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Theorem fniunfv 6383
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 6133 . . 3 (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)})
21unieqd 4372 . 2 (𝐹 Fn 𝐴 ran 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)})
3 fvex 6094 . . 3 (𝐹𝑥) ∈ V
43dfiun2 4480 . 2 𝑥𝐴 (𝐹𝑥) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)}
52, 4syl6reqr 2658 1 (𝐹 Fn 𝐴 𝑥𝐴 (𝐹𝑥) = ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  {cab 2591  wrex 2892   cuni 4362   ciun 4445  ran crn 5025   Fn wfn 5781  cfv 5786
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-sep 4699  ax-nul 4708  ax-pr 4824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ral 2896  df-rex 2897  df-rab 2900  df-v 3170  df-sbc 3398  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-sn 4121  df-pr 4123  df-op 4127  df-uni 4363  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-id 4939  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-iota 5750  df-fun 5788  df-fn 5789  df-fv 5794
This theorem is referenced by:  funiunfv  6384  dffi3  8193  jech9.3  8533  hsmexlem5  9108  wuncval2  9421  dprdspan  18191  tgcmp  20952  txcmplem1  21192  txcmplem2  21193  xkococnlem  21210  alexsubALT  21603  bcth3  22849  ovolfioo  22956  ovolficc  22957  voliunlem2  23039  voliunlem3  23040  volsup  23044  uniiccdif  23065  uniioovol  23066  uniiccvol  23067  uniioombllem2  23070  uniioombllem4  23073  volsup2  23092  itg1climres  23200  itg2monolem1  23236  itg2gt0  23246  sigapildsys  29354  omssubadd  29491  carsgclctunlem3  29511  dftrpred2  30765  volsupnfl  32423  hbt  36518  ovolval4lem1  39339  ovolval5lem3  39344  ovnovollem1  39346  ovnovollem2  39347
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