Theorem fnrnafv 42058

Description: The range of a function expressed as a collection of the function's values, analogous to fnrnfv 6493. (Contributed by Alexander van der Vekens, 25-May-2017.)

Assertion

Ref	Expression
fnrnafv	⊢ (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹'''𝑥)})

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

Proof of Theorem fnrnafv

Step	Hyp	Ref	Expression
1		dfafn5a 42056	. . 3 ⊢ (𝐹 Fn 𝐴 → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹'''𝑥)))
2	1	rneqd 5589	. 2 ⊢ (𝐹 Fn 𝐴 → ran 𝐹 = ran (𝑥 ∈ 𝐴 ↦ (𝐹'''𝑥)))
3		eqid 2825	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐹'''𝑥)) = (𝑥 ∈ 𝐴 ↦ (𝐹'''𝑥))
4	3	rnmpt 5608	. 2 ⊢ ran (𝑥 ∈ 𝐴 ↦ (𝐹'''𝑥)) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹'''𝑥)}
5	2, 4	syl6eq 2877	1 ⊢ (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹'''𝑥)})

Syntax hints:   → wi 4   = wceq 1656  {cab 2811  ∃wrex 3118   ↦ cmpt 4954  ran crn 5347   Fn wfn 6122  '''cafv 42013

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-fal 1670  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-int 4700  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-iota 6090  df-fun 6129  df-fn 6130  df-fv 6135  df-aiota 41976  df-dfat 42015  df-afv 42016

This theorem is referenced by:  afvelrnb  42059  afvelrnb0  42060