Theorem cnvun 6133

Description: The converse of a union is the union of converses. Theorem 16 of [Suppes] p. 62. (Contributed by NM, 25-Mar-1998.) (Proof shortened by Andrew Salmon, 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		df-cnv 5663	. . 3 ⊢ ◡(𝐴 ∪ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ 𝑦(𝐴 ∪ 𝐵)𝑥}
2		unopab 5185	. . . 4 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥} ∪ {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥}) = {⟨𝑥, 𝑦⟩ ∣ (𝑦𝐴𝑥 ∨ 𝑦𝐵𝑥)}
3		brun 5156	. . . . 5 ⊢ (𝑦(𝐴 ∪ 𝐵)𝑥 ↔ (𝑦𝐴𝑥 ∨ 𝑦𝐵𝑥))
4	3	opabbii 5172	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑦(𝐴 ∪ 𝐵)𝑥} = {⟨𝑥, 𝑦⟩ ∣ (𝑦𝐴𝑥 ∨ 𝑦𝐵𝑥)}
5	2, 4	eqtr4i 2786	. . 3 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥} ∪ {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥}) = {⟨𝑥, 𝑦⟩ ∣ 𝑦(𝐴 ∪ 𝐵)𝑥}
6	1, 5	eqtr4i 2786	. 2 ⊢ ◡(𝐴 ∪ 𝐵) = ({⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥} ∪ {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥})
7		df-cnv 5663	. . 3 ⊢ ◡𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥}
8		df-cnv 5663	. . 3 ⊢ ◡𝐵 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥}
9	7, 8	uneq12i 4113	. 2 ⊢ (◡𝐴 ∪ ◡𝐵) = ({⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥} ∪ {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥})
10	6, 9	eqtr4i 2786	1 ⊢ ◡(𝐴 ∪ 𝐵) = (◡𝐴 ∪ ◡𝐵)

This proof depends on syntax axioms:   ∨ wo 861   = wceq 1570   ∪ cun 3897   class class class wbr 5103  {copab 5167  ^◡ccnv 5654

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-un 3904  df-br 5104  df-opab 5168  df-cnv 5663

This theorem is used by:  rnun  6136  funcnvpr  6595  funcnvtp  6596  funcnvqp  6597  f1oun  6837  f1oprswap  6863  suppun  8182  sbthlem8  9092  domss2  9134  cnvfi  9170  fsuppun  9357  fpwwe2lem12  10651  trclublem  15068  mbfres2  25873  ex-cnv  30917  suppun2  33156  cnvprop  33168  padct  33189  cycpmconjslem2  33595  eulerpartlemt  34882  mthmpps  36161  clcnvlem  44463  frege131d  44604