MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fnunirn Structured version   Visualization version   GIF version

Theorem fnunirn 6735
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 6199 . . 3 (𝐹 Fn 𝐼 → Fun 𝐹)
2 elunirn 6733 . . 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 6201 . . 3 (𝐹 Fn 𝐼 → dom 𝐹 = 𝐼)
54rexeqdv 3334 . 2 (𝐹 Fn 𝐼 → (∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹𝑥) ↔ ∃𝑥𝐼 𝐴 ∈ (𝐹𝑥)))
63, 5bitrd 270 1 (𝐹 Fn 𝐼 → (𝐴 ran 𝐹 ↔ ∃𝑥𝐼 𝐴 ∈ (𝐹𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wcel 2156  wrex 3097   cuni 4630  dom cdm 5311  ran crn 5312  Fun wfun 6095   Fn wfn 6096  cfv 6101
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pr 5096
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-iota 6064  df-fun 6103  df-fn 6104  df-fv 6109
This theorem is referenced by:  itunitc  9528  wunex2  9845  mreunirn  16466  arwhoma  16899  filunirn  21899  xmetunirn  22355  abfmpunirn  29779  cmpcref  30242  neibastop2lem  32676  stoweidlem59  40755
  Copyright terms: Public domain W3C validator