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Theorem fnrnfv 6467
Description: The range of a function expressed as a collection of the function's values. (Contributed by NM, 20-Oct-2005.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
Assertion
Ref Expression
fnrnfv (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)})
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐹,𝑦

Proof of Theorem fnrnfv
StepHypRef Expression
1 dffn5 6466 . . 3 (𝐹 Fn 𝐴𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
2 rneq 5554 . . 3 (𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)) → ran 𝐹 = ran (𝑥𝐴 ↦ (𝐹𝑥)))
31, 2sylbi 209 . 2 (𝐹 Fn 𝐴 → ran 𝐹 = ran (𝑥𝐴 ↦ (𝐹𝑥)))
4 eqid 2799 . . 3 (𝑥𝐴 ↦ (𝐹𝑥)) = (𝑥𝐴 ↦ (𝐹𝑥))
54rnmpt 5575 . 2 ran (𝑥𝐴 ↦ (𝐹𝑥)) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)}
63, 5syl6eq 2849 1 (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1653  {cab 2785  wrex 3090  cmpt 4922  ran crn 5313   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 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-iota 6064  df-fun 6103  df-fn 6104  df-fv 6109
This theorem is referenced by:  fvelrnb  6468  fniinfv  6482  dffo3  6600  fniunfv  6733  fnrnov  7041  pwcfsdom  9693  hauscmplem  21538  grpoinvf  27912  fpwrelmapffslem  30025  meascnbl  30798  omssubadd  30878  dffo3f  40114  rnfdmpr  42132  fargshiftfo  42214
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