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Theorem dmuni 4472
Description: The domain of a union. Part of Exercise 8 of [Enderton] p. 41. (Contributed by NM, 3-Feb-2004.)
Assertion
Ref Expression
dmuni dom A = x A dom x
Distinct variable group:   x,A

Proof of Theorem dmuni
Dummy variables y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 excom 1536 . . . . 5 (zx(⟨y, z x x A) ↔ xz(⟨y, z x x A))
2 ancom 253 . . . . . . 7 ((zy, z x x A) ↔ (x A zy, z x))
3 19.41v 1764 . . . . . . 7 (z(⟨y, z x x A) ↔ (zy, z x x A))
4 vex 2538 . . . . . . . . 9 y V
54eldm2 4460 . . . . . . . 8 (y dom xzy, z x)
65anbi2i 433 . . . . . . 7 ((x A y dom x) ↔ (x A zy, z x))
72, 3, 63bitr4i 201 . . . . . 6 (z(⟨y, z x x A) ↔ (x A y dom x))
87exbii 1478 . . . . 5 (xz(⟨y, z x x A) ↔ x(x A y dom x))
91, 8bitri 173 . . . 4 (zx(⟨y, z x x A) ↔ x(x A y dom x))
10 eluni 3557 . . . . 5 (⟨y, z Ax(⟨y, z x x A))
1110exbii 1478 . . . 4 (zy, z Azx(⟨y, z x x A))
12 df-rex 2290 . . . 4 (x A y dom xx(x A y dom x))
139, 11, 123bitr4i 201 . . 3 (zy, z Ax A y dom x)
144eldm2 4460 . . 3 (y dom Azy, z A)
15 eliun 3635 . . 3 (y x A dom xx A y dom x)
1613, 14, 153bitr4i 201 . 2 (y dom Ay x A dom x)
1716eqriv 2019 1 dom A = x A dom x
Colors of variables: wff set class
Syntax hints:   wa 97   = wceq 1228  wex 1362   wcel 1374  wrex 2285  cop 3353   cuni 3554   ciun 3631  dom cdm 4272
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-un 2899  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-iun 3633  df-br 3739  df-dm 4282
This theorem is referenced by:  tfrlem8  5856  tfrlemi14d  5868  tfrlemi14  5869
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