Theorem dmxpid 2553

Description: The domain of a cross product.

Assertion

Ref	Expression
dmxpid	⊢ dom (A × A) = A

Proof of Theorem dmxpid

Step	Hyp	Ref	Expression
1		xpeq1 2440	. . . . . 6 ⊢ (A = ∅ → (A × A) = (∅ × A))
2		xp0r 2474	. . . . . 6 ⊢ (∅ × A) = ∅
3	1, 2	syl6eq 1140	. . . . 5 ⊢ (A = ∅ → (A × A) = ∅)
4	3	dmeqd 2533	. . . 4 ⊢ (A = ∅ → dom (A × A) = dom ∅)
5		dm0 2542	. . . 4 ⊢ dom ∅ = ∅
6	4, 5	syl6eq 1140	. . 3 ⊢ (A = ∅ → dom (A × A) = ∅)
7		id 9	. . 3 ⊢ (A = ∅ → A = ∅)
8	6, 7	eqtr4d 1131	. 2 ⊢ (A = ∅ → dom (A × A) = A)
9		dmxp 2552	. 2 ⊢ (¬ A = ∅ → dom (A × A) = A)
10	8, 9	pm2.61i 110	1 ⊢ dom (A × A) = A

Syntax hints:   = wceq 1091  ∅c0 1707   × cxp 2408  dom cdm 2410

This theorem is referenced by:  ecopoprdm 3245

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp b6  ax-4 673  ax-5 674  ax-6&n sp;675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-xp 2424  df-dm 2428

metamath.org