Theorem dmxpin 4808

Description: The domain of the intersection of two square Cartesian products. Unlike dmin 4794, 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 4720	. . 3 ⊢ ((𝐴 × 𝐴) ∩ (𝐵 × 𝐵)) = ((𝐴 ∩ 𝐵) × (𝐴 ∩ 𝐵))
2	1	dmeqi 4787	. 2 ⊢ dom ((𝐴 × 𝐴) ∩ (𝐵 × 𝐵)) = dom ((𝐴 ∩ 𝐵) × (𝐴 ∩ 𝐵))
3		dmxpid 4807	. 2 ⊢ dom ((𝐴 ∩ 𝐵) × (𝐴 ∩ 𝐵)) = (𝐴 ∩ 𝐵)
4	2, 3	eqtri 2178	1 ⊢ dom ((𝐴 × 𝐴) ∩ (𝐵 × 𝐵)) = (𝐴 ∩ 𝐵)

Syntax hints:   = wceq 1335   ∩ cin 3101   × cxp 4584  dom cdm 4586

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-br 3966  df-opab 4026  df-xp 4592  df-rel 4593  df-dm 4596

This theorem is referenced by: (None)