Theorem rnuni 6139

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 5016	. . 3 ⊢ ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝑥
2	1	rneqi 5918	. 2 ⊢ ran ∪ 𝐴 = ran ∪ 𝑥 ∈ 𝐴 𝑥
3		rniun 6138	. 2 ⊢ ran ∪ 𝑥 ∈ 𝐴 𝑥 = ∪ 𝑥 ∈ 𝐴 ran 𝑥
4	2, 3	eqtri 2783	1 ⊢ ran ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 ran 𝑥

This proof depends on syntax axioms:   = wceq 1569  ∪ cuni 4866  ∪ ciun 4950  ran crn 5651

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-11 2193  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-cnv 5658  df-dm 5660  df-rn 5661

This theorem is used by:  ackbij2  10323  axdc3lem2  10532  unirnmap  46314  unirnmapsn  46320