Theorem fnrnfv 6902

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 6901	. . 3 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
2		rneq 5889	. . 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 2729	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))
5	4	rnmpt 5910	. 2 ⊢ ran (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)}
6	3, 5	eqtrdi 2780	1 ⊢ (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)})

Syntax hints:   → wi 4   = wceq 1540  {cab 2707  ∃wrex 3053   ↦ cmpt 5183  ran crn 5632   Fn wfn 6494  ‘cfv 6499

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-iota 6452  df-fun 6501  df-fn 6502  df-fv 6507

This theorem is referenced by:  fvelrnb  6903  fniinfv  6921  dffo3  7056  dffo3f  7060  fniunfv  7203  fnrnov  7542  pwcfsdom  10512  hauscmplem  23269  madef  27740  grpoinvf  30434  fpwrelmapffslem  32628  cshwrnid  32856  meascnbl  34182  omssubadd  34264  fvineqsneu  37372  fvineqsneq  37373  tfsconcatrn  43304  rnfdmpr  47255  fargshiftfo  47416