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Theorem fniunfv 5342
 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 5133 . . . . 5 ((Fun 𝐹 x dom 𝐹) → (𝐹x) V)
21funfni 4940 . . . 4 ((𝐹 Fn A x A) → (𝐹x) V)
32ralrimiva 2386 . . 3 (𝐹 Fn Ax A (𝐹x) V)
4 dfiun2g 3679 . . 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 5161 . . 3 (𝐹 Fn A → ran 𝐹 = {yx A y = (𝐹x)})
76unieqd 3581 . 2 (𝐹 Fn A ran 𝐹 = {yx A y = (𝐹x)})
85, 7eqtr4d 2072 1 (𝐹 Fn A x A (𝐹x) = ran 𝐹)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1242   ∈ wcel 1390  {cab 2023  ∀wral 2300  ∃wrex 2301  Vcvv 2551  ∪ cuni 3570  ∪ ciun 3647  ran crn 4288   Fn wfn 4839  ‘cfv 4844 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3865  ax-pow 3917  ax-pr 3934 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3352  df-sn 3372  df-pr 3373  df-op 3375  df-uni 3571  df-iun 3649  df-br 3755  df-opab 3809  df-mpt 3810  df-id 4020  df-xp 4293  df-rel 4294  df-cnv 4295  df-co 4296  df-dm 4297  df-rn 4298  df-iota 4809  df-fun 4846  df-fn 4847  df-fv 4852 This theorem is referenced by:  funiunfvdm  5343
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