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 32195
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 3448 . . . 4 𝑥 ∈ V
2 elecg 8694 . . . 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 5682 . . . 4 (𝑋(𝐴 × 𝐴)𝑥 ↔ (𝑋𝐴𝑥𝐴))
54baib 537 . . 3 (𝑋𝐴 → (𝑋(𝐴 × 𝐴)𝑥𝑥𝐴))
63, 5bitrd 279 . 2 (𝑋𝐴 → (𝑥 ∈ [𝑋](𝐴 × 𝐴) ↔ 𝑥𝐴))
76eqrdv 2731 1 (𝑋𝐴 → [𝑋](𝐴 × 𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1542  wcel 2107  Vcvv 3444   class class class wbr 5106   × cxp 5632  [cec 8649
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-xp 5640  df-cnv 5642  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-ec 8653
This theorem is referenced by:  qsxpid  32197
  Copyright terms: Public domain W3C validator