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Theorem fniinfv 6986
Description: The indexed intersection of a function's values is the intersection of its range. (Contributed by NM, 20-Oct-2005.)
Assertion
Ref Expression
fniinfv (𝐹 Fn 𝐴 𝑥𝐴 (𝐹𝑥) = ran 𝐹)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹

Proof of Theorem fniinfv
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 fvex 6918 . . 3 (𝐹𝑥) ∈ V
21dfiin2 5033 . 2 𝑥𝐴 (𝐹𝑥) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)}
3 fnrnfv 6967 . . 3 (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)})
43inteqd 4950 . 2 (𝐹 Fn 𝐴 ran 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)})
52, 4eqtr4id 2795 1 (𝐹 Fn 𝐴 𝑥𝐴 (𝐹𝑥) = ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  {cab 2713  wrex 3069   cint 4945   ciin 4991  ran crn 5685   Fn wfn 6555  cfv 6560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iin 4993  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-iota 6513  df-fun 6562  df-fn 6563  df-fv 6568
This theorem is referenced by:  firest  17478  pnrmopn  23352  txtube  23649  bcth3  25366  diaintclN  41061  dibintclN  41170  dihintcl  41347  imaiinfv  42709
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