Theorem uniiun 3967

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 3838	. 2 ⊢ ∪ 𝐴 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥}
2		df-iun 3915	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 𝑥 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥}
3	1, 2	eqtr4i 2217	1 ⊢ ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝑥

Syntax hints:   = wceq 1364  {cab 2179  ∃wrex 2473  ∪ cuni 3836  ∪ ciun 3913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-rex 2478  df-uni 3837  df-iun 3915

This theorem is referenced by:  iunpwss  4005  truni  4142  iunpw  4512  reluni  4783  rnuni  5078  imauni  5805  hashuni  11628  tgidm  14253  unicld  14295  tgrest  14348  txbasval  14446