Theorem rnuni 5544

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 4573	. . 3 ⊢ ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝑥
2	1	rneqi 5352	. 2 ⊢ ran ∪ 𝐴 = ran ∪ 𝑥 ∈ 𝐴 𝑥
3		rniun 5543	. 2 ⊢ ran ∪ 𝑥 ∈ 𝐴 𝑥 = ∪ 𝑥 ∈ 𝐴 ran 𝑥
4	2, 3	eqtri 2644	1 ⊢ ran ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 ran 𝑥

Syntax hints:   = wceq 1483  ∪ cuni 4436  ∪ ciun 4520  ran crn 5115

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-cnv 5122  df-dm 5124  df-rn 5125

This theorem is referenced by:  ackbij2  9065  axdc3lem2  9273  unirnmap  39400  unirnmapsn  39406