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Theorem ecxpid 33086
Description: The equivalence class of a cartesian product is the whole set. (Contributed by Thierry Arnoux, 15-Jan-2024.)
Assertion
Ref Expression
ecxpid (𝑋𝐴 → [𝑋](𝐴 × 𝐴) = 𝐴)

Proof of Theorem ecxpid
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 vex 3475 . . . 4 𝑥 ∈ V
2 elecg 8768 . . . 4 ((𝑥 ∈ V ∧ 𝑋𝐴) → (𝑥 ∈ [𝑋](𝐴 × 𝐴) ↔ 𝑋(𝐴 × 𝐴)𝑥))
31, 2mpan 689 . . 3 (𝑋𝐴 → (𝑥 ∈ [𝑋](𝐴 × 𝐴) ↔ 𝑋(𝐴 × 𝐴)𝑥))
4 brxp 5727 . . . 4 (𝑋(𝐴 × 𝐴)𝑥 ↔ (𝑋𝐴𝑥𝐴))
54baib 535 . . 3 (𝑋𝐴 → (𝑋(𝐴 × 𝐴)𝑥𝑥𝐴))
63, 5bitrd 279 . 2 (𝑋𝐴 → (𝑥 ∈ [𝑋](𝐴 × 𝐴) ↔ 𝑥𝐴))
76eqrdv 2726 1 (𝑋𝐴 → [𝑋](𝐴 × 𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1534  wcel 2099  Vcvv 3471   class class class wbr 5148   × cxp 5676  [cec 8723
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5149  df-opab 5211  df-xp 5684  df-cnv 5686  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-ec 8727
This theorem is referenced by:  qsxpid  33087
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