Theorem fnimage 35911

Description: Image𝑅 is a function over the set-like portion of 𝑅. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Assertion

Ref	Expression
fnimage	⊢ Image𝑅 Fn {𝑥 ∣ (𝑅 “ 𝑥) ∈ V}

Distinct variable group:   𝑥,𝑅

Proof of Theorem fnimage

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		funimage 35910	. 2 ⊢ Fun Image𝑅
2		vex 3482	. . . . . . . 8 ⊢ 𝑦 ∈ V
3		vex 3482	. . . . . . . 8 ⊢ 𝑥 ∈ V
4	2, 3	brimage 35908	. . . . . . 7 ⊢ (𝑦Image𝑅𝑥 ↔ 𝑥 = (𝑅 “ 𝑦))
5		eqvisset 3498	. . . . . . 7 ⊢ (𝑥 = (𝑅 “ 𝑦) → (𝑅 “ 𝑦) ∈ V)
6	4, 5	sylbi 217	. . . . . 6 ⊢ (𝑦Image𝑅𝑥 → (𝑅 “ 𝑦) ∈ V)
7	6	exlimiv 1928	. . . . 5 ⊢ (∃𝑥 𝑦Image𝑅𝑥 → (𝑅 “ 𝑦) ∈ V)
8		eqid 2735	. . . . . . 7 ⊢ (𝑅 “ 𝑦) = (𝑅 “ 𝑦)
9		brimageg 35909	. . . . . . . 8 ⊢ ((𝑦 ∈ V ∧ (𝑅 “ 𝑦) ∈ V) → (𝑦Image𝑅(𝑅 “ 𝑦) ↔ (𝑅 “ 𝑦) = (𝑅 “ 𝑦)))
10	2, 9	mpan 690	. . . . . . 7 ⊢ ((𝑅 “ 𝑦) ∈ V → (𝑦Image𝑅(𝑅 “ 𝑦) ↔ (𝑅 “ 𝑦) = (𝑅 “ 𝑦)))
11	8, 10	mpbiri 258	. . . . . 6 ⊢ ((𝑅 “ 𝑦) ∈ V → 𝑦Image𝑅(𝑅 “ 𝑦))
12		breq2 5152	. . . . . . 7 ⊢ (𝑥 = (𝑅 “ 𝑦) → (𝑦Image𝑅𝑥 ↔ 𝑦Image𝑅(𝑅 “ 𝑦)))
13	12	spcegv 3597	. . . . . 6 ⊢ ((𝑅 “ 𝑦) ∈ V → (𝑦Image𝑅(𝑅 “ 𝑦) → ∃𝑥 𝑦Image𝑅𝑥))
14	11, 13	mpd 15	. . . . 5 ⊢ ((𝑅 “ 𝑦) ∈ V → ∃𝑥 𝑦Image𝑅𝑥)
15	7, 14	impbii 209	. . . 4 ⊢ (∃𝑥 𝑦Image𝑅𝑥 ↔ (𝑅 “ 𝑦) ∈ V)
16	2	eldm 5914	. . . 4 ⊢ (𝑦 ∈ dom Image𝑅 ↔ ∃𝑥 𝑦Image𝑅𝑥)
17		imaeq2 6076	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑅 “ 𝑥) = (𝑅 “ 𝑦))
18	17	eleq1d 2824	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑅 “ 𝑥) ∈ V ↔ (𝑅 “ 𝑦) ∈ V))
19	2, 18	elab 3681	. . . 4 ⊢ (𝑦 ∈ {𝑥 ∣ (𝑅 “ 𝑥) ∈ V} ↔ (𝑅 “ 𝑦) ∈ V)
20	15, 16, 19	3bitr4i 303	. . 3 ⊢ (𝑦 ∈ dom Image𝑅 ↔ 𝑦 ∈ {𝑥 ∣ (𝑅 “ 𝑥) ∈ V})
21	20	eqriv 2732	. 2 ⊢ dom Image𝑅 = {𝑥 ∣ (𝑅 “ 𝑥) ∈ V}
22		df-fn 6566	. 2 ⊢ (Image𝑅 Fn {𝑥 ∣ (𝑅 “ 𝑥) ∈ V} ↔ (Fun Image𝑅 ∧ dom Image𝑅 = {𝑥 ∣ (𝑅 “ 𝑥) ∈ V}))
23	1, 21, 22	mpbir2an 711	1 ⊢ Image𝑅 Fn {𝑥 ∣ (𝑅 “ 𝑥) ∈ V}

Syntax hints:   ↔ wb 206   = wceq 1537  ∃wex 1776   ∈ wcel 2106  {cab 2712  Vcvv 3478   class class class wbr 5148  dom cdm 5689   “ cima 5692  Fun wfun 6557   Fn wfn 6558  Imagecimage 35822

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-symdif 4259  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-eprel 5589  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fo 6569  df-fv 6571  df-1st 8013  df-2nd 8014  df-txp 35836  df-image 35846

This theorem is referenced by:  imageval  35912