Theorem rnuni 6095

Description: The range of a union. Part of Exercise 8 of [Enderton] p. 41. (Contributed by NM, 17-Mar-2004.) (Revised by Mario Carneiro, 29-May-2015.)

Assertion

Ref	Expression
rnuni	⊢ ran ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 ran 𝑥

Distinct variable group:   𝑥,𝐴

Proof of Theorem rnuni

Step	Hyp	Ref	Expression
1		uniiun 5007	. . 3 ⊢ ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝑥
2	1	rneqi 5877	. 2 ⊢ ran ∪ 𝐴 = ran ∪ 𝑥 ∈ 𝐴 𝑥
3		rniun 6094	. 2 ⊢ ran ∪ 𝑥 ∈ 𝐴 𝑥 = ∪ 𝑥 ∈ 𝐴 ran 𝑥
4	2, 3	eqtri 2754	1 ⊢ ran ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 ran 𝑥

Syntax hints:   = wceq 1541  ∪ cuni 4859  ∪ ciun 4941  ran crn 5617

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-11 2160  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-cnv 5624  df-dm 5626  df-rn 5627

This theorem is referenced by:  ackbij2  10130  axdc3lem2  10339  unirnmap  45244  unirnmapsn  45250