Theorem rnuni 5139

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 4018	. . 3 |- U. A = U_ x e. A x
2	1	rneqi 4951	. 2 |- ran U. A = ran U_ x e. A x
3		rniun 5138	. 2 |- ran U_ x e. A x = U_ x e. A ran x
4	2, 3	eqtri 2250	1 |- ran U. A = U_ x e. A ran x

Syntax hints:   = wceq 1395   U.cuni 3887   U_ciun 3964   ran crn 4719

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-iun 3966  df-br 4083  df-opab 4145  df-cnv 4726  df-dm 4728  df-rn 4729

This theorem is referenced by:  ennnfonelemf1  12984