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

Theorem fniunfv 7246
Description: The indexed union of a function's values is the union of its range. Compare Definition 5.4 of [Monk1] p. 50. (Contributed by NM, 27-Sep-2004.)
Assertion
Ref Expression
fniunfv (𝐹 Fn 𝐴 𝑥𝐴 (𝐹𝑥) = ran 𝐹)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹

Proof of Theorem fniunfv
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 fvex 6905 . . 3 (𝐹𝑥) ∈ V
21dfiun2 5037 . 2 𝑥𝐴 (𝐹𝑥) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)}
3 fnrnfv 6952 . . 3 (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)})
43unieqd 4923 . 2 (𝐹 Fn 𝐴 ran 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)})
52, 4eqtr4id 2792 1 (𝐹 Fn 𝐴 𝑥𝐴 (𝐹𝑥) = ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  {cab 2710  wrex 3071   cuni 4909   ciun 4998  ran crn 5678   Fn wfn 6539  cfv 6544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fn 6547  df-fv 6552
This theorem is referenced by:  funiunfv  7247  dffi3  9426  jech9.3  9809  hsmexlem5  10425  wuncval2  10742  dprdspan  19897  tgcmp  22905  txcmplem1  23145  txcmplem2  23146  xkococnlem  23163  alexsubALT  23555  bcth3  24848  ovolfioo  24984  ovolficc  24985  voliunlem2  25068  voliunlem3  25069  volsup  25073  uniiccdif  25095  uniioovol  25096  uniiccvol  25097  uniioombllem2  25100  uniioombllem4  25103  volsup2  25122  itg1climres  25232  itg2monolem1  25268  itg2gt0  25278  sigapildsys  33160  omssubadd  33299  carsgclctunlem3  33319  pibt2  36298  volsupnfl  36533  hbt  41872  ovolval4lem1  45365  ovolval5lem3  45370  ovnovollem1  45372  ovnovollem2  45373
  Copyright terms: Public domain W3C validator