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

Theorem nfiun 4519
Description: Bound-variable hypothesis builder for indexed union. (Contributed by Mario Carneiro, 25-Jan-2014.)
Hypotheses
Ref Expression
nfiun.1 𝑦𝐴
nfiun.2 𝑦𝐵
Assertion
Ref Expression
nfiun 𝑦 𝑥𝐴 𝐵

Proof of Theorem nfiun
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-iun 4492 . 2 𝑥𝐴 𝐵 = {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵}
2 nfiun.1 . . . 4 𝑦𝐴
3 nfiun.2 . . . . 5 𝑦𝐵
43nfcri 2761 . . . 4 𝑦 𝑧𝐵
52, 4nfrex 3006 . . 3 𝑦𝑥𝐴 𝑧𝐵
65nfab 2771 . 2 𝑦{𝑧 ∣ ∃𝑥𝐴 𝑧𝐵}
71, 6nfcxfr 2765 1 𝑦 𝑥𝐴 𝐵
Colors of variables: wff setvar class
Syntax hints:  wcel 1992  {cab 2612  wnfc 2754  wrex 2913   ciun 4490
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 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606
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 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-iun 4492
This theorem is referenced by:  iunab  4537  disjxiun  4614  disjxiunOLD  4615  ovoliunnul  23177  iundisjf  29238  iundisj2f  29239  iundisjfi  29388  iundisj2fi  29389  bnj1498  30829  trpredlem1  31420  trpredrec  31431  ss2iundf  37418  fnlimcnv  39290  fnlimfvre  39297  fnlimabslt  39302  smfaddlem1  40265  smflimlem6  40278  smflim  40279  smfmullem4  40295  smflim2  40306  smflimsup  40328
  Copyright terms: Public domain W3C validator