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Theorem fnunirn 7196
Description: Membership in a union of some function-defined family of sets. (Contributed by Stefan O'Rear, 30-Jan-2015.)
Assertion
Ref Expression
fnunirn (𝐹 Fn 𝐼 → (𝐴 ran 𝐹 ↔ ∃𝑥𝐼 𝐴 ∈ (𝐹𝑥)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐼   𝑥,𝐹

Proof of Theorem fnunirn
StepHypRef Expression
1 fnfun 6598 . . 3 (𝐹 Fn 𝐼 → Fun 𝐹)
2 elunirn 7193 . . 3 (Fun 𝐹 → (𝐴 ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹𝑥)))
31, 2syl 17 . 2 (𝐹 Fn 𝐼 → (𝐴 ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹𝑥)))
4 fndm 6601 . . 3 (𝐹 Fn 𝐼 → dom 𝐹 = 𝐼)
54rexeqdv 3313 . 2 (𝐹 Fn 𝐼 → (∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹𝑥) ↔ ∃𝑥𝐼 𝐴 ∈ (𝐹𝑥)))
63, 5bitrd 279 1 (𝐹 Fn 𝐼 → (𝐴 ran 𝐹 ↔ ∃𝑥𝐼 𝐴 ∈ (𝐹𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wcel 2107  wrex 3072   cuni 4864  dom cdm 5631  ran crn 5632  Fun wfun 6486   Fn wfn 6487  cfv 6492
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 2709  ax-sep 5255  ax-nul 5262  ax-pr 5383
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 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-br 5105  df-opab 5167  df-mpt 5188  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-iota 6444  df-fun 6494  df-fn 6495  df-fv 6500
This theorem is referenced by:  itunitc  10291  wunex2  10608  mreunirn  17416  arwhoma  17866  filunirn  23156  xmetunirn  23613  abfmpunirn  31366  cmpcref  32205  neibastop2lem  34728  stoweidlem59  44055
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