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Theorem iinuni 4049
 Description: A relationship involving union and indexed intersection. Exercise 23 of [Enderton] p. 33. (Contributed by NM, 25-Nov-2003.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
Assertion
Ref Expression
iinuni (AB) = x B (Ax)
Distinct variable groups:   x,A   x,B

Proof of Theorem iinuni
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 r19.32v 2757 . . . 4 (x B (y A y x) ↔ (y A x B y x))
2 elun 3220 . . . . 5 (y (Ax) ↔ (y A y x))
32ralbii 2638 . . . 4 (x B y (Ax) ↔ x B (y A y x))
4 vex 2862 . . . . . 6 y V
54elint2 3933 . . . . 5 (y Bx B y x)
65orbi2i 505 . . . 4 ((y A y B) ↔ (y A x B y x))
71, 3, 63bitr4ri 269 . . 3 ((y A y B) ↔ x B y (Ax))
8 elun 3220 . . 3 (y (AB) ↔ (y A y B))
9 eliin 3974 . . . 4 (y V → (y x B (Ax) ↔ x B y (Ax)))
104, 9ax-mp 5 . . 3 (y x B (Ax) ↔ x B y (Ax))
117, 8, 103bitr4i 268 . 2 (y (AB) ↔ y x B (Ax))
1211eqriv 2350 1 (AB) = x B (Ax)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∨ wo 357   = wceq 1642   ∈ wcel 1710  ∀wral 2614  Vcvv 2859   ∪ cun 3207  ∩cint 3926  ∩ciin 3970 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-v 2861  df-nin 3211  df-compl 3212  df-un 3214  df-int 3927  df-iin 3972 This theorem is referenced by: (None)
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