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Theorem fniunfv 5293
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 5084 . . . . 5 ((Fun 𝐹 x dom 𝐹) → (𝐹x) V)
21funfni 4892 . . . 4 ((𝐹 Fn A x A) → (𝐹x) V)
32ralrimiva 2368 . . 3 (𝐹 Fn Ax A (𝐹x) V)
4 dfiun2g 3641 . . 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 5112 . . 3 (𝐹 Fn A → ran 𝐹 = {yx A y = (𝐹x)})
76unieqd 3543 . 2 (𝐹 Fn A ran 𝐹 = {yx A y = (𝐹x)})
85, 7eqtr4d 2057 1 (𝐹 Fn A x A (𝐹x) = ran 𝐹)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1373   wcel 1375  {cab 2008  wral 2282  wrex 2283  Vcvv 2533   cuni 3532   ciun 3609  ran crn 4239   Fn wfn 4791  cfv 4796
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 1315  ax-7 1316  ax-gen 1317  ax-ie1 1362  ax-ie2 1363  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 2004  ax-sep 3827  ax-pow 3879  ax-pr 3896
This theorem depends on definitions:  df-bi 110  df-3an 877  df-tru 1231  df-nf 1329  df-sb 1628  df-eu 1884  df-mo 1885  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2287  df-rex 2288  df-v 2535  df-sbc 2740  df-un 2900  df-in 2902  df-ss 2909  df-pw 3313  df-sn 3333  df-pr 3334  df-op 3336  df-uni 3533  df-iun 3611  df-br 3717  df-opab 3771  df-mpt 3772  df-id 3983  df-xp 4244  df-rel 4245  df-cnv 4246  df-co 4247  df-dm 4248  df-rn 4249  df-iota 4761  df-fun 4798  df-fn 4799  df-fv 4804
This theorem is referenced by:  funiunfvdm  5294
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