New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  dfiun2g GIF version

Theorem dfiun2g 3999
 Description: Alternate definition of indexed union when B is a set. Definition 15(a) of [Suppes] p. 44. (Contributed by NM, 23-Mar-2006.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
dfiun2g (x A B Cx A B = {y x A y = B})
Distinct variable groups:   y,A   y,B   x,y
Allowed substitution hints:   A(x)   B(x)   C(x,y)

Proof of Theorem dfiun2g
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 nfra1 2664 . . . . . 6 xx A B C
2 rsp 2674 . . . . . . . 8 (x A B C → (x AB C))
3 clel3g 2976 . . . . . . . 8 (B C → (z By(y = B z y)))
42, 3syl6 29 . . . . . . 7 (x A B C → (x A → (z By(y = B z y))))
54imp 418 . . . . . 6 ((x A B C x A) → (z By(y = B z y)))
61, 5rexbida 2629 . . . . 5 (x A B C → (x A z Bx A y(y = B z y)))
7 rexcom4 2878 . . . . 5 (x A y(y = B z y) ↔ yx A (y = B z y))
86, 7syl6bb 252 . . . 4 (x A B C → (x A z Byx A (y = B z y)))
9 r19.41v 2764 . . . . . 6 (x A (y = B z y) ↔ (x A y = B z y))
109exbii 1582 . . . . 5 (yx A (y = B z y) ↔ y(x A y = B z y))
11 exancom 1586 . . . . 5 (y(x A y = B z y) ↔ y(z y x A y = B))
1210, 11bitri 240 . . . 4 (yx A (y = B z y) ↔ y(z y x A y = B))
138, 12syl6bb 252 . . 3 (x A B C → (x A z By(z y x A y = B)))
14 eliun 3973 . . 3 (z x A Bx A z B)
15 eluniab 3903 . . 3 (z {y x A y = B} ↔ y(z y x A y = B))
1613, 14, 153bitr4g 279 . 2 (x A B C → (z x A Bz {y x A y = B}))
1716eqrdv 2351 1 (x A B Cx A B = {y x A y = B})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339  ∀wral 2614  ∃wrex 2615  ∪cuni 3891  ∪ciun 3969 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-rex 2620  df-v 2861  df-uni 3892  df-iun 3971 This theorem is referenced by:  dfiun2  4001  uniqs  5984
 Copyright terms: Public domain W3C validator