Theorem xpexcnv 7943

Description: A condition where the converse of xpex 7774 holds as well. Corollary 6.9(2) in [TakeutiZaring] p. 26. (Contributed by Andrew Salmon, 13-Nov-2011.)

Assertion

Ref	Expression
xpexcnv	⊢ ((𝐵 ≠ ∅ ∧ (𝐴 × 𝐵) ∈ V) → 𝐴 ∈ V)

Proof of Theorem xpexcnv

Step	Hyp	Ref	Expression
1		dmexg 7924	. . 3 ⊢ ((𝐴 × 𝐵) ∈ V → dom (𝐴 × 𝐵) ∈ V)
2		dmxp 5938	. . . 4 ⊢ (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴)
3	2	eleq1d 2825	. . 3 ⊢ (𝐵 ≠ ∅ → (dom (𝐴 × 𝐵) ∈ V ↔ 𝐴 ∈ V))
4	1, 3	imbitrid 244	. 2 ⊢ (𝐵 ≠ ∅ → ((𝐴 × 𝐵) ∈ V → 𝐴 ∈ V))
5	4	imp 406	1 ⊢ ((𝐵 ≠ ∅ ∧ (𝐴 × 𝐵) ∈ V) → 𝐴 ∈ V)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2107   ≠ wne 2939  Vcvv 3479  ∅c0 4332   × cxp 5682  dom cdm 5684

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-xp 5690  df-cnv 5692  df-dm 5694  df-rn 5695

This theorem is referenced by:  fczsupp0  8219