Theorem dmxpinm 4556

Description: The domain of the intersection of two square cross products. Unlike dmin 4543, equality holds. (Contributed by NM, 29-Jan-2008.)

Assertion

Ref	Expression
dmxpinm	⊢ (∃𝑥 𝑥 ∈ (𝐴 ∩ 𝐵) → dom ((𝐴 × 𝐴) ∩ (𝐵 × 𝐵)) = (𝐴 ∩ 𝐵))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem dmxpinm

Step	Hyp	Ref	Expression
1		inxp 4470	. . . 4 ⊢ ((𝐴 × 𝐴) ∩ (𝐵 × 𝐵)) = ((𝐴 ∩ 𝐵) × (𝐴 ∩ 𝐵))
2	1	dmeqi 4536	. . 3 ⊢ dom ((𝐴 × 𝐴) ∩ (𝐵 × 𝐵)) = dom ((𝐴 ∩ 𝐵) × (𝐴 ∩ 𝐵))
3	2	a1i 9	. 2 ⊢ (∃𝑥 𝑥 ∈ (𝐴 ∩ 𝐵) → dom ((𝐴 × 𝐴) ∩ (𝐵 × 𝐵)) = dom ((𝐴 ∩ 𝐵) × (𝐴 ∩ 𝐵)))
4		dmxpm 4555	. 2 ⊢ (∃𝑥 𝑥 ∈ (𝐴 ∩ 𝐵) → dom ((𝐴 ∩ 𝐵) × (𝐴 ∩ 𝐵)) = (𝐴 ∩ 𝐵))
5	3, 4	eqtrd 2072	1 ⊢ (∃𝑥 𝑥 ∈ (𝐴 ∩ 𝐵) → dom ((𝐴 × 𝐴) ∩ (𝐵 × 𝐵)) = (𝐴 ∩ 𝐵))

Syntax hints:   → wi 4   = wceq 1243  ∃wex 1381   ∈ wcel 1393   ∩ cin 2916   × cxp 4343  dom cdm 4345

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-dm 4355

This theorem is referenced by: (None)