Theorem ecxpid 19243

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 3459	. . . 4 ⊢ 𝑥 ∈ V
2		elecg 8740	. . . 4 ⊢ ((𝑥 ∈ V ∧ 𝑋 ∈ 𝐴) → (𝑥 ∈ [𝑋](𝐴 × 𝐴) ↔ 𝑋(𝐴 × 𝐴)𝑥))
3	1, 2	mpan 702	. . 3 ⊢ (𝑋 ∈ 𝐴 → (𝑥 ∈ [𝑋](𝐴 × 𝐴) ↔ 𝑋(𝐴 × 𝐴)𝑥))
4		brxp 5712	. . . 4 ⊢ (𝑋(𝐴 × 𝐴)𝑥 ↔ (𝑋 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴))
5	4	baib 544	. . 3 ⊢ (𝑋 ∈ 𝐴 → (𝑋(𝐴 × 𝐴)𝑥 ↔ 𝑥 ∈ 𝐴))
6	3, 5	bitrd 282	. 2 ⊢ (𝑋 ∈ 𝐴 → (𝑥 ∈ [𝑋](𝐴 × 𝐴) ↔ 𝑥 ∈ 𝐴))
7	6	eqrdv 2761	1 ⊢ (𝑋 ∈ 𝐴 → [𝑋](𝐴 × 𝐴) = 𝐴)

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2143  Vcvv 3455   class class class wbr 5110   × cxp 5661  [cec 8693

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5258  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-br 5111  df-opab 5175  df-xp 5669  df-cnv 5671  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-ec 8697

This theorem is referenced by:  qsxpid  19244