Theorem cnvun 6118

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 5649	. . 3 ⊢ ◡(𝐴 ∪ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ 𝑦(𝐴 ∪ 𝐵)𝑥}
2		unopab 5190	. . . 4 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥} ∪ {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥}) = {⟨𝑥, 𝑦⟩ ∣ (𝑦𝐴𝑥 ∨ 𝑦𝐵𝑥)}
3		brun 5161	. . . . 5 ⊢ (𝑦(𝐴 ∪ 𝐵)𝑥 ↔ (𝑦𝐴𝑥 ∨ 𝑦𝐵𝑥))
4	3	opabbii 5177	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑦(𝐴 ∪ 𝐵)𝑥} = {⟨𝑥, 𝑦⟩ ∣ (𝑦𝐴𝑥 ∨ 𝑦𝐵𝑥)}
5	2, 4	eqtr4i 2756	. . 3 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥} ∪ {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥}) = {⟨𝑥, 𝑦⟩ ∣ 𝑦(𝐴 ∪ 𝐵)𝑥}
6	1, 5	eqtr4i 2756	. 2 ⊢ ◡(𝐴 ∪ 𝐵) = ({⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥} ∪ {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥})
7		df-cnv 5649	. . 3 ⊢ ◡𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥}
8		df-cnv 5649	. . 3 ⊢ ◡𝐵 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥}
9	7, 8	uneq12i 4132	. 2 ⊢ (◡𝐴 ∪ ◡𝐵) = ({⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥} ∪ {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥})
10	6, 9	eqtr4i 2756	1 ⊢ ◡(𝐴 ∪ 𝐵) = (◡𝐴 ∪ ◡𝐵)

Syntax hints:   ∨ wo 847   = wceq 1540   ∪ cun 3915   class class class wbr 5110  {copab 5172  ^◡ccnv 5640

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-v 3452  df-un 3922  df-br 5111  df-opab 5173  df-cnv 5649

This theorem is referenced by:  rnun  6121  funcnvpr  6581  funcnvtp  6582  funcnvqp  6583  f1oun  6822  f1oprswap  6847  suppun  8166  sbthlem8  9064  domss2  9106  cnvfi  9146  1sdomOLD  9203  fsuppun  9345  fpwwe2lem12  10602  trclublem  14968  mbfres2  25553  ex-cnv  30373  suppun2  32614  cnvprop  32626  padct  32650  cycpmconjslem2  33119  eulerpartlemt  34369  mthmpps  35576  clcnvlem  43619  frege131d  43760