Theorem fnrnfv 6876

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 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)})

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐹,𝑦

Proof of Theorem fnrnfv

Step	Hyp	Ref	Expression
1		dffn5 6875	. . 3 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
2		rneq 5871	. . 3 ⊢ (𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) → ran 𝐹 = ran (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
3	1, 2	sylbi 217	. 2 ⊢ (𝐹 Fn 𝐴 → ran 𝐹 = ran (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
4		eqid 2731	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))
5	4	rnmpt 5892	. 2 ⊢ ran (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)}
6	3, 5	eqtrdi 2782	1 ⊢ (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)})

Syntax hints:   → wi 4   = wceq 1541  {cab 2709  ∃wrex 3056   ↦ cmpt 5167  ran crn 5612   Fn wfn 6471  ‘cfv 6476

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pr 5365

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-iota 6432  df-fun 6478  df-fn 6479  df-fv 6484

This theorem is referenced by:  fvelrnb  6877  fniinfv  6895  dffo3  7030  dffo3f  7034  fniunfv  7176  fnrnov  7514  pwcfsdom  10469  hauscmplem  23316  madef  27792  grpoinvf  30504  fpwrelmapffslem  32707  cshwrnid  32934  meascnbl  34224  omssubadd  34305  fvineqsneu  37445  fvineqsneq  37446  tfsconcatrn  43375  rnfdmpr  47312  fargshiftfo  47473