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Theorem fniinfv 5259
 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 funfvex 5220 . . . . 5 ((Fun 𝐹𝑥 ∈ dom 𝐹) → (𝐹𝑥) ∈ V)
21funfni 5027 . . . 4 ((𝐹 Fn 𝐴𝑥𝐴) → (𝐹𝑥) ∈ V)
32ralrimiva 2409 . . 3 (𝐹 Fn 𝐴 → ∀𝑥𝐴 (𝐹𝑥) ∈ V)
4 dfiin2g 3718 . . 3 (∀𝑥𝐴 (𝐹𝑥) ∈ V → 𝑥𝐴 (𝐹𝑥) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)})
53, 4syl 14 . 2 (𝐹 Fn 𝐴 𝑥𝐴 (𝐹𝑥) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)})
6 fnrnfv 5248 . . 3 (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)})
76inteqd 3648 . 2 (𝐹 Fn 𝐴 ran 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)})
85, 7eqtr4d 2091 1 (𝐹 Fn 𝐴 𝑥𝐴 (𝐹𝑥) = ran 𝐹)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1259   ∈ wcel 1409  {cab 2042  ∀wral 2323  ∃wrex 2324  Vcvv 2574  ∩ cint 3643  ∩ ciin 3686  ran crn 4374   Fn wfn 4925  ‘cfv 4930 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972 This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2788  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iin 3688  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-iota 4895  df-fun 4932  df-fn 4933  df-fv 4938 This theorem is referenced by: (None)
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