Theorem dmxpin 5773

Description: The domain of the intersection of two Cartesian squares. Unlike in dmin 5752, 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 5673	. . 3 ⊢ ((𝐴 × 𝐴) ∩ (𝐵 × 𝐵)) = ((𝐴 ∩ 𝐵) × (𝐴 ∩ 𝐵))
2	1	dmeqi 5745	. 2 ⊢ dom ((𝐴 × 𝐴) ∩ (𝐵 × 𝐵)) = dom ((𝐴 ∩ 𝐵) × (𝐴 ∩ 𝐵))
3		dmxpid 5772	. 2 ⊢ dom ((𝐴 ∩ 𝐵) × (𝐴 ∩ 𝐵)) = (𝐴 ∩ 𝐵)
4	2, 3	eqtri 2782	1 ⊢ dom ((𝐴 × 𝐴) ∩ (𝐵 × 𝐵)) = (𝐴 ∩ 𝐵)

Syntax hints:   = wceq 1539   ∩ cin 3858   × cxp 5523  dom cdm 5525

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pr 5299

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rab 3080  df-v 3412  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-sn 4524  df-pr 4526  df-op 4530  df-br 5034  df-opab 5096  df-xp 5531  df-rel 5532  df-dm 5535

This theorem is referenced by: (None)