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Mirrors > Home > NFE Home > Th. List > cnvkxpk | GIF version |
Description: The converse of a Kuratowski cross product. (Contributed by SF, 13-Jan-2015.) |
Ref | Expression |
---|---|
cnvkxpk | ⊢ ◡k(A ×k B) = (B ×k A) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvkssvvk 4275 | . 2 ⊢ ◡k(A ×k B) ⊆ (V ×k V) | |
2 | xpkssvvk 4210 | . 2 ⊢ (B ×k A) ⊆ (V ×k V) | |
3 | ancom 437 | . . 3 ⊢ ((y ∈ A ∧ x ∈ B) ↔ (x ∈ B ∧ y ∈ A)) | |
4 | vex 2862 | . . . . 5 ⊢ x ∈ V | |
5 | vex 2862 | . . . . 5 ⊢ y ∈ V | |
6 | 4, 5 | opkelcnvk 4250 | . . . 4 ⊢ (⟪x, y⟫ ∈ ◡k(A ×k B) ↔ ⟪y, x⟫ ∈ (A ×k B)) |
7 | 5, 4 | opkelxpk 4248 | . . . 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 4248 | . . 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 4213 | 1 ⊢ ◡k(A ×k B) = (B ×k A) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 358 = wceq 1642 ∈ wcel 1710 ⟪copk 4057 ×k cxpk 4174 ◡kccnvk 4175 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 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 4078 ax-sn 4087 |
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 2478 df-ne 2518 df-ral 2619 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-ss 3259 df-nul 3551 df-sn 3741 df-pr 3742 df-opk 4058 df-xpk 4185 df-cnvk 4186 |
This theorem is referenced by: xpkexg 4288 |
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