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Theorem inuni 3900
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) = {xy A 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 3575 . . . . 5 (z Ay A z y)
21anbi1i 431 . . . 4 ((z A z B) ↔ (y A z y z B))
3 elin 3120 . . . 4 (z ( AB) ↔ (z A z B))
4 ancom 253 . . . . . . . 8 ((z x y A x = (yB)) ↔ (y A x = (yB) z x))
5 r19.41v 2460 . . . . . . . 8 (y A (x = (yB) z x) ↔ (y A x = (yB) z x))
64, 5bitr4i 176 . . . . . . 7 ((z x y A x = (yB)) ↔ y A (x = (yB) z x))
76exbii 1493 . . . . . 6 (x(z x y A x = (yB)) ↔ xy A (x = (yB) z x))
8 rexcom4 2571 . . . . . 6 (y A x(x = (yB) z x) ↔ xy A (x = (yB) z x))
97, 8bitr4i 176 . . . . 5 (x(z x y A x = (yB)) ↔ y A x(x = (yB) z x))
10 vex 2554 . . . . . . . . . 10 y V
1110inex1 3882 . . . . . . . . 9 (yB) V
12 eleq2 2098 . . . . . . . . 9 (x = (yB) → (z xz (yB)))
1311, 12ceqsexv 2587 . . . . . . . 8 (x(x = (yB) z x) ↔ z (yB))
14 elin 3120 . . . . . . . 8 (z (yB) ↔ (z y z B))
1513, 14bitri 173 . . . . . . 7 (x(x = (yB) z x) ↔ (z y z B))
1615rexbii 2325 . . . . . 6 (y A x(x = (yB) z x) ↔ y A (z y z B))
17 r19.41v 2460 . . . . . 6 (y A (z y z B) ↔ (y A z y z B))
1816, 17bitri 173 . . . . 5 (y A x(x = (yB) z x) ↔ (y A z y z B))
199, 18bitri 173 . . . 4 (x(z x y A x = (yB)) ↔ (y A z y z B))
202, 3, 193bitr4i 201 . . 3 (z ( AB) ↔ x(z x y A x = (yB)))
21 eluniab 3583 . . 3 (z {xy A x = (yB)} ↔ x(z x y A x = (yB)))
2220, 21bitr4i 176 . 2 (z ( AB) ↔ z {xy A x = (yB)})
2322eqriv 2034 1 ( AB) = {xy A x = (yB)}
Colors of variables: wff set class
Syntax hints:   wa 97   = wceq 1242  wex 1378   wcel 1390  {cab 2023  wrex 2301  cin 2910   cuni 3571
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553  df-in 2918  df-uni 3572
This theorem is referenced by: (None)
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