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Theorem xpexcnv 7932
Description: A condition where the converse of xpex 7759 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 7913 . . 3 ((𝐴 × 𝐵) ∈ V → dom (𝐴 × 𝐵) ∈ V)
2 dmxp 5933 . . . 4 (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴)
32eleq1d 2813 . . 3 (𝐵 ≠ ∅ → (dom (𝐴 × 𝐵) ∈ V ↔ 𝐴 ∈ V))
41, 3imbitrid 243 . 2 (𝐵 ≠ ∅ → ((𝐴 × 𝐵) ∈ V → 𝐴 ∈ V))
54imp 405 1 ((𝐵 ≠ ∅ ∧ (𝐴 × 𝐵) ∈ V) → 𝐴 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wcel 2098  wne 2936  Vcvv 3471  c0 4324   × cxp 5678  dom cdm 5680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pr 5431  ax-un 7744
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-br 5151  df-opab 5213  df-xp 5686  df-cnv 5688  df-dm 5690  df-rn 5691
This theorem is referenced by:  fczsupp0  8202
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