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Theorem fniunfv 6668
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 fnrnfv 6404 . . 3 (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)})
21unieqd 4598 . 2 (𝐹 Fn 𝐴 ran 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)})
3 fvex 6362 . . 3 (𝐹𝑥) ∈ V
43dfiun2 4706 . 2 𝑥𝐴 (𝐹𝑥) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)}
52, 4syl6reqr 2813 1 (𝐹 Fn 𝐴 𝑥𝐴 (𝐹𝑥) = ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  {cab 2746  wrex 3051   cuni 4588   ciun 4672  ran crn 5267   Fn wfn 6044  cfv 6049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-iota 6012  df-fun 6051  df-fn 6052  df-fv 6057
This theorem is referenced by:  funiunfv  6669  dffi3  8502  jech9.3  8850  hsmexlem5  9444  wuncval2  9761  dprdspan  18626  tgcmp  21406  txcmplem1  21646  txcmplem2  21647  xkococnlem  21664  alexsubALT  22056  bcth3  23328  ovolfioo  23436  ovolficc  23437  voliunlem2  23519  voliunlem3  23520  volsup  23524  uniiccdif  23546  uniioovol  23547  uniiccvol  23548  uniioombllem2  23551  uniioombllem4  23554  volsup2  23573  itg1climres  23680  itg2monolem1  23716  itg2gt0  23726  sigapildsys  30534  omssubadd  30671  carsgclctunlem3  30691  dftrpred2  32024  volsupnfl  33767  hbt  38202  ovolval4lem1  41369  ovolval5lem3  41374  ovnovollem1  41376  ovnovollem2  41377
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