Theorem ecxpid 19273

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 3462	. . . 4 ⊢ 𝑥 ∈ V
2		elecg 8748	. . . 4 ⊢ ((𝑥 ∈ V ∧ 𝑋 ∈ 𝐴) → (𝑥 ∈ [𝑋](𝐴 × 𝐴) ↔ 𝑋(𝐴 × 𝐴)𝑥))
3	1, 2	mpan 703	. . 3 ⊢ (𝑋 ∈ 𝐴 → (𝑥 ∈ [𝑋](𝐴 × 𝐴) ↔ 𝑋(𝐴 × 𝐴)𝑥))
4		brxp 5715	. . . 4 ⊢ (𝑋(𝐴 × 𝐴)𝑥 ↔ (𝑋 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴))
5	4	baib 545	. . 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 2764	1 ⊢ (𝑋 ∈ 𝐴 → [𝑋](𝐴 × 𝐴) = 𝐴)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2146  Vcvv 3458   class class class wbr 5114   × cxp 5664  [cec 8701

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738  ax-sep 5262  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115  df-opab 5179  df-xp 5672  df-cnv 5674  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-ec 8705

This theorem is used by:  qsxpid  19274