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Theorem inuni 5339
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.) (Proof shortened by Wolf Lammen, 15-May-2025.)
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 ancom 460 . . . . 5 ((𝑧𝑥 ∧ ∃𝑦𝐴 𝑥 = (𝑦𝐵)) ↔ (∃𝑦𝐴 𝑥 = (𝑦𝐵) ∧ 𝑧𝑥))
2 r19.41v 3184 . . . . 5 (∃𝑦𝐴 (𝑥 = (𝑦𝐵) ∧ 𝑧𝑥) ↔ (∃𝑦𝐴 𝑥 = (𝑦𝐵) ∧ 𝑧𝑥))
31, 2bitr4i 278 . . . 4 ((𝑧𝑥 ∧ ∃𝑦𝐴 𝑥 = (𝑦𝐵)) ↔ ∃𝑦𝐴 (𝑥 = (𝑦𝐵) ∧ 𝑧𝑥))
43exbii 1843 . . 3 (∃𝑥(𝑧𝑥 ∧ ∃𝑦𝐴 𝑥 = (𝑦𝐵)) ↔ ∃𝑥𝑦𝐴 (𝑥 = (𝑦𝐵) ∧ 𝑧𝑥))
5 eluniab 4917 . . 3 (𝑧 {𝑥 ∣ ∃𝑦𝐴 𝑥 = (𝑦𝐵)} ↔ ∃𝑥(𝑧𝑥 ∧ ∃𝑦𝐴 𝑥 = (𝑦𝐵)))
6 eluni2 4907 . . . . . 6 (𝑧 𝐴 ↔ ∃𝑦𝐴 𝑧𝑦)
76anbi1i 623 . . . . 5 ((𝑧 𝐴𝑧𝐵) ↔ (∃𝑦𝐴 𝑧𝑦𝑧𝐵))
8 elin 3961 . . . . 5 (𝑧 ∈ ( 𝐴𝐵) ↔ (𝑧 𝐴𝑧𝐵))
9 r19.41v 3184 . . . . 5 (∃𝑦𝐴 (𝑧𝑦𝑧𝐵) ↔ (∃𝑦𝐴 𝑧𝑦𝑧𝐵))
107, 8, 93bitr4i 303 . . . 4 (𝑧 ∈ ( 𝐴𝐵) ↔ ∃𝑦𝐴 (𝑧𝑦𝑧𝐵))
11 vex 3474 . . . . . . . 8 𝑦 ∈ V
1211inex1 5311 . . . . . . 7 (𝑦𝐵) ∈ V
13 eleq2 2818 . . . . . . 7 (𝑥 = (𝑦𝐵) → (𝑧𝑥𝑧 ∈ (𝑦𝐵)))
1412, 13ceqsexv 3522 . . . . . 6 (∃𝑥(𝑥 = (𝑦𝐵) ∧ 𝑧𝑥) ↔ 𝑧 ∈ (𝑦𝐵))
15 elin 3961 . . . . . 6 (𝑧 ∈ (𝑦𝐵) ↔ (𝑧𝑦𝑧𝐵))
1614, 15bitri 275 . . . . 5 (∃𝑥(𝑥 = (𝑦𝐵) ∧ 𝑧𝑥) ↔ (𝑧𝑦𝑧𝐵))
1716rexbii 3090 . . . 4 (∃𝑦𝐴𝑥(𝑥 = (𝑦𝐵) ∧ 𝑧𝑥) ↔ ∃𝑦𝐴 (𝑧𝑦𝑧𝐵))
18 rexcom4 3281 . . . 4 (∃𝑦𝐴𝑥(𝑥 = (𝑦𝐵) ∧ 𝑧𝑥) ↔ ∃𝑥𝑦𝐴 (𝑥 = (𝑦𝐵) ∧ 𝑧𝑥))
1910, 17, 183bitr2i 299 . . 3 (𝑧 ∈ ( 𝐴𝐵) ↔ ∃𝑥𝑦𝐴 (𝑥 = (𝑦𝐵) ∧ 𝑧𝑥))
204, 5, 193bitr4ri 304 . 2 (𝑧 ∈ ( 𝐴𝐵) ↔ 𝑧 {𝑥 ∣ ∃𝑦𝐴 𝑥 = (𝑦𝐵)})
2120eqriv 2725 1 ( 𝐴𝐵) = {𝑥 ∣ ∃𝑦𝐴 𝑥 = (𝑦𝐵)}
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1534  wex 1774  wcel 2099  {cab 2705  wrex 3066  cin 3944   cuni 4903
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5293
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-rex 3067  df-v 3472  df-in 3952  df-uni 4904
This theorem is referenced by: (None)
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