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Theorem rnun 4675
 Description: Distributive law for range over union. Theorem 8 of [Suppes] p. 60. (Contributed by NM, 24-Mar-1998.)
Assertion
Ref Expression
rnun ran (AB) = (ran A ∪ ran B)

Proof of Theorem rnun
StepHypRef Expression
1 cnvun 4672 . . . 4 (AB) = (AB)
21dmeqi 4479 . . 3 dom (AB) = dom (AB)
3 dmun 4485 . . 3 dom (AB) = (dom A ∪ dom B)
42, 3eqtri 2057 . 2 dom (AB) = (dom A ∪ dom B)
5 df-rn 4299 . 2 ran (AB) = dom (AB)
6 df-rn 4299 . . 3 ran A = dom A
7 df-rn 4299 . . 3 ran B = dom B
86, 7uneq12i 3089 . 2 (ran A ∪ ran B) = (dom A ∪ dom B)
94, 5, 83eqtr4i 2067 1 ran (AB) = (ran A ∪ ran B)
 Colors of variables: wff set class Syntax hints:   = wceq 1242   ∪ cun 2909  ◡ccnv 4287  dom cdm 4288  ran crn 4289 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-cnv 4296  df-dm 4298  df-rn 4299 This theorem is referenced by:  imaundi  4679  imaundir  4680  rnpropg  4743  fun  5006  foun  5088  fpr  5288  fprg  5289
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