Theorem rnuni 5015

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 U. A = U_ x e. A ran x

Distinct variable group:   x, A

Proof of Theorem rnuni

Step	Hyp	Ref	Expression
1		uniiun 3919	. . 3 |- U. A = U_ x e. A x
2	1	rneqi 4832	. 2 |- ran U. A = ran U_ x e. A x
3		rniun 5014	. 2 |- ran U_ x e. A x = U_ x e. A ran x
4	2, 3	eqtri 2186	1 |- ran U. A = U_ x e. A ran x

Syntax hints:   = wceq 1343   U.cuni 3789   U_ciun 3866   ran crn 4605

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-iun 3868  df-br 3983  df-opab 4044  df-cnv 4612  df-dm 4614  df-rn 4615

This theorem is referenced by:  ennnfonelemf1  12351