Theorem rnuni 6052

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 4988	. . 3 ⊢ ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝑥
2	1	rneqi 5846	. 2 ⊢ ran ∪ 𝐴 = ran ∪ 𝑥 ∈ 𝐴 𝑥
3		rniun 6051	. 2 ⊢ ran ∪ 𝑥 ∈ 𝐴 𝑥 = ∪ 𝑥 ∈ 𝐴 ran 𝑥
4	2, 3	eqtri 2766	1 ⊢ ran ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 ran 𝑥

Syntax hints:   = wceq 1539  ∪ cuni 4839  ∪ ciun 4924  ran crn 5590

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-cnv 5597  df-dm 5599  df-rn 5600

This theorem is referenced by:  ackbij2  9999  axdc3lem2  10207  unirnmap  42748  unirnmapsn  42754