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Theorem rnun 4760
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 4757 . . . 4 (𝐴𝐵) = (𝐴𝐵)
21dmeqi 4564 . . 3 dom (𝐴𝐵) = dom (𝐴𝐵)
3 dmun 4570 . . 3 dom (𝐴𝐵) = (dom 𝐴 ∪ dom 𝐵)
42, 3eqtri 2076 . 2 dom (𝐴𝐵) = (dom 𝐴 ∪ dom 𝐵)
5 df-rn 4384 . 2 ran (𝐴𝐵) = dom (𝐴𝐵)
6 df-rn 4384 . . 3 ran 𝐴 = dom 𝐴
7 df-rn 4384 . . 3 ran 𝐵 = dom 𝐵
86, 7uneq12i 3123 . 2 (ran 𝐴 ∪ ran 𝐵) = (dom 𝐴 ∪ dom 𝐵)
94, 5, 83eqtr4i 2086 1 ran (𝐴𝐵) = (ran 𝐴 ∪ ran 𝐵)
Colors of variables: wff set class
Syntax hints:   = wceq 1259  cun 2943  ccnv 4372  dom cdm 4373  ran crn 4374
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-cnv 4381  df-dm 4383  df-rn 4384
This theorem is referenced by:  imaundi  4764  imaundir  4765  rnpropg  4828  fun  5091  foun  5173  fpr  5373  fprg  5374
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