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Theorem fnrnfv 6722
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 6721 . . 3 (𝐹 Fn 𝐴𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
2 rneq 5805 . . 3 (𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)) → ran 𝐹 = ran (𝑥𝐴 ↦ (𝐹𝑥)))
31, 2sylbi 218 . 2 (𝐹 Fn 𝐴 → ran 𝐹 = ran (𝑥𝐴 ↦ (𝐹𝑥)))
4 eqid 2826 . . 3 (𝑥𝐴 ↦ (𝐹𝑥)) = (𝑥𝐴 ↦ (𝐹𝑥))
54rnmpt 5826 . 2 ran (𝑥𝐴 ↦ (𝐹𝑥)) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)}
63, 5syl6eq 2877 1 (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1530  {cab 2804  wrex 3144  cmpt 5143  ran crn 5555   Fn wfn 6347  cfv 6352
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-iota 6312  df-fun 6354  df-fn 6355  df-fv 6360
This theorem is referenced by:  fvelrnb  6723  fniinfv  6739  dffo3  6864  fniunfv  7000  fnrnov  7311  pwcfsdom  9994  hauscmplem  21930  grpoinvf  28223  fpwrelmapffslem  30381  cshwrnid  30549  meascnbl  31364  omssubadd  31444  fvineqsneu  34561  fvineqsneq  34562  dffo3f  41303  rnfdmpr  43346  fargshiftfo  43434
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