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Theorem dmxpOLD 5940
Description: Obsolete version of dmxp 5939 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 5691 . . 3 (𝐴 × 𝐵) = {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝑥𝐵)}
21dmeqi 5915 . 2 dom (𝐴 × 𝐵) = dom {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝑥𝐵)}
3 n0 4353 . . . . 5 (𝐵 ≠ ∅ ↔ ∃𝑥 𝑥𝐵)
43biimpi 216 . . . 4 (𝐵 ≠ ∅ → ∃𝑥 𝑥𝐵)
54ralrimivw 3150 . . 3 (𝐵 ≠ ∅ → ∀𝑦𝐴𝑥 𝑥𝐵)
6 dmopab3 5930 . . 3 (∀𝑦𝐴𝑥 𝑥𝐵 ↔ dom {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝑥𝐵)} = 𝐴)
75, 6sylib 218 . 2 (𝐵 ≠ ∅ → dom {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝑥𝐵)} = 𝐴)
82, 7eqtrid 2789 1 (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wex 1779  wcel 2108  wne 2940  wral 3061  c0 4333  {copab 5205   × cxp 5683  dom cdm 5685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-dm 5695
This theorem is referenced by: (None)
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