Theorem fnrnfv 6981

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 6980	. . 3 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
2		rneq 5961	. . 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 2740	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))
5	4	rnmpt 5980	. 2 ⊢ ran (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)}
6	3, 5	eqtrdi 2796	1 ⊢ (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)})

Syntax hints:   → wi 4   = wceq 1537  {cab 2717  ∃wrex 3076   ↦ cmpt 5249  ran crn 5701   Fn wfn 6568  ‘cfv 6573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-iota 6525  df-fun 6575  df-fn 6576  df-fv 6581

This theorem is referenced by:  fvelrnb  6982  fniinfv  7000  dffo3  7136  dffo3f  7140  fniunfv  7284  fnrnov  7623  pwcfsdom  10652  hauscmplem  23435  madef  27913  grpoinvf  30564  fpwrelmapffslem  32746  cshwrnid  32928  meascnbl  34183  omssubadd  34265  fvineqsneu  37377  fvineqsneq  37378  tfsconcatrn  43304  rnfdmpr  47196  fargshiftfo  47316