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Theorem xpexcnv 7858
Description: A condition where the converse of xpex 7688 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
StepHypRef Expression
1 dmexg 7841 . . 3 ((𝐴 × 𝐵) ∈ V → dom (𝐴 × 𝐵) ∈ V)
2 dmxp 5885 . . . 4 (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴)
32eleq1d 2819 . . 3 (𝐵 ≠ ∅ → (dom (𝐴 × 𝐵) ∈ V ↔ 𝐴 ∈ V))
41, 3imbitrid 243 . 2 (𝐵 ≠ ∅ → ((𝐴 × 𝐵) ∈ V → 𝐴 ∈ V))
54imp 408 1 ((𝐵 ≠ ∅ ∧ (𝐴 × 𝐵) ∈ V) → 𝐴 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wcel 2107  wne 2940  Vcvv 3444  c0 4283   × cxp 5632  dom cdm 5634
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-xp 5640  df-cnv 5642  df-dm 5644  df-rn 5645
This theorem is referenced by:  fczsupp0  8125
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