.a, b>. <-> (t = a /\ u = b))
15	13, 14	bitri 240	. . . . . . . . . . . 12 |- (<.a, b>. = <.t, u>. <-> (t = a /\ u = b))
16	15	anbi1i 676	. . . . . . . . . . 11 |- ((<.a, b>. = <.t, u>. /\ (x = <.c, d>. /\ (tAc /\ uBd))) <-> ((t = a /\ u = b) /\ (x = <.c, d>. /\ (tAc /\ uBd))))
17	12, 16	bitri 240	. . . . . . . . . 10 |- ((<.a, b>. = <.t, u>. /\ x = <.c, d>. /\ (tAc /\ uBd)) <-> ((t = a /\ u = b) /\ (x = <.c, d>. /\ (tAc /\ uBd))))
18	17	2exbii 1583	. . . . . . . . 9 |- (E.cE.d(<.a, b>. = <.t, u>. /\ x = <.c, d>. /\ (tAc /\ uBd)) <-> E.cE.d((t = a /\ u = b) /\ (x = <.c, d>. /\ (tAc /\ uBd))))
19		df-3an 936	. . . . . . . . 9 |- ((t = a /\ u = b /\ E.cE.d(x = <.c, d>. /\ (tAc /\ uBd))) <-> ((t = a /\ u = b) /\ E.cE.d(x = <.c, d>. /\ (tAc /\ uBd))))
20	11, 18, 19	3bitr4i 268	. . . . . . . 8 |- (E.cE.d(<.a, b>. = <.t, u>. /\ x = <.c, d>. /\ (tAc /\ uBd)) <-> (t = a /\ u = b /\ E.cE.d(x = <.c, d>. /\ (tAc /\ uBd))))
21	20	2exbii 1583	. . . . . . 7 |- (E.tE.uE.cE.d(<.a, b>. = <.t, u>. /\ x = <.c, d>. /\ (tAc /\ uBd)) <-> E.tE.u(t = a /\ u = b /\ E.cE.d(x = <.c, d>. /\ (tAc /\ uBd))))
22		vex 2863	. . . . . . . 8 |- a e. _V
23		vex 2863	. . . . . . . 8 |- b e. _V
24		breq1 4643	. . . . . . . . . . 11 |- (t = a -> (tAc <-> aAc))
25	24	anbi1d 685	. . . . . . . . . 10 |- (t = a -> ((tAc /\ uBd) <-> (aAc /\ uBd)))
26	25	anbi2d 684	. . . . . . . . 9 |- (t = a -> ((x = <.c, d>. /\ (tAc /\ uBd)) <-> (x = <.c, d>. /\ (aAc /\ uBd))))
27	26	2exbidv 1628	. . . . . . . 8 |- (t = a -> (E.cE.d(x = <.c, d>. /\ (tAc /\ uBd)) <-> E.cE.d(x = <.c, d>. /\ (aAc /\ uBd))))
28		breq1 4643	. . . . . . . . . . 11 |- (u = b -> (uBd <-> bBd))
29	28	anbi2d 684	. . . . . . . . . 10 |- (u = b -> ((aAc /\ uBd) <-> (aAc /\ bBd)))
30	29	anbi2d 684	. . . . . . . . 9 |- (u = b -> ((x = <.c, d>. /\ (aAc /\ uBd)) <-> (x = <.c, d>. /\ (aAc /\ bBd))))
31	30	2exbidv 1628	. . . . . . . 8 |- (u = b -> (E.cE.d(x = <.c, d>. /\ (aAc /\ uBd)) <-> E.cE.d(x = <.c, d>. /\ (aAc /\ bBd))))
32	22, 23, 27, 31	ceqsex2v 2897	. . . . . . 7 |- (E.tE.u(t = a /\ u = b /\ E.cE.d(x = <.c, d>. /\ (tAc /\ uBd))) <-> E.cE.d(x = <.c, d>. /\ (aAc /\ bBd)))
33	10, 21, 32	3bitri 262	. . . . . 6 |- (<.a, b>. PProd (A, B)x <-> E.cE.d(x = <.c, d>. /\ (aAc /\ bBd)))
34	33	exbii 1582	. . . . 5 |- (E.x<.a, b>. PProd (A, B)x <-> E.xE.cE.d(x = <.c, d>. /\ (aAc /\ bBd)))
35		exrot3 1744	. . . . 5 |- (E.xE.cE.d(x = <.c, d>. /\ (aAc /\ bBd)) <-> E.cE.dE.x(x = <.c, d>. /\ (aAc /\ bBd)))
36	34, 35	bitri 240	. . . 4 |- (E.x<.a, b>. PProd (A, B)x <-> E.cE.dE.x(x = <.c, d>. /\ (aAc /\ bBd)))
37	8, 9, 36	3bitri 262	. . 3 |- (adom PProd (A, B)b <-> E.cE.dE.x(x = <.c, d>. /\ (aAc /\ bBd)))
38		eldm 4899	. . . . 5 |- (a e. dom A <-> E.c aAc)
39		eldm 4899	. . . . 5 |- (b e. dom B <-> E.d bBd)
40	38, 39	anbi12i 678	. . . 4 |- ((a e. dom A /\ b e. dom B) <-> (E.c aAc /\ E.d bBd))
41		brxp 4813	. . . 4 |- (a(dom A X. dom B)b <-> (a e. dom A /\ b e. dom B))
42		eeanv 1913	. . . 4 |- (E.cE.d(aAc /\ bBd) <-> (E.c aAc /\ E.d bBd))
43	40, 41, 42	3bitr4i 268	. . 3 |- (a(dom A X. dom B)b <-> E.cE.d(aAc /\ bBd))
44	7, 37, 43	3bitr4i 268	. 2 |- (adom PProd (A, B)b <-> a(dom A X. dom B)b)
45	44	eqbrriv 4852	1 |- dom PProd (A, B) = (dom A X. dom B)

Syntax hints:   /\ wa 358   /\ w3a 934  E.wex 1541   = wceq 1642   e. wcel 1710  <.cop 4562   class class class wbr 4640   X. cxp 4771  dom cdm 4773   PProd cpprod 5738

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 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-1st 4724  df-co 4727  df-ima 4728  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-2nd 4798  df-txp 5737  df-pprod 5739

This theorem is referenced by:  rnpprod  5843  fnpprod  5844