Theorem inxp 4863

Description: The intersection of two cross products. Exercise 9 of [TakeutiZaring] p. 25. (The proof was shortened by Andrew Salmon, 27-Aug-2011.) (Contributed by NM, 3-Aug-1994.) (Revised by set.mm contributors, 27-Aug-2011.)

Assertion

Ref	Expression
inxp	⊢ ((A × B) ∩ (C × D)) = ((A ∩ C) × (B ∩ D))

Proof of Theorem inxp

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		inopab 4862	. . 3 ⊢ ({⟨x, y⟩ ∣ (x ∈ A ∧ y ∈ B)} ∩ {⟨x, y⟩ ∣ (x ∈ C ∧ y ∈ D)}) = {⟨x, y⟩ ∣ ((x ∈ A ∧ y ∈ B) ∧ (x ∈ C ∧ y ∈ D))}
2		an4 797	. . . . 5 ⊢ (((x ∈ A ∧ y ∈ B) ∧ (x ∈ C ∧ y ∈ D)) ↔ ((x ∈ A ∧ x ∈ C) ∧ (y ∈ B ∧ y ∈ D)))
3		elin 3219	. . . . . 6 ⊢ (x ∈ (A ∩ C) ↔ (x ∈ A ∧ x ∈ C))
4		elin 3219	. . . . . 6 ⊢ (y ∈ (B ∩ D) ↔ (y ∈ B ∧ y ∈ D))
5	3, 4	anbi12i 678	. . . . 5 ⊢ ((x ∈ (A ∩ C) ∧ y ∈ (B ∩ D)) ↔ ((x ∈ A ∧ x ∈ C) ∧ (y ∈ B ∧ y ∈ D)))
6	2, 5	bitr4i 243	. . . 4 ⊢ (((x ∈ A ∧ y ∈ B) ∧ (x ∈ C ∧ y ∈ D)) ↔ (x ∈ (A ∩ C) ∧ y ∈ (B ∩ D)))
7	6	opabbii 4626	. . 3 ⊢ {⟨x, y⟩ ∣ ((x ∈ A ∧ y ∈ B) ∧ (x ∈ C ∧ y ∈ D))} = {⟨x, y⟩ ∣ (x ∈ (A ∩ C) ∧ y ∈ (B ∩ D))}
8	1, 7	eqtri 2373	. 2 ⊢ ({⟨x, y⟩ ∣ (x ∈ A ∧ y ∈ B)} ∩ {⟨x, y⟩ ∣ (x ∈ C ∧ y ∈ D)}) = {⟨x, y⟩ ∣ (x ∈ (A ∩ C) ∧ y ∈ (B ∩ D))}
9		df-xp 4784	. . 3 ⊢ (A × B) = {⟨x, y⟩ ∣ (x ∈ A ∧ y ∈ B)}
10		df-xp 4784	. . 3 ⊢ (C × D) = {⟨x, y⟩ ∣ (x ∈ C ∧ y ∈ D)}
11	9, 10	ineq12i 3455	. 2 ⊢ ((A × B) ∩ (C × D)) = ({⟨x, y⟩ ∣ (x ∈ A ∧ y ∈ B)} ∩ {⟨x, y⟩ ∣ (x ∈ C ∧ y ∈ D)})
12		df-xp 4784	. 2 ⊢ ((A ∩ C) × (B ∩ D)) = {⟨x, y⟩ ∣ (x ∈ (A ∩ C) ∧ y ∈ (B ∩ D))}
13	8, 11, 12	3eqtr4i 2383	1 ⊢ ((A × B) ∩ (C × D)) = ((A ∩ C) × (B ∩ D))

Syntax hints:   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ∩ cin 3208  {copab 4622   × cxp 4770

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-xp 4784

This theorem is referenced by:  xpindi  4864  xpindir  4865  dmxpin  4925  xpdisj1  5047  xpdisj2  5048