Theorem ecxpid 32467

Description: The equivalence class of a cartesian product is the whole set. (Contributed by Thierry Arnoux, 15-Jan-2024.)

Assertion

Ref	Expression
ecxpid	⊢ (𝑋 ∈ 𝐴 → [𝑋](𝐴 × 𝐴) = 𝐴)

Proof of Theorem ecxpid

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3478	. . . 4 ⊢ 𝑥 ∈ V
2		elecg 8745	. . . 4 ⊢ ((𝑥 ∈ V ∧ 𝑋 ∈ 𝐴) → (𝑥 ∈ [𝑋](𝐴 × 𝐴) ↔ 𝑋(𝐴 × 𝐴)𝑥))
3	1, 2	mpan 688	. . 3 ⊢ (𝑋 ∈ 𝐴 → (𝑥 ∈ [𝑋](𝐴 × 𝐴) ↔ 𝑋(𝐴 × 𝐴)𝑥))
4		brxp 5725	. . . 4 ⊢ (𝑋(𝐴 × 𝐴)𝑥 ↔ (𝑋 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴))
5	4	baib 536	. . 3 ⊢ (𝑋 ∈ 𝐴 → (𝑋(𝐴 × 𝐴)𝑥 ↔ 𝑥 ∈ 𝐴))
6	3, 5	bitrd 278	. 2 ⊢ (𝑋 ∈ 𝐴 → (𝑥 ∈ [𝑋](𝐴 × 𝐴) ↔ 𝑥 ∈ 𝐴))
7	6	eqrdv 2730	1 ⊢ (𝑋 ∈ 𝐴 → [𝑋](𝐴 × 𝐴) = 𝐴)

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1541   ∈ wcel 2106  Vcvv 3474   class class class wbr 5148   × cxp 5674  [cec 8700

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-cnv 5684  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ec 8704

This theorem is referenced by:  qsxpid  32469