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Theorem dmun 4485
Description: The domain of a union is the union of domains. Exercise 56(a) of [Enderton] p. 65. (Contributed by NM, 12-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
dmun dom (AB) = (dom A ∪ dom B)

Proof of Theorem dmun
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 unab 3198 . . 3 ({yx yAx} ∪ {yx yBx}) = {y ∣ (x yAx x yBx)}
2 brun 3801 . . . . . 6 (y(AB)x ↔ (yAx yBx))
32exbii 1493 . . . . 5 (x y(AB)xx(yAx yBx))
4 19.43 1516 . . . . 5 (x(yAx yBx) ↔ (x yAx x yBx))
53, 4bitr2i 174 . . . 4 ((x yAx x yBx) ↔ x y(AB)x)
65abbii 2150 . . 3 {y ∣ (x yAx x yBx)} = {yx y(AB)x}
71, 6eqtri 2057 . 2 ({yx yAx} ∪ {yx yBx}) = {yx y(AB)x}
8 df-dm 4298 . . 3 dom A = {yx yAx}
9 df-dm 4298 . . 3 dom B = {yx yBx}
108, 9uneq12i 3089 . 2 (dom A ∪ dom B) = ({yx yAx} ∪ {yx yBx})
11 df-dm 4298 . 2 dom (AB) = {yx y(AB)x}
127, 10, 113eqtr4ri 2068 1 dom (AB) = (dom A ∪ dom B)
Colors of variables: wff set class
Syntax hints:   wo 628   = wceq 1242  wex 1378  {cab 2023  cun 2909   class class class wbr 3755  dom cdm 4288
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-bnd 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-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-br 3756  df-dm 4298
This theorem is referenced by:  rnun  4675  dmpropg  4736  dmtpop  4739  fntpg  4898  fnun  4948
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