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Theorem inuni 4786
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 4406 . . . . 5 (𝑧 𝐴 ↔ ∃𝑦𝐴 𝑧𝑦)
21anbi1i 730 . . . 4 ((𝑧 𝐴𝑧𝐵) ↔ (∃𝑦𝐴 𝑧𝑦𝑧𝐵))
3 elin 3774 . . . 4 (𝑧 ∈ ( 𝐴𝐵) ↔ (𝑧 𝐴𝑧𝐵))
4 ancom 466 . . . . . . . 8 ((𝑧𝑥 ∧ ∃𝑦𝐴 𝑥 = (𝑦𝐵)) ↔ (∃𝑦𝐴 𝑥 = (𝑦𝐵) ∧ 𝑧𝑥))
5 r19.41v 3081 . . . . . . . 8 (∃𝑦𝐴 (𝑥 = (𝑦𝐵) ∧ 𝑧𝑥) ↔ (∃𝑦𝐴 𝑥 = (𝑦𝐵) ∧ 𝑧𝑥))
64, 5bitr4i 267 . . . . . . 7 ((𝑧𝑥 ∧ ∃𝑦𝐴 𝑥 = (𝑦𝐵)) ↔ ∃𝑦𝐴 (𝑥 = (𝑦𝐵) ∧ 𝑧𝑥))
76exbii 1771 . . . . . 6 (∃𝑥(𝑧𝑥 ∧ ∃𝑦𝐴 𝑥 = (𝑦𝐵)) ↔ ∃𝑥𝑦𝐴 (𝑥 = (𝑦𝐵) ∧ 𝑧𝑥))
8 rexcom4 3211 . . . . . 6 (∃𝑦𝐴𝑥(𝑥 = (𝑦𝐵) ∧ 𝑧𝑥) ↔ ∃𝑥𝑦𝐴 (𝑥 = (𝑦𝐵) ∧ 𝑧𝑥))
97, 8bitr4i 267 . . . . 5 (∃𝑥(𝑧𝑥 ∧ ∃𝑦𝐴 𝑥 = (𝑦𝐵)) ↔ ∃𝑦𝐴𝑥(𝑥 = (𝑦𝐵) ∧ 𝑧𝑥))
10 vex 3189 . . . . . . . . . 10 𝑦 ∈ V
1110inex1 4759 . . . . . . . . 9 (𝑦𝐵) ∈ V
12 eleq2 2687 . . . . . . . . 9 (𝑥 = (𝑦𝐵) → (𝑧𝑥𝑧 ∈ (𝑦𝐵)))
1311, 12ceqsexv 3228 . . . . . . . 8 (∃𝑥(𝑥 = (𝑦𝐵) ∧ 𝑧𝑥) ↔ 𝑧 ∈ (𝑦𝐵))
14 elin 3774 . . . . . . . 8 (𝑧 ∈ (𝑦𝐵) ↔ (𝑧𝑦𝑧𝐵))
1513, 14bitri 264 . . . . . . 7 (∃𝑥(𝑥 = (𝑦𝐵) ∧ 𝑧𝑥) ↔ (𝑧𝑦𝑧𝐵))
1615rexbii 3034 . . . . . 6 (∃𝑦𝐴𝑥(𝑥 = (𝑦𝐵) ∧ 𝑧𝑥) ↔ ∃𝑦𝐴 (𝑧𝑦𝑧𝐵))
17 r19.41v 3081 . . . . . 6 (∃𝑦𝐴 (𝑧𝑦𝑧𝐵) ↔ (∃𝑦𝐴 𝑧𝑦𝑧𝐵))
1816, 17bitri 264 . . . . 5 (∃𝑦𝐴𝑥(𝑥 = (𝑦𝐵) ∧ 𝑧𝑥) ↔ (∃𝑦𝐴 𝑧𝑦𝑧𝐵))
199, 18bitri 264 . . . 4 (∃𝑥(𝑧𝑥 ∧ ∃𝑦𝐴 𝑥 = (𝑦𝐵)) ↔ (∃𝑦𝐴 𝑧𝑦𝑧𝐵))
202, 3, 193bitr4i 292 . . 3 (𝑧 ∈ ( 𝐴𝐵) ↔ ∃𝑥(𝑧𝑥 ∧ ∃𝑦𝐴 𝑥 = (𝑦𝐵)))
21 eluniab 4413 . . 3 (𝑧 {𝑥 ∣ ∃𝑦𝐴 𝑥 = (𝑦𝐵)} ↔ ∃𝑥(𝑧𝑥 ∧ ∃𝑦𝐴 𝑥 = (𝑦𝐵)))
2220, 21bitr4i 267 . 2 (𝑧 ∈ ( 𝐴𝐵) ↔ 𝑧 {𝑥 ∣ ∃𝑦𝐴 𝑥 = (𝑦𝐵)})
2322eqriv 2618 1 ( 𝐴𝐵) = {𝑥 ∣ ∃𝑦𝐴 𝑥 = (𝑦𝐵)}
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1480  wex 1701  wcel 1987  {cab 2607  wrex 2908  cin 3554   cuni 4402
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-v 3188  df-in 3562  df-uni 4403
This theorem is referenced by: (None)
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