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Theorem dfxp2 5113
 Description: Define cross product via the set construction functions. (Contributed by SF, 8-Jan-2015.)
Assertion
Ref Expression
dfxp2 (A × B) = ((1stA) ∩ (2ndB))

Proof of Theorem dfxp2
Dummy variables x y z w v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eeanv 1913 . . . . 5 (wv(x = y, w x = v, z) ↔ (w x = y, w v x = v, z))
2 vex 2862 . . . . . . . 8 z V
3 vex 2862 . . . . . . . 8 y V
4 opeq2 4579 . . . . . . . . . 10 (w = zy, w = y, z)
54eqeq2d 2364 . . . . . . . . 9 (w = z → (x = y, wx = y, z))
6 opeq1 4578 . . . . . . . . . 10 (v = yv, z = y, z)
76eqeq2d 2364 . . . . . . . . 9 (v = y → (x = v, zx = y, z))
85, 7bi2anan9 843 . . . . . . . 8 ((w = z v = y) → ((x = y, w x = v, z) ↔ (x = y, z x = y, z)))
92, 3, 8spc2ev 2947 . . . . . . 7 ((x = y, z x = y, z) → wv(x = y, w x = v, z))
109anidms 626 . . . . . 6 (x = y, zwv(x = y, w x = v, z))
11 simpl 443 . . . . . . . 8 ((x = y, w x = v, z) → x = y, w)
12 eqtr2 2371 . . . . . . . . 9 ((x = y, w x = v, z) → y, w = v, z)
13 opth 4602 . . . . . . . . . 10 (y, w = v, z ↔ (y = v w = z))
144adantl 452 . . . . . . . . . 10 ((y = v w = z) → y, w = y, z)
1513, 14sylbi 187 . . . . . . . . 9 (y, w = v, zy, w = y, z)
1612, 15syl 15 . . . . . . . 8 ((x = y, w x = v, z) → y, w = y, z)
1711, 16eqtrd 2385 . . . . . . 7 ((x = y, w x = v, z) → x = y, z)
1817exlimivv 1635 . . . . . 6 (wv(x = y, w x = v, z) → x = y, z)
1910, 18impbii 180 . . . . 5 (x = y, zwv(x = y, w x = v, z))
20 brcnv 4892 . . . . . . 7 (y1st xx1st y)
213br1st 4858 . . . . . . 7 (x1st yw x = y, w)
2220, 21bitri 240 . . . . . 6 (y1st xw x = y, w)
23 brcnv 4892 . . . . . . 7 (z2nd xx2nd z)
242br2nd 4859 . . . . . . 7 (x2nd zv x = v, z)
2523, 24bitri 240 . . . . . 6 (z2nd xv x = v, z)
2622, 25anbi12i 678 . . . . 5 ((y1st x z2nd x) ↔ (w x = y, w v x = v, z))
271, 19, 263bitr4i 268 . . . 4 (x = y, z ↔ (y1st x z2nd x))
28272rexbii 2641 . . 3 (y A z B x = y, zy A z B (y1st x z2nd x))
29 elxp2 4802 . . 3 (x (A × B) ↔ y A z B x = y, z)
30 elima 4754 . . . . 5 (x (1stA) ↔ y A y1st x)
31 elima 4754 . . . . 5 (x (2ndB) ↔ z B z2nd x)
3230, 31anbi12i 678 . . . 4 ((x (1stA) x (2ndB)) ↔ (y A y1st x z B z2nd x))
33 elin 3219 . . . 4 (x ((1stA) ∩ (2ndB)) ↔ (x (1stA) x (2ndB)))
34 reeanv 2778 . . . 4 (y A z B (y1st x z2nd x) ↔ (y A y1st x z B z2nd x))
3532, 33, 343bitr4i 268 . . 3 (x ((1stA) ∩ (2ndB)) ↔ y A z B (y1st x z2nd x))
3628, 29, 353bitr4i 268 . 2 (x (A × B) ↔ x ((1stA) ∩ (2ndB)))
3736eqriv 2350 1 (A × B) = ((1stA) ∩ (2ndB))
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2615   ∩ cin 3208  ⟨cop 4561   class class class wbr 4639  1st c1st 4717   “ cima 4722   × cxp 4770  ◡ccnv 4771  2nd c2nd 4783 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-ima 4727  df-xp 4784  df-cnv 4785  df-2nd 4797 This theorem is referenced by:  xpexg  5114
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