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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.
StepHypRef Expression
1 vex 2862 . . . . . . 7 c V
2 vex 2862 . . . . . . 7 d V
31, 2opex 4588 . . . . . 6 c, d V
43isseti 2865 . . . . 5 x x = c, d
5 19.41v 1901 . . . . 5 (x(x = c, d (aAc bBd)) ↔ (x x = c, d (aAc bBd)))
64, 5mpbiran 884 . . . 4 (x(x = c, d (aAc bBd)) ↔ (aAc bBd))
762exbii 1583 . . 3 (cdx(x = c, d (aAc bBd)) ↔ cd(aAc bBd))
8 df-br 4640 . . . 4 (adom PProd (A, B)ba, b dom PProd (A, B))
9 eldm 4898 . . . 4 (a, b dom PProd (A, B) ↔ xa, b PProd (A, B)x)
10 brpprod 5839 . . . . . . 7 (a, b PProd (A, B)xtucd(a, b = t, u x = c, d (tAc uBd)))
11 19.42vv 1907 . . . . . . . . 9 (cd((t = a u = b) (x = c, d (tAc uBd))) ↔ ((t = a u = b) cd(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, ut, u = a, b)
14 opth 4602 . . . . . . . . . . . . 13 (t, u = a, b ↔ (t = a u = b))
1513, 14bitri 240 . . . . . . . . . . . 12 (a, b = t, u ↔ (t = a u = b))
1615anbi1i 676 . . . . . . . . . . 11 ((a, b = t, u (x = c, d (tAc uBd))) ↔ ((t = a u = b) (x = c, d (tAc uBd))))
1712, 16bitri 240 . . . . . . . . . 10 ((a, b = t, u x = c, d (tAc uBd)) ↔ ((t = a u = b) (x = c, d (tAc uBd))))
18172exbii 1583 . . . . . . . . 9 (cd(a, b = t, u x = c, d (tAc uBd)) ↔ cd((t = a u = b) (x = c, d (tAc uBd))))
19 df-3an 936 . . . . . . . . 9 ((t = a u = b cd(x = c, d (tAc uBd))) ↔ ((t = a u = b) cd(x = c, d (tAc uBd))))
2011, 18, 193bitr4i 268 . . . . . . . 8 (cd(a, b = t, u x = c, d (tAc uBd)) ↔ (t = a u = b cd(x = c, d (tAc uBd))))
21202exbii 1583 . . . . . . 7 (tucd(a, b = t, u x = c, d (tAc uBd)) ↔ tu(t = a u = b cd(x = c, d (tAc uBd))))
22 vex 2862 . . . . . . . 8 a V
23 vex 2862 . . . . . . . 8 b V
24 breq1 4642 . . . . . . . . . . 11 (t = a → (tAcaAc))
2524anbi1d 685 . . . . . . . . . 10 (t = a → ((tAc uBd) ↔ (aAc uBd)))
2625anbi2d 684 . . . . . . . . 9 (t = a → ((x = c, d (tAc uBd)) ↔ (x = c, d (aAc uBd))))
27262exbidv 1628 . . . . . . . 8 (t = a → (cd(x = c, d (tAc uBd)) ↔ cd(x = c, d (aAc uBd))))
28 breq1 4642 . . . . . . . . . . 11 (u = b → (uBdbBd))
2928anbi2d 684 . . . . . . . . . 10 (u = b → ((aAc uBd) ↔ (aAc bBd)))
3029anbi2d 684 . . . . . . . . 9 (u = b → ((x = c, d (aAc uBd)) ↔ (x = c, d (aAc bBd))))
31302exbidv 1628 . . . . . . . 8 (u = b → (cd(x = c, d (aAc uBd)) ↔ cd(x = c, d (aAc bBd))))
3222, 23, 27, 31ceqsex2v 2896 . . . . . . 7 (tu(t = a u = b cd(x = c, d (tAc uBd))) ↔ cd(x = c, d (aAc bBd)))
3310, 21, 323bitri 262 . . . . . 6 (a, b PProd (A, B)xcd(x = c, d (aAc bBd)))
3433exbii 1582 . . . . 5 (xa, b PProd (A, B)xxcd(x = c, d (aAc bBd)))
35 exrot3 1744 . . . . 5 (xcd(x = c, d (aAc bBd)) ↔ cdx(x = c, d (aAc bBd)))
3634, 35bitri 240 . . . 4 (xa, b PProd (A, B)xcdx(x = c, d (aAc bBd)))
378, 9, 363bitri 262 . . 3 (adom PProd (A, B)bcdx(x = c, d (aAc bBd)))
38 eldm 4898 . . . . 5 (a dom Ac aAc)
39 eldm 4898 . . . . 5 (b dom Bd bBd)
4038, 39anbi12i 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 (cd(aAc bBd) ↔ (c aAc d bBd))
4340, 41, 423bitr4i 268 . . 3 (a(dom A × dom B)bcd(aAc bBd))
447, 37, 433bitr4i 268 . 2 (adom PProd (A, B)ba(dom A × dom B)b)
4544eqbrriv 4851 1 dom PProd (A, B) = (dom A × dom B)
 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  dom 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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