Theorem cnvxp 2651

Description: The converse of a cross product. Exercise 11 of [Suppes] p. 67.

Assertion

Ref	Expression
cnvxp	⊢ ◡(A × B) = (B × A)

Proof of Theorem cnvxp

Step	Hyp	Ref	Expression
1		relcnv 2624	. 2 ⊢ Rel ◡(A × B)
2		relxp 2486	. 2 ⊢ Rel (B × A)
3		visset 1350	. . . 4 ⊢ x ∈ V
4		visset 1350	. . . 4 ⊢ y ∈ V
5	3, 4	opelcnv 2518	. . 3 ⊢ (⟨x, y⟩ ∈ ◡(A × B) ↔ ⟨y, x⟩ ∈ (A × B))
6		ancom 333	. . . 4 ⊢ ((y ∈ A ∧ x ∈ B) ↔ (x ∈ B ∧ y ∈ A))
7	3	opelxp 2452	. . . 4 ⊢ (⟨y, x⟩ ∈ (A × B) ↔ (y ∈ A ∧ x ∈ B))
8	4	opelxp 2452	. . . 4 ⊢ (⟨x, y⟩ ∈ (B × A) ↔ (x ∈ B ∧ y ∈ A))
9	6, 7, 8	3bitr4 158	. . 3 ⊢ (⟨y, x⟩ ∈ (A × B) ↔ ⟨x, y⟩ ∈ (B × A))
10	5, 9	bitr 151	. 2 ⊢ (⟨x, y⟩ ∈ ◡(A × B) ↔ ⟨x, y⟩ ∈ (B × A))
11	1, 2, 10	cleqreli 2484	1 ⊢ ◡(A × B) = (B × A)

Syntax hints:   ∧ wa 196   = wceq 1091   ∈ wcel 1092  ⟨cop 1810   × cxp 2408  ^◡ccnv 2409

This theorem is referenced by:  xp0 2652  rnxp 2657  fconst 2774

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-xp 2424  df-rel 2425  df-cnv 2426

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