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Theorem xpexcnv 7849
Description: A condition where the converse of xpex 7679 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 7832 . . 3 ((𝐴 × 𝐵) ∈ V → dom (𝐴 × 𝐵) ∈ V)
2 dmxp 5882 . . . 4 (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴)
32eleq1d 2822 . . 3 (𝐵 ≠ ∅ → (dom (𝐴 × 𝐵) ∈ V ↔ 𝐴 ∈ V))
41, 3imbitrid 243 . 2 (𝐵 ≠ ∅ → ((𝐴 × 𝐵) ∈ V → 𝐴 ∈ V))
54imp 407 1 ((𝐵 ≠ ∅ ∧ (𝐴 × 𝐵) ∈ V) → 𝐴 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2106  wne 2941  Vcvv 3443  c0 4280   × cxp 5629  dom cdm 5631
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-sep 5254  ax-nul 5261  ax-pr 5382  ax-un 7664
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-xp 5637  df-cnv 5639  df-dm 5641  df-rn 5642
This theorem is referenced by:  fczsupp0  8116
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