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Theorem cnvkxpk 4277

Description: The converse of a Kuratowski cross product. (Contributed by SF, 13-Jan-2015.)

**Assertion**
Ref	Expression
cnvkxpk	⊢ ^◡_k(A ×_k B) = (B ×_k A)

Proof of Theorem cnvkxpk Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnvkssvvk 4276	. 2 ⊢ ^◡_k(A ×_k B) ⊆ (V ×_k V)
2		xpkssvvk 4211	. 2 ⊢ (B ×_k A) ⊆ (V ×_k V)
3		ancom 437	. . 3 ⊢ ((y ∈ A ∧ x ∈ B) ↔ (x ∈ B ∧ y ∈ A))
4		vex 2863	. . . . 5 ⊢ x ∈ V
5		vex 2863	. . . . 5 ⊢ y ∈ V
6	4, 5	opkelcnvk 4251	. . . 4 ⊢ (⟪x, y⟫ ∈ ^◡_k(A ×_k B) ↔ ⟪y, x⟫ ∈ (A ×_k B))
7	5, 4	opkelxpk 4249	. . . 4 ⊢ (⟪y, x⟫ ∈ (A ×_k B) ↔ (y ∈ A ∧ x ∈ B))
8	6, 7	bitri 240	. . 3 ⊢ (⟪x, y⟫ ∈ ^◡_k(A ×_k B) ↔ (y ∈ A ∧ x ∈ B))
9	4, 5	opkelxpk 4249	. . 3 ⊢ (⟪x, y⟫ ∈ (B ×_k A) ↔ (x ∈ B ∧ y ∈ A))
10	3, 8, 9	3bitr4i 268	. 2 ⊢ (⟪x, y⟫ ∈ ^◡_k(A ×_k B) ↔ ⟪x, y⟫ ∈ (B ×_k A))
11	1, 2, 10	eqrelkriiv 4214	1 ⊢ ^◡_k(A ×_k B) = (B ×_k A)

Colors of variables: wff setvar class

Syntax hints: ∧ wa 358 = wceq 1642 ∈ wcel 1710 ⟪copk 4058 ×_k cxpk 4175 ^◡_kccnvk 4176

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-3an 936 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-ral 2620 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 df-cnvk 4187

This theorem is referenced by: xpkexg 4289