Theorem cnvun 5034

Description: The converse of a union is the union of converses. Theorem 16 of [Suppes] p. 62. (The proof was shortened by Andrew Salmon, 27-Aug-2011.) (Contributed by set.mm contributors, 25-Mar-1998.) (Revised by set.mm contributors, 27-Aug-2011.)

Assertion

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

Proof of Theorem cnvun

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

Step	Hyp	Ref	Expression
1		unopab 4639	. . 3 ⊢ ({⟨x, y⟩ ∣ yAx} ∪ {⟨x, y⟩ ∣ yBx}) = {⟨x, y⟩ ∣ (yAx ∨ yBx)}
2		brun 4693	. . . 4 ⊢ (y(A ∪ B)x ↔ (yAx ∨ yBx))
3	2	opabbii 4627	. . 3 ⊢ {⟨x, y⟩ ∣ y(A ∪ B)x} = {⟨x, y⟩ ∣ (yAx ∨ yBx)}
4	1, 3	eqtr4i 2376	. 2 ⊢ ({⟨x, y⟩ ∣ yAx} ∪ {⟨x, y⟩ ∣ yBx}) = {⟨x, y⟩ ∣ y(A ∪ B)x}
5		df-cnv 4786	. . 3 ⊢ ◡A = {⟨x, y⟩ ∣ yAx}
6		df-cnv 4786	. . 3 ⊢ ◡B = {⟨x, y⟩ ∣ yBx}
7	5, 6	uneq12i 3417	. 2 ⊢ (◡A ∪ ◡B) = ({⟨x, y⟩ ∣ yAx} ∪ {⟨x, y⟩ ∣ yBx})
8		df-cnv 4786	. 2 ⊢ ◡(A ∪ B) = {⟨x, y⟩ ∣ y(A ∪ B)x}
9	4, 7, 8	3eqtr4ri 2384	1 ⊢ ◡(A ∪ B) = (◡A ∪ ◡B)

Syntax hints:   ∨ wo 357   = wceq 1642   ∪ cun 3208  {copab 4623   class class class wbr 4640  ^◡ccnv 4772

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

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-v 2862  df-nin 3212  df-compl 3213  df-un 3215  df-opab 4624  df-br 4641  df-cnv 4786

This theorem is referenced by:  rnun  5037  f1oun  5305