ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  rnun GIF version

Theorem rnun 4917
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 4914 . . . 4 (𝐴𝐵) = (𝐴𝐵)
21dmeqi 4710 . . 3 dom (𝐴𝐵) = dom (𝐴𝐵)
3 dmun 4716 . . 3 dom (𝐴𝐵) = (dom 𝐴 ∪ dom 𝐵)
42, 3eqtri 2138 . 2 dom (𝐴𝐵) = (dom 𝐴 ∪ dom 𝐵)
5 df-rn 4520 . 2 ran (𝐴𝐵) = dom (𝐴𝐵)
6 df-rn 4520 . . 3 ran 𝐴 = dom 𝐴
7 df-rn 4520 . . 3 ran 𝐵 = dom 𝐵
86, 7uneq12i 3198 . 2 (ran 𝐴 ∪ ran 𝐵) = (dom 𝐴 ∪ dom 𝐵)
94, 5, 83eqtr4i 2148 1 ran (𝐴𝐵) = (ran 𝐴 ∪ ran 𝐵)
Colors of variables: wff set class
Syntax hints:   = wceq 1316  cun 3039  ccnv 4508  dom cdm 4509  ran crn 4510
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-cnv 4517  df-dm 4519  df-rn 4520
This theorem is referenced by:  imaundi  4921  imaundir  4922  rnpropg  4988  fun  5265  foun  5354  fpr  5570  fprg  5571  sbthlemi6  6818  exmidfodomrlemim  7025
  Copyright terms: Public domain W3C validator