MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nfiun Structured version   Unicode version

Theorem nfiun 4298
Description: Bound-variable hypothesis builder for indexed union. (Contributed by Mario Carneiro, 25-Jan-2014.)
Hypotheses
Ref Expression
nfiun.1  |-  F/_ y A
nfiun.2  |-  F/_ y B
Assertion
Ref Expression
nfiun  |-  F/_ y U_ x  e.  A  B

Proof of Theorem nfiun
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 df-iun 4273 . 2  |-  U_ x  e.  A  B  =  { z  |  E. x  e.  A  z  e.  B }
2 nfiun.1 . . . 4  |-  F/_ y A
3 nfiun.2 . . . . 5  |-  F/_ y B
43nfcri 2606 . . . 4  |-  F/ y  z  e.  B
52, 4nfrex 2882 . . 3  |-  F/ y E. x  e.  A  z  e.  B
65nfab 2617 . 2  |-  F/_ y { z  |  E. x  e.  A  z  e.  B }
71, 6nfcxfr 2611 1  |-  F/_ y U_ x  e.  A  B
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1758   {cab 2436   F/_wnfc 2599   E.wrex 2796   U_ciun 4271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ral 2800  df-rex 2801  df-iun 4273
This theorem is referenced by:  iunab  4316  disjxiun  4389  ovoliunnul  21108  iundisjf  26067  iundisj2f  26068  iundisjfi  26216  iundisj2fi  26217  trpredlem1  27827  trpredrec  27838  bnj1498  32354
  Copyright terms: Public domain W3C validator