Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ecxpid Structured version   Visualization version   GIF version

Theorem ecxpid 32979
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 3472 . . . 4 𝑥 ∈ V
2 elecg 8745 . . . 4 ((𝑥 ∈ V ∧ 𝑋𝐴) → (𝑥 ∈ [𝑋](𝐴 × 𝐴) ↔ 𝑋(𝐴 × 𝐴)𝑥))
31, 2mpan 687 . . 3 (𝑋𝐴 → (𝑥 ∈ [𝑋](𝐴 × 𝐴) ↔ 𝑋(𝐴 × 𝐴)𝑥))
4 brxp 5718 . . . 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 2724 1 (𝑋𝐴 → [𝑋](𝐴 × 𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1533  wcel 2098  Vcvv 3468   class class class wbr 5141   × cxp 5667  [cec 8700
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-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-xp 5675  df-cnv 5677  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ec 8704
This theorem is referenced by:  qsxpid  32981
  Copyright terms: Public domain W3C validator