Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  inuni Structured version   Visualization version   GIF version

Theorem inuni 4753
 Description: The intersection of a union ∪ 𝐴 with a class 𝐵 is equal to the union of the intersections of each element of 𝐴 with 𝐵. (Contributed by FL, 24-Mar-2007.)
Assertion
Ref Expression
inuni ( 𝐴𝐵) = {𝑥 ∣ ∃𝑦𝐴 𝑥 = (𝑦𝐵)}
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦

Proof of Theorem inuni
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 eluni2 4376 . . . . 5 (𝑧 𝐴 ↔ ∃𝑦𝐴 𝑧𝑦)
21anbi1i 727 . . . 4 ((𝑧 𝐴𝑧𝐵) ↔ (∃𝑦𝐴 𝑧𝑦𝑧𝐵))
3 elin 3758 . . . 4 (𝑧 ∈ ( 𝐴𝐵) ↔ (𝑧 𝐴𝑧𝐵))
4 ancom 465 . . . . . . . 8 ((𝑧𝑥 ∧ ∃𝑦𝐴 𝑥 = (𝑦𝐵)) ↔ (∃𝑦𝐴 𝑥 = (𝑦𝐵) ∧ 𝑧𝑥))
5 r19.41v 3070 . . . . . . . 8 (∃𝑦𝐴 (𝑥 = (𝑦𝐵) ∧ 𝑧𝑥) ↔ (∃𝑦𝐴 𝑥 = (𝑦𝐵) ∧ 𝑧𝑥))
64, 5bitr4i 266 . . . . . . 7 ((𝑧𝑥 ∧ ∃𝑦𝐴 𝑥 = (𝑦𝐵)) ↔ ∃𝑦𝐴 (𝑥 = (𝑦𝐵) ∧ 𝑧𝑥))
76exbii 1764 . . . . . 6 (∃𝑥(𝑧𝑥 ∧ ∃𝑦𝐴 𝑥 = (𝑦𝐵)) ↔ ∃𝑥𝑦𝐴 (𝑥 = (𝑦𝐵) ∧ 𝑧𝑥))
8 rexcom4 3198 . . . . . 6 (∃𝑦𝐴𝑥(𝑥 = (𝑦𝐵) ∧ 𝑧𝑥) ↔ ∃𝑥𝑦𝐴 (𝑥 = (𝑦𝐵) ∧ 𝑧𝑥))
97, 8bitr4i 266 . . . . 5 (∃𝑥(𝑧𝑥 ∧ ∃𝑦𝐴 𝑥 = (𝑦𝐵)) ↔ ∃𝑦𝐴𝑥(𝑥 = (𝑦𝐵) ∧ 𝑧𝑥))
10 vex 3176 . . . . . . . . . 10 𝑦 ∈ V
1110inex1 4727 . . . . . . . . 9 (𝑦𝐵) ∈ V
12 eleq2 2677 . . . . . . . . 9 (𝑥 = (𝑦𝐵) → (𝑧𝑥𝑧 ∈ (𝑦𝐵)))
1311, 12ceqsexv 3215 . . . . . . . 8 (∃𝑥(𝑥 = (𝑦𝐵) ∧ 𝑧𝑥) ↔ 𝑧 ∈ (𝑦𝐵))
14 elin 3758 . . . . . . . 8 (𝑧 ∈ (𝑦𝐵) ↔ (𝑧𝑦𝑧𝐵))
1513, 14bitri 263 . . . . . . 7 (∃𝑥(𝑥 = (𝑦𝐵) ∧ 𝑧𝑥) ↔ (𝑧𝑦𝑧𝐵))
1615rexbii 3023 . . . . . 6 (∃𝑦𝐴𝑥(𝑥 = (𝑦𝐵) ∧ 𝑧𝑥) ↔ ∃𝑦𝐴 (𝑧𝑦𝑧𝐵))
17 r19.41v 3070 . . . . . 6 (∃𝑦𝐴 (𝑧𝑦𝑧𝐵) ↔ (∃𝑦𝐴 𝑧𝑦𝑧𝐵))
1816, 17bitri 263 . . . . 5 (∃𝑦𝐴𝑥(𝑥 = (𝑦𝐵) ∧ 𝑧𝑥) ↔ (∃𝑦𝐴 𝑧𝑦𝑧𝐵))
199, 18bitri 263 . . . 4 (∃𝑥(𝑧𝑥 ∧ ∃𝑦𝐴 𝑥 = (𝑦𝐵)) ↔ (∃𝑦𝐴 𝑧𝑦𝑧𝐵))
202, 3, 193bitr4i 291 . . 3 (𝑧 ∈ ( 𝐴𝐵) ↔ ∃𝑥(𝑧𝑥 ∧ ∃𝑦𝐴 𝑥 = (𝑦𝐵)))
21 eluniab 4383 . . 3 (𝑧 {𝑥 ∣ ∃𝑦𝐴 𝑥 = (𝑦𝐵)} ↔ ∃𝑥(𝑧𝑥 ∧ ∃𝑦𝐴 𝑥 = (𝑦𝐵)))
2220, 21bitr4i 266 . 2 (𝑧 ∈ ( 𝐴𝐵) ↔ 𝑧 {𝑥 ∣ ∃𝑦𝐴 𝑥 = (𝑦𝐵)})
2322eqriv 2607 1 ( 𝐴𝐵) = {𝑥 ∣ ∃𝑦𝐴 𝑥 = (𝑦𝐵)}
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {cab 2596  ∃wrex 2897   ∩ cin 3539  ∪ cuni 4372 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175  df-in 3547  df-uni 4373 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator