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Theorem dmxpOLD 5942
Description: Obsolete version of dmxp 5941 as of 19-Dec-2024. (Contributed by NM, 28-Jul-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
dmxpOLD (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴)

Proof of Theorem dmxpOLD
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-xp 5694 . . 3 (𝐴 × 𝐵) = {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝑥𝐵)}
21dmeqi 5917 . 2 dom (𝐴 × 𝐵) = dom {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝑥𝐵)}
3 n0 4358 . . . . 5 (𝐵 ≠ ∅ ↔ ∃𝑥 𝑥𝐵)
43biimpi 216 . . . 4 (𝐵 ≠ ∅ → ∃𝑥 𝑥𝐵)
54ralrimivw 3147 . . 3 (𝐵 ≠ ∅ → ∀𝑦𝐴𝑥 𝑥𝐵)
6 dmopab3 5932 . . 3 (∀𝑦𝐴𝑥 𝑥𝐵 ↔ dom {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝑥𝐵)} = 𝐴)
75, 6sylib 218 . 2 (𝐵 ≠ ∅ → dom {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝑥𝐵)} = 𝐴)
82, 7eqtrid 2786 1 (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1536  wex 1775  wcel 2105  wne 2937  wral 3058  c0 4338  {copab 5209   × cxp 5686  dom cdm 5688
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210  df-xp 5694  df-dm 5698
This theorem is referenced by: (None)
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