Theorem fnimage 35902

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 35901	. 2 ⊢ Fun Image𝑅
2		vex 3442	. . . . . . . 8 ⊢ 𝑦 ∈ V
3		vex 3442	. . . . . . . 8 ⊢ 𝑥 ∈ V
4	2, 3	brimage 35899	. . . . . . 7 ⊢ (𝑦Image𝑅𝑥 ↔ 𝑥 = (𝑅 “ 𝑦))
5		eqvisset 3458	. . . . . . 7 ⊢ (𝑥 = (𝑅 “ 𝑦) → (𝑅 “ 𝑦) ∈ V)
6	4, 5	sylbi 217	. . . . . 6 ⊢ (𝑦Image𝑅𝑥 → (𝑅 “ 𝑦) ∈ V)
7	6	exlimiv 1930	. . . . 5 ⊢ (∃𝑥 𝑦Image𝑅𝑥 → (𝑅 “ 𝑦) ∈ V)
8		eqid 2729	. . . . . . 7 ⊢ (𝑅 “ 𝑦) = (𝑅 “ 𝑦)
9		brimageg 35900	. . . . . . . 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 5099	. . . . . . 7 ⊢ (𝑥 = (𝑅 “ 𝑦) → (𝑦Image𝑅𝑥 ↔ 𝑦Image𝑅(𝑅 “ 𝑦)))
13	12	spcegv 3554	. . . . . 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 5847	. . . 4 ⊢ (𝑦 ∈ dom Image𝑅 ↔ ∃𝑥 𝑦Image𝑅𝑥)
17		imaeq2 6011	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑅 “ 𝑥) = (𝑅 “ 𝑦))
18	17	eleq1d 2813	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑅 “ 𝑥) ∈ V ↔ (𝑅 “ 𝑦) ∈ V))
19	2, 18	elab 3637	. . . 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 2726	. 2 ⊢ dom Image𝑅 = {𝑥 ∣ (𝑅 “ 𝑥) ∈ V}
22		df-fn 6489	. 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 1540  ∃wex 1779   ∈ wcel 2109  {cab 2707  Vcvv 3438   class class class wbr 5095  dom cdm 5623   “ cima 5626  Fun wfun 6480   Fn wfn 6481  Imagecimage 35813

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 5238  ax-nul 5248  ax-pr 5374  ax-un 7675

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 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-symdif 4206  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-eprel 5523  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-fo 6492  df-fv 6494  df-1st 7931  df-2nd 7932  df-txp 35827  df-image 35837

This theorem is referenced by:  imageval  35903