Theorem rnuni 5199

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 4066	. . 3 |- U. A = U_ x e. A x
2	1	rneqi 5010	. 2 |- ran U. A = ran U_ x e. A x
3		rniun 5198	. 2 |- ran U_ x e. A x = U_ x e. A ran x
4	2, 3	eqtri 2259	1 |- ran U. A = U_ x e. A ran x

This proof depends on syntax axioms:   = wceq 1402   U.cuni 3935   U_ciun 4012   ran crn 4775

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-iun 4014  df-br 4131  df-opab 4193  df-cnv 4782  df-dm 4784  df-rn 4785

This theorem is used by:  ennnfonelemf1  13309