Theorem xpkeq2 4200

Description: Equality theorem for Kuratowski cross product. (Contributed by SF, 12-Jan-2015.)

Assertion

Ref	Expression
xpkeq2	⊢ (A = B → (C ×k A) = (C ×k B))

Proof of Theorem xpkeq2

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rexeq 2809	. . . 4 ⊢ (A = B → (∃z ∈ A x = ⟪y, z⟫ ↔ ∃z ∈ B x = ⟪y, z⟫))
2	1	rexbidv 2636	. . 3 ⊢ (A = B → (∃y ∈ C ∃z ∈ A x = ⟪y, z⟫ ↔ ∃y ∈ C ∃z ∈ B x = ⟪y, z⟫))
3		elxpk2 4198	. . 3 ⊢ (x ∈ (C ×k A) ↔ ∃y ∈ C ∃z ∈ A x = ⟪y, z⟫)
4		elxpk2 4198	. . 3 ⊢ (x ∈ (C ×k B) ↔ ∃y ∈ C ∃z ∈ B x = ⟪y, z⟫)
5	2, 3, 4	3bitr4g 279	. 2 ⊢ (A = B → (x ∈ (C ×k A) ↔ x ∈ (C ×k B)))
6	5	eqrdv 2351	1 ⊢ (A = B → (C ×k A) = (C ×k B))

Syntax hints:   → wi 4   = wceq 1642   ∈ wcel 1710  ∃wrex 2616  ⟪copk 4058   ×_k cxpk 4175

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-ss 3260  df-nul 3552  df-sn 3742  df-pr 3743  df-opk 4059  df-xpk 4186

This theorem is referenced by:  xpkeq12  4201  xpkeq2i  4203  xpkeq2d  4206  xpkvexg  4286