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Theorem dmuni 4488
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 1551 . . . . 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 1779 . . . . . . 7 (z(⟨y, z x x A) ↔ (zy, z x x A))
4 vex 2554 . . . . . . . . 9 y V
54eldm2 4476 . . . . . . . 8 (y dom xzy, z x)
65anbi2i 430 . . . . . . 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 1493 . . . . 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 3574 . . . . 5 (⟨y, z Ax(⟨y, z x x A))
1110exbii 1493 . . . 4 (zy, z Azx(⟨y, z x x A))
12 df-rex 2306 . . . 4 (x A y dom xx(x A y dom x))
139, 11, 123bitr4i 201 . . 3 (zy, z Ax A y dom x)
144eldm2 4476 . . 3 (y dom Azy, z A)
15 eliun 3652 . . 3 (y x A dom xx A y dom x)
1613, 14, 153bitr4i 201 . 2 (y dom Ay x A dom x)
1716eqriv 2034 1 dom A = x A dom x
Colors of variables: wff set class
Syntax hints:   wa 97   = wceq 1242  wex 1378   wcel 1390  wrex 2301  cop 3370   cuni 3571   ciun 3648  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-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-dm 4298
This theorem is referenced by:  tfrlem8  5875  tfrlemi14d  5888
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