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Theorem fniunfv 5324
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 x A (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 funfvex 5115 . . . . 5 ((Fun 𝐹 x dom 𝐹) → (𝐹x) V)
21funfni 4923 . . . 4 ((𝐹 Fn A x A) → (𝐹x) V)
32ralrimiva 2369 . . 3 (𝐹 Fn Ax A (𝐹x) V)
4 dfiun2g 3662 . . 3 (x A (𝐹x) V → x A (𝐹x) = {yx A y = (𝐹x)})
53, 4syl 14 . 2 (𝐹 Fn A x A (𝐹x) = {yx A y = (𝐹x)})
6 fnrnfv 5143 . . 3 (𝐹 Fn A → ran 𝐹 = {yx A y = (𝐹x)})
76unieqd 3564 . 2 (𝐹 Fn A ran 𝐹 = {yx A y = (𝐹x)})
85, 7eqtr4d 2058 1 (𝐹 Fn A x A (𝐹x) = ran 𝐹)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1228   wcel 1375  {cab 2009  wral 2283  wrex 2284  Vcvv 2534   cuni 3553   ciun 3630  ran crn 4271   Fn wfn 4822  cfv 4827
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1364  ax-ie2 1365  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-14 1387  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2005  ax-sep 3848  ax-pow 3900  ax-pr 3917
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1629  df-eu 1886  df-mo 1887  df-clab 2010  df-cleq 2016  df-clel 2019  df-nfc 2150  df-ral 2288  df-rex 2289  df-v 2536  df-sbc 2741  df-un 2898  df-in 2900  df-ss 2907  df-pw 3335  df-sn 3355  df-pr 3356  df-op 3358  df-uni 3554  df-iun 3632  df-br 3738  df-opab 3792  df-mpt 3793  df-id 4003  df-xp 4276  df-rel 4277  df-cnv 4278  df-co 4279  df-dm 4280  df-rn 4281  df-iota 4792  df-fun 4829  df-fn 4830  df-fv 4835
This theorem is referenced by:  funiunfvdm  5325
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