Theorem dmxpin 5923

Description: The domain of the intersection of two Cartesian squares. Unlike in dmin 5903, 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 5820	. . 3 ⊢ ((𝐴 × 𝐴) ∩ (𝐵 × 𝐵)) = ((𝐴 ∩ 𝐵) × (𝐴 ∩ 𝐵))
2	1	dmeqi 5896	. 2 ⊢ dom ((𝐴 × 𝐴) ∩ (𝐵 × 𝐵)) = dom ((𝐴 ∩ 𝐵) × (𝐴 ∩ 𝐵))
3		dmxpid 5922	. 2 ⊢ dom ((𝐴 ∩ 𝐵) × (𝐴 ∩ 𝐵)) = (𝐴 ∩ 𝐵)
4	2, 3	eqtri 2786	1 ⊢ dom ((𝐴 × 𝐴) ∩ (𝐵 × 𝐵)) = (𝐴 ∩ 𝐵)

Syntax hints:   = wceq 1570   ∩ cin 3905   × cxp 5661  dom cdm 5663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5258  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-br 5111  df-opab 5175  df-xp 5669  df-rel 5670  df-dm 5673

This theorem is referenced by: (None)