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Theorem inuni 5060
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 4675 . . . . 5 (𝑧 𝐴 ↔ ∃𝑦𝐴 𝑧𝑦)
21anbi1i 617 . . . 4 ((𝑧 𝐴𝑧𝐵) ↔ (∃𝑦𝐴 𝑧𝑦𝑧𝐵))
3 elin 4019 . . . 4 (𝑧 ∈ ( 𝐴𝐵) ↔ (𝑧 𝐴𝑧𝐵))
4 ancom 454 . . . . . . . 8 ((𝑧𝑥 ∧ ∃𝑦𝐴 𝑥 = (𝑦𝐵)) ↔ (∃𝑦𝐴 𝑥 = (𝑦𝐵) ∧ 𝑧𝑥))
5 r19.41v 3275 . . . . . . . 8 (∃𝑦𝐴 (𝑥 = (𝑦𝐵) ∧ 𝑧𝑥) ↔ (∃𝑦𝐴 𝑥 = (𝑦𝐵) ∧ 𝑧𝑥))
64, 5bitr4i 270 . . . . . . 7 ((𝑧𝑥 ∧ ∃𝑦𝐴 𝑥 = (𝑦𝐵)) ↔ ∃𝑦𝐴 (𝑥 = (𝑦𝐵) ∧ 𝑧𝑥))
76exbii 1892 . . . . . 6 (∃𝑥(𝑧𝑥 ∧ ∃𝑦𝐴 𝑥 = (𝑦𝐵)) ↔ ∃𝑥𝑦𝐴 (𝑥 = (𝑦𝐵) ∧ 𝑧𝑥))
8 rexcom4 3427 . . . . . 6 (∃𝑦𝐴𝑥(𝑥 = (𝑦𝐵) ∧ 𝑧𝑥) ↔ ∃𝑥𝑦𝐴 (𝑥 = (𝑦𝐵) ∧ 𝑧𝑥))
97, 8bitr4i 270 . . . . 5 (∃𝑥(𝑧𝑥 ∧ ∃𝑦𝐴 𝑥 = (𝑦𝐵)) ↔ ∃𝑦𝐴𝑥(𝑥 = (𝑦𝐵) ∧ 𝑧𝑥))
10 vex 3401 . . . . . . . . . 10 𝑦 ∈ V
1110inex1 5036 . . . . . . . . 9 (𝑦𝐵) ∈ V
12 eleq2 2848 . . . . . . . . 9 (𝑥 = (𝑦𝐵) → (𝑧𝑥𝑧 ∈ (𝑦𝐵)))
1311, 12ceqsexv 3444 . . . . . . . 8 (∃𝑥(𝑥 = (𝑦𝐵) ∧ 𝑧𝑥) ↔ 𝑧 ∈ (𝑦𝐵))
14 elin 4019 . . . . . . . 8 (𝑧 ∈ (𝑦𝐵) ↔ (𝑧𝑦𝑧𝐵))
1513, 14bitri 267 . . . . . . 7 (∃𝑥(𝑥 = (𝑦𝐵) ∧ 𝑧𝑥) ↔ (𝑧𝑦𝑧𝐵))
1615rexbii 3224 . . . . . 6 (∃𝑦𝐴𝑥(𝑥 = (𝑦𝐵) ∧ 𝑧𝑥) ↔ ∃𝑦𝐴 (𝑧𝑦𝑧𝐵))
17 r19.41v 3275 . . . . . 6 (∃𝑦𝐴 (𝑧𝑦𝑧𝐵) ↔ (∃𝑦𝐴 𝑧𝑦𝑧𝐵))
1816, 17bitri 267 . . . . 5 (∃𝑦𝐴𝑥(𝑥 = (𝑦𝐵) ∧ 𝑧𝑥) ↔ (∃𝑦𝐴 𝑧𝑦𝑧𝐵))
199, 18bitri 267 . . . 4 (∃𝑥(𝑧𝑥 ∧ ∃𝑦𝐴 𝑥 = (𝑦𝐵)) ↔ (∃𝑦𝐴 𝑧𝑦𝑧𝐵))
202, 3, 193bitr4i 295 . . 3 (𝑧 ∈ ( 𝐴𝐵) ↔ ∃𝑥(𝑧𝑥 ∧ ∃𝑦𝐴 𝑥 = (𝑦𝐵)))
21 eluniab 4682 . . 3 (𝑧 {𝑥 ∣ ∃𝑦𝐴 𝑥 = (𝑦𝐵)} ↔ ∃𝑥(𝑧𝑥 ∧ ∃𝑦𝐴 𝑥 = (𝑦𝐵)))
2220, 21bitr4i 270 . 2 (𝑧 ∈ ( 𝐴𝐵) ↔ 𝑧 {𝑥 ∣ ∃𝑦𝐴 𝑥 = (𝑦𝐵)})
2322eqriv 2775 1 ( 𝐴𝐵) = {𝑥 ∣ ∃𝑦𝐴 𝑥 = (𝑦𝐵)}
Colors of variables: wff setvar class
Syntax hints:  wa 386   = wceq 1601  wex 1823  wcel 2107  {cab 2763  wrex 3091  cin 3791   cuni 4671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-v 3400  df-in 3799  df-uni 4672
This theorem is referenced by: (None)
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