Theorem cnvun 2642

Description: The converse of a union is the union of converses. Theorem 16 of [Suppes] p. 62.

Assertion

Ref	Expression
cnvun	⊢ ◡(A ∪ B) = (◡A ∪ ◡B)

Proof of Theorem cnvun

Step	Hyp	Ref	Expression
1		relcnv 2624	. 2 ⊢ Rel ◡(A ∪ B)
2		relcnv 2624	. . . 4 ⊢ Rel ◡A
3		relcnv 2624	. . . 4 ⊢ Rel ◡B
4	2, 3	pm3.2i 234	. . 3 ⊢ (Rel ◡A ∧ Rel ◡B)
5		relun 2490	. . 3 ⊢ (Rel (◡A ∪ ◡B) ↔ (Rel ◡A ∧ Rel ◡B))
6	4, 5	mpbir 165	. 2 ⊢ Rel (◡A ∪ ◡B)
7		elun 1601	. . . 4 ⊢ (⟨y, x⟩ ∈ (A ∪ B) ↔ (⟨y, x⟩ ∈ A ∨ ⟨y, x⟩ ∈ B))
8		visset 1350	. . . . . 6 ⊢ x ∈ V
9		visset 1350	. . . . . 6 ⊢ y ∈ V
10	8, 9	opelcnv 2518	. . . . 5 ⊢ (⟨x, y⟩ ∈ ◡A ↔ ⟨y, x⟩ ∈ A)
11	8, 9	opelcnv 2518	. . . . 5 ⊢ (⟨x, y⟩ ∈ ◡B ↔ ⟨y, x⟩ ∈ B)
12	10, 11	orbi12i 216	. . . 4 ⊢ ((⟨x, y⟩ ∈ ◡A ∨ ⟨x, y⟩ ∈ ◡B) ↔ (⟨y, x⟩ ∈ A ∨ ⟨y, x⟩ ∈ B))
13	7, 12	bitr4 154	. . 3 ⊢ (⟨y, x⟩ ∈ (A ∪ B) ↔ (⟨x, y⟩ ∈ ◡A ∨ ⟨x, y⟩ ∈ ◡B))
14	8, 9	opelcnv 2518	. . 3 ⊢ (⟨x, y⟩ ∈ ◡(A ∪ B) ↔ ⟨y, x⟩ ∈ (A ∪ B))
15		elun 1601	. . 3 ⊢ (⟨x, y⟩ ∈ (◡A ∪ ◡B) ↔ (⟨x, y⟩ ∈ ◡A ∨ ⟨x, y⟩ ∈ ◡B))
16	13, 14, 15	3bitr4 158	. 2 ⊢ (⟨x, y⟩ ∈ ◡(A ∪ B) ↔ ⟨x, y⟩ ∈ (◡A ∪ ◡B))
17	1, 6, 16	cleqreli 2484	1 ⊢ ◡(A ∪ B) = (◡A ∪ ◡B)

Syntax hints:   ∨ wo 195   ∧ wa 196   = wceq 1091   ∈ wcel 1092   ∪ cun 1485  ⟨cop 1810  ^◡ccnv 2409  Rel wrel 2415

This theorem is referenced by:  rnun 2644  f1oun 2815  sbthlem8 3356

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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