Theorem dmpprod 5840

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

Assertion

Ref	Expression
dmpprod	⊢ dom PProd (A, B) = (dom A × 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 2862	. . . . . . 7 ⊢ c ∈ V
2		vex 2862	. . . . . . 7 ⊢ d ∈ V
3	1, 2	opex 4588	. . . . . 6 ⊢ ⟨c, d⟩ ∈ V
4	3	isseti 2865	. . . . 5 ⊢ ∃x x = ⟨c, d⟩
5		19.41v 1901	. . . . 5 ⊢ (∃x(x = ⟨c, d⟩ ∧ (aAc ∧ bBd)) ↔ (∃x x = ⟨c, d⟩ ∧ (aAc ∧ bBd)))
6	4, 5	mpbiran 884	. . . 4 ⊢ (∃x(x = ⟨c, d⟩ ∧ (aAc ∧ bBd)) ↔ (aAc ∧ bBd))
7	6	2exbii 1583	. . 3 ⊢ (∃c∃d∃x(x = ⟨c, d⟩ ∧ (aAc ∧ bBd)) ↔ ∃c∃d(aAc ∧ bBd))
8		df-br 4640	. . . 4 ⊢ (adom PProd (A, B)b ↔ ⟨a, b⟩ ∈ dom PProd (A, B))
9		eldm 4898	. . . 4 ⊢ (⟨a, b⟩ ∈ dom PProd (A, B) ↔ ∃x⟨a, b⟩ PProd (A, B)x)
10		brpprod 5839	. . . . . . 7 ⊢ (⟨a, b⟩ PProd (A, B)x ↔ ∃t∃u∃c∃d(⟨a, b⟩ = ⟨t, u⟩ ∧ x = ⟨c, d⟩ ∧ (tAc ∧ uBd)))
11		19.42vv 1907	. . . . . . . . 9 ⊢ (∃c∃d((t = a ∧ u = b) ∧ (x = ⟨c, d⟩ ∧ (tAc ∧ uBd))) ↔ ((t = a ∧ u = b) ∧ ∃c∃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 4602	. . . . . . . . . . . . 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 ⊢ (∃c∃d(⟨a, b⟩ = ⟨t, u⟩ ∧ x = ⟨c, d⟩ ∧ (tAc ∧ uBd)) ↔ ∃c∃d((t = a ∧ u = b) ∧ (x = ⟨c, d⟩ ∧ (tAc ∧ uBd))))
19		df-3an 936	. . . . . . . . 9 ⊢ ((t = a ∧ u = b ∧ ∃c∃d(x = ⟨c, d⟩ ∧ (tAc ∧ uBd))) ↔ ((t = a ∧ u = b) ∧ ∃c∃d(x = ⟨c, d⟩ ∧ (tAc ∧ uBd))))
20	11, 18, 19	3bitr4i 268	. . . . . . . 8 ⊢ (∃c∃d(⟨a, b⟩ = ⟨t, u⟩ ∧ x = ⟨c, d⟩ ∧ (tAc ∧ uBd)) ↔ (t = a ∧ u = b ∧ ∃c∃d(x = ⟨c, d⟩ ∧ (tAc ∧ uBd))))
21	20	2exbii 1583	. . . . . . 7 ⊢ (∃t∃u∃c∃d(⟨a, b⟩ = ⟨t, u⟩ ∧ x = ⟨c, d⟩ ∧ (tAc ∧ uBd)) ↔ ∃t∃u(t = a ∧ u = b ∧ ∃c∃d(x = ⟨c, d⟩ ∧ (tAc ∧ uBd))))
22		vex 2862	. . . . . . . 8 ⊢ a ∈ V
23		vex 2862	. . . . . . . 8 ⊢ b ∈ V
24		breq1 4642	. . . . . . . . . . 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 → (∃c∃d(x = ⟨c, d⟩ ∧ (tAc ∧ uBd)) ↔ ∃c∃d(x = ⟨c, d⟩ ∧ (aAc ∧ uBd))))
28		breq1 4642	. . . . . . . . . . 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 → (∃c∃d(x = ⟨c, d⟩ ∧ (aAc ∧ uBd)) ↔ ∃c∃d(x = ⟨c, d⟩ ∧ (aAc ∧ bBd))))
32	22, 23, 27, 31	ceqsex2v 2896	. . . . . . 7 ⊢ (∃t∃u(t = a ∧ u = b ∧ ∃c∃d(x = ⟨c, d⟩ ∧ (tAc ∧ uBd))) ↔ ∃c∃d(x = ⟨c, d⟩ ∧ (aAc ∧ bBd)))
33	10, 21, 32	3bitri 262	. . . . . 6 ⊢ (⟨a, b⟩ PProd (A, B)x ↔ ∃c∃d(x = ⟨c, d⟩ ∧ (aAc ∧ bBd)))
34	33	exbii 1582	. . . . 5 ⊢ (∃x⟨a, b⟩ PProd (A, B)x ↔ ∃x∃c∃d(x = ⟨c, d⟩ ∧ (aAc ∧ bBd)))
35		exrot3 1744	. . . . 5 ⊢ (∃x∃c∃d(x = ⟨c, d⟩ ∧ (aAc ∧ bBd)) ↔ ∃c∃d∃x(x = ⟨c, d⟩ ∧ (aAc ∧ bBd)))
36	34, 35	bitri 240	. . . 4 ⊢ (∃x⟨a, b⟩ PProd (A, B)x ↔ ∃c∃d∃x(x = ⟨c, d⟩ ∧ (aAc ∧ bBd)))
37	8, 9, 36	3bitri 262	. . 3 ⊢ (adom PProd (A, B)b ↔ ∃c∃d∃x(x = ⟨c, d⟩ ∧ (aAc ∧ bBd)))
38		eldm 4898	. . . . 5 ⊢ (a ∈ dom A ↔ ∃c aAc)
39		eldm 4898	. . . . 5 ⊢ (b ∈ dom B ↔ ∃d bBd)
40	38, 39	anbi12i 678	. . . 4 ⊢ ((a ∈ dom A ∧ b ∈ dom B) ↔ (∃c aAc ∧ ∃d bBd))
41		brxp 4812	. . . 4 ⊢ (a(dom A × dom B)b ↔ (a ∈ dom A ∧ b ∈ dom B))
42		eeanv 1913	. . . 4 ⊢ (∃c∃d(aAc ∧ bBd) ↔ (∃c aAc ∧ ∃d bBd))
43	40, 41, 42	3bitr4i 268	. . 3 ⊢ (a(dom A × dom B)b ↔ ∃c∃d(aAc ∧ bBd))
44	7, 37, 43	3bitr4i 268	. 2 ⊢ (adom PProd (A, B)b ↔ a(dom A × dom B)b)
45	44	eqbrriv 4851	1 ⊢ dom PProd (A, B) = (dom A × dom B)

Syntax hints:   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ⟨cop 4561   class class class wbr 4639   × cxp 4770  dom cdm 4772   PProd cpprod 5737

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-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