Theorem uniiun 4020

Description: Class union in terms of indexed union. Definition in [Stoll] p. 43. (Contributed by NM, 28-Jun-1998.)

Assertion

Ref	Expression
uniiun	⊢ ∪A = ∪x ∈ A x

Distinct variable group:   x,A

Proof of Theorem uniiun

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfuni2 3894	. 2 ⊢ ∪A = {y ∣ ∃x ∈ A y ∈ x}
2		df-iun 3972	. 2 ⊢ ∪x ∈ A x = {y ∣ ∃x ∈ A y ∈ x}
3	1, 2	eqtr4i 2376	1 ⊢ ∪A = ∪x ∈ A x

Syntax hints:   = wceq 1642   ∈ wcel 1710  {cab 2339  ∃wrex 2616  ∪cuni 3892  ∪ciun 3970

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-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-rex 2621  df-uni 3893  df-iun 3972

This theorem is referenced by:  iununi  4051  iunpwss  4056  imauni  5466