Theorem ecxpid 33354

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 3492	. . . 4 ⊢ 𝑥 ∈ V
2		elecg 8807	. . . 4 ⊢ ((𝑥 ∈ V ∧ 𝑋 ∈ 𝐴) → (𝑥 ∈ [𝑋](𝐴 × 𝐴) ↔ 𝑋(𝐴 × 𝐴)𝑥))
3	1, 2	mpan 689	. . 3 ⊢ (𝑋 ∈ 𝐴 → (𝑥 ∈ [𝑋](𝐴 × 𝐴) ↔ 𝑋(𝐴 × 𝐴)𝑥))
4		brxp 5749	. . . 4 ⊢ (𝑋(𝐴 × 𝐴)𝑥 ↔ (𝑋 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴))
5	4	baib 535	. . 3 ⊢ (𝑋 ∈ 𝐴 → (𝑋(𝐴 × 𝐴)𝑥 ↔ 𝑥 ∈ 𝐴))
6	3, 5	bitrd 279	. 2 ⊢ (𝑋 ∈ 𝐴 → (𝑥 ∈ [𝑋](𝐴 × 𝐴) ↔ 𝑥 ∈ 𝐴))
7	6	eqrdv 2738	1 ⊢ (𝑋 ∈ 𝐴 → [𝑋](𝐴 × 𝐴) = 𝐴)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1537   ∈ wcel 2108  Vcvv 3488   class class class wbr 5166   × cxp 5698  [cec 8761

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-xp 5706  df-cnv 5708  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ec 8765

This theorem is referenced by:  qsxpid  33355