Theorem uniiun 3731

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 3603	. 2 ⊢ ∪ 𝐴 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥}
2		df-iun 3680	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 𝑥 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥}
3	1, 2	eqtr4i 2104	1 ⊢ ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝑥

Syntax hints:   = wceq 1284  {cab 2067  ∃wrex 2349  ∪ cuni 3601  ∪ ciun 3678

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-rex 2354  df-uni 3602  df-iun 3680

This theorem is referenced by:  iunpwss  3764  truni  3889  iunpw  4229  reluni  4478  rnuni  4755  imauni  5421