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Theorem fniunfv 7199
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 6860 . . 3 (𝐹𝑥) ∈ V
21dfiun2 4998 . 2 𝑥𝐴 (𝐹𝑥) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)}
3 fnrnfv 6907 . . 3 (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)})
43unieqd 4884 . 2 (𝐹 Fn 𝐴 ran 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)})
52, 4eqtr4id 2796 1 (𝐹 Fn 𝐴 𝑥𝐴 (𝐹𝑥) = ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  {cab 2714  wrex 3074   cuni 4870   ciun 4959  ran crn 5639   Fn wfn 6496  cfv 6501
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 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-iota 6453  df-fun 6503  df-fn 6504  df-fv 6509
This theorem is referenced by:  funiunfv  7200  dffi3  9374  jech9.3  9757  hsmexlem5  10373  wuncval2  10690  dprdspan  19813  tgcmp  22768  txcmplem1  23008  txcmplem2  23009  xkococnlem  23026  alexsubALT  23418  bcth3  24711  ovolfioo  24847  ovolficc  24848  voliunlem2  24931  voliunlem3  24932  volsup  24936  uniiccdif  24958  uniioovol  24959  uniiccvol  24960  uniioombllem2  24963  uniioombllem4  24966  volsup2  24985  itg1climres  25095  itg2monolem1  25131  itg2gt0  25141  sigapildsys  32801  omssubadd  32940  carsgclctunlem3  32960  pibt2  35917  volsupnfl  36152  hbt  41486  ovolval4lem1  44964  ovolval5lem3  44969  ovnovollem1  44971  ovnovollem2  44972
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