Theorem dmxpin 5254

Description: The domain of the intersection of two square Cartesian products. Unlike dmin 5241, equality holds. (Contributed by NM, 29-Jan-2008.)

Assertion

Ref	Expression
dmxpin	⊢ dom ((𝐴 × 𝐴) ∩ (𝐵 × 𝐵)) = (𝐴 ∩ 𝐵)

Proof of Theorem dmxpin

Step	Hyp	Ref	Expression
1		inxp 5164	. . 3 ⊢ ((𝐴 × 𝐴) ∩ (𝐵 × 𝐵)) = ((𝐴 ∩ 𝐵) × (𝐴 ∩ 𝐵))
2	1	dmeqi 5234	. 2 ⊢ dom ((𝐴 × 𝐴) ∩ (𝐵 × 𝐵)) = dom ((𝐴 ∩ 𝐵) × (𝐴 ∩ 𝐵))
3		dmxpid 5253	. 2 ⊢ dom ((𝐴 ∩ 𝐵) × (𝐴 ∩ 𝐵)) = (𝐴 ∩ 𝐵)
4	2, 3	eqtri 2631	1 ⊢ dom ((𝐴 × 𝐴) ∩ (𝐵 × 𝐵)) = (𝐴 ∩ 𝐵)

Syntax hints:   = wceq 1474   ∩ cin 3538   × cxp 5026  dom cdm 5028

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828

This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-br 4578  df-opab 4638  df-xp 5034  df-rel 5035  df-dm 5038

This theorem is referenced by: (None)