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Theorem inuni 3956
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 3631 . . . . 5 (𝑧 𝐴 ↔ ∃𝑦𝐴 𝑧𝑦)
21anbi1i 446 . . . 4 ((𝑧 𝐴𝑧𝐵) ↔ (∃𝑦𝐴 𝑧𝑦𝑧𝐵))
3 elin 3167 . . . 4 (𝑧 ∈ ( 𝐴𝐵) ↔ (𝑧 𝐴𝑧𝐵))
4 ancom 262 . . . . . . . 8 ((𝑧𝑥 ∧ ∃𝑦𝐴 𝑥 = (𝑦𝐵)) ↔ (∃𝑦𝐴 𝑥 = (𝑦𝐵) ∧ 𝑧𝑥))
5 r19.41v 2516 . . . . . . . 8 (∃𝑦𝐴 (𝑥 = (𝑦𝐵) ∧ 𝑧𝑥) ↔ (∃𝑦𝐴 𝑥 = (𝑦𝐵) ∧ 𝑧𝑥))
64, 5bitr4i 185 . . . . . . 7 ((𝑧𝑥 ∧ ∃𝑦𝐴 𝑥 = (𝑦𝐵)) ↔ ∃𝑦𝐴 (𝑥 = (𝑦𝐵) ∧ 𝑧𝑥))
76exbii 1537 . . . . . 6 (∃𝑥(𝑧𝑥 ∧ ∃𝑦𝐴 𝑥 = (𝑦𝐵)) ↔ ∃𝑥𝑦𝐴 (𝑥 = (𝑦𝐵) ∧ 𝑧𝑥))
8 rexcom4 2633 . . . . . 6 (∃𝑦𝐴𝑥(𝑥 = (𝑦𝐵) ∧ 𝑧𝑥) ↔ ∃𝑥𝑦𝐴 (𝑥 = (𝑦𝐵) ∧ 𝑧𝑥))
97, 8bitr4i 185 . . . . 5 (∃𝑥(𝑧𝑥 ∧ ∃𝑦𝐴 𝑥 = (𝑦𝐵)) ↔ ∃𝑦𝐴𝑥(𝑥 = (𝑦𝐵) ∧ 𝑧𝑥))
10 vex 2615 . . . . . . . . . 10 𝑦 ∈ V
1110inex1 3938 . . . . . . . . 9 (𝑦𝐵) ∈ V
12 eleq2 2146 . . . . . . . . 9 (𝑥 = (𝑦𝐵) → (𝑧𝑥𝑧 ∈ (𝑦𝐵)))
1311, 12ceqsexv 2649 . . . . . . . 8 (∃𝑥(𝑥 = (𝑦𝐵) ∧ 𝑧𝑥) ↔ 𝑧 ∈ (𝑦𝐵))
14 elin 3167 . . . . . . . 8 (𝑧 ∈ (𝑦𝐵) ↔ (𝑧𝑦𝑧𝐵))
1513, 14bitri 182 . . . . . . 7 (∃𝑥(𝑥 = (𝑦𝐵) ∧ 𝑧𝑥) ↔ (𝑧𝑦𝑧𝐵))
1615rexbii 2379 . . . . . 6 (∃𝑦𝐴𝑥(𝑥 = (𝑦𝐵) ∧ 𝑧𝑥) ↔ ∃𝑦𝐴 (𝑧𝑦𝑧𝐵))
17 r19.41v 2516 . . . . . 6 (∃𝑦𝐴 (𝑧𝑦𝑧𝐵) ↔ (∃𝑦𝐴 𝑧𝑦𝑧𝐵))
1816, 17bitri 182 . . . . 5 (∃𝑦𝐴𝑥(𝑥 = (𝑦𝐵) ∧ 𝑧𝑥) ↔ (∃𝑦𝐴 𝑧𝑦𝑧𝐵))
199, 18bitri 182 . . . 4 (∃𝑥(𝑧𝑥 ∧ ∃𝑦𝐴 𝑥 = (𝑦𝐵)) ↔ (∃𝑦𝐴 𝑧𝑦𝑧𝐵))
202, 3, 193bitr4i 210 . . 3 (𝑧 ∈ ( 𝐴𝐵) ↔ ∃𝑥(𝑧𝑥 ∧ ∃𝑦𝐴 𝑥 = (𝑦𝐵)))
21 eluniab 3639 . . 3 (𝑧 {𝑥 ∣ ∃𝑦𝐴 𝑥 = (𝑦𝐵)} ↔ ∃𝑥(𝑧𝑥 ∧ ∃𝑦𝐴 𝑥 = (𝑦𝐵)))
2220, 21bitr4i 185 . 2 (𝑧 ∈ ( 𝐴𝐵) ↔ 𝑧 {𝑥 ∣ ∃𝑦𝐴 𝑥 = (𝑦𝐵)})
2322eqriv 2080 1 ( 𝐴𝐵) = {𝑥 ∣ ∃𝑦𝐴 𝑥 = (𝑦𝐵)}
Colors of variables: wff set class
Syntax hints:  wa 102   = wceq 1285  wex 1422  wcel 1434  {cab 2069  wrex 2354  cin 2983   cuni 3627
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-rex 2359  df-v 2614  df-in 2990  df-uni 3628
This theorem is referenced by: (None)
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