Theorem imageval 34159

Description: The image functor in maps-to notation. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Assertion

Ref	Expression
imageval	⊢ Image𝑅 = (𝑥 ∈ V ↦ (𝑅 “ 𝑥))

Distinct variable group:   𝑥,𝑅

Proof of Theorem imageval

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		funimage 34157	. . 3 ⊢ Fun Image𝑅
2		funrel 6435	. . 3 ⊢ (Fun Image𝑅 → Rel Image𝑅)
3	1, 2	ax-mp 5	. 2 ⊢ Rel Image𝑅
4		mptrel 5724	. 2 ⊢ Rel (𝑥 ∈ V ↦ (𝑅 “ 𝑥))
5		vex 3426	. . . . 5 ⊢ 𝑦 ∈ V
6		vex 3426	. . . . 5 ⊢ 𝑧 ∈ V
7	5, 6	breldm 5806	. . . 4 ⊢ (𝑦Image𝑅𝑧 → 𝑦 ∈ dom Image𝑅)
8		fnimage 34158	. . . . 5 ⊢ Image𝑅 Fn {𝑥 ∣ (𝑅 “ 𝑥) ∈ V}
9	8	fndmi 6521	. . . 4 ⊢ dom Image𝑅 = {𝑥 ∣ (𝑅 “ 𝑥) ∈ V}
10	7, 9	eleqtrdi 2849	. . 3 ⊢ (𝑦Image𝑅𝑧 → 𝑦 ∈ {𝑥 ∣ (𝑅 “ 𝑥) ∈ V})
11	5, 6	breldm 5806	. . . 4 ⊢ (𝑦(𝑥 ∈ V ↦ (𝑅 “ 𝑥))𝑧 → 𝑦 ∈ dom (𝑥 ∈ V ↦ (𝑅 “ 𝑥)))
12		eqid 2738	. . . . . 6 ⊢ (𝑥 ∈ V ↦ (𝑅 “ 𝑥)) = (𝑥 ∈ V ↦ (𝑅 “ 𝑥))
13	12	dmmpt 6132	. . . . 5 ⊢ dom (𝑥 ∈ V ↦ (𝑅 “ 𝑥)) = {𝑥 ∈ V ∣ (𝑅 “ 𝑥) ∈ V}
14		rabab 3450	. . . . 5 ⊢ {𝑥 ∈ V ∣ (𝑅 “ 𝑥) ∈ V} = {𝑥 ∣ (𝑅 “ 𝑥) ∈ V}
15	13, 14	eqtri 2766	. . . 4 ⊢ dom (𝑥 ∈ V ↦ (𝑅 “ 𝑥)) = {𝑥 ∣ (𝑅 “ 𝑥) ∈ V}
16	11, 15	eleqtrdi 2849	. . 3 ⊢ (𝑦(𝑥 ∈ V ↦ (𝑅 “ 𝑥))𝑧 → 𝑦 ∈ {𝑥 ∣ (𝑅 “ 𝑥) ∈ V})
17		imaeq2 5954	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑅 “ 𝑥) = (𝑅 “ 𝑦))
18	17	eleq1d 2823	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑅 “ 𝑥) ∈ V ↔ (𝑅 “ 𝑦) ∈ V))
19	5, 18	elab 3602	. . . 4 ⊢ (𝑦 ∈ {𝑥 ∣ (𝑅 “ 𝑥) ∈ V} ↔ (𝑅 “ 𝑦) ∈ V)
20	5, 6	brimage 34155	. . . . 5 ⊢ (𝑦Image𝑅𝑧 ↔ 𝑧 = (𝑅 “ 𝑦))
21		eqcom 2745	. . . . . 6 ⊢ (𝑧 = (𝑅 “ 𝑦) ↔ (𝑅 “ 𝑦) = 𝑧)
22	17, 12	fvmptg 6855	. . . . . . . . 9 ⊢ ((𝑦 ∈ V ∧ (𝑅 “ 𝑦) ∈ V) → ((𝑥 ∈ V ↦ (𝑅 “ 𝑥))‘𝑦) = (𝑅 “ 𝑦))
23	5, 22	mpan 686	. . . . . . . 8 ⊢ ((𝑅 “ 𝑦) ∈ V → ((𝑥 ∈ V ↦ (𝑅 “ 𝑥))‘𝑦) = (𝑅 “ 𝑦))
24	23	eqeq1d 2740	. . . . . . 7 ⊢ ((𝑅 “ 𝑦) ∈ V → (((𝑥 ∈ V ↦ (𝑅 “ 𝑥))‘𝑦) = 𝑧 ↔ (𝑅 “ 𝑦) = 𝑧))
25		funmpt 6456	. . . . . . . . 9 ⊢ Fun (𝑥 ∈ V ↦ (𝑅 “ 𝑥))
26		df-fn 6421	. . . . . . . . 9 ⊢ ((𝑥 ∈ V ↦ (𝑅 “ 𝑥)) Fn {𝑥 ∣ (𝑅 “ 𝑥) ∈ V} ↔ (Fun (𝑥 ∈ V ↦ (𝑅 “ 𝑥)) ∧ dom (𝑥 ∈ V ↦ (𝑅 “ 𝑥)) = {𝑥 ∣ (𝑅 “ 𝑥) ∈ V}))
27	25, 15, 26	mpbir2an 707	. . . . . . . 8 ⊢ (𝑥 ∈ V ↦ (𝑅 “ 𝑥)) Fn {𝑥 ∣ (𝑅 “ 𝑥) ∈ V}
28	19	biimpri 227	. . . . . . . 8 ⊢ ((𝑅 “ 𝑦) ∈ V → 𝑦 ∈ {𝑥 ∣ (𝑅 “ 𝑥) ∈ V})
29		fnbrfvb 6804	. . . . . . . 8 ⊢ (((𝑥 ∈ V ↦ (𝑅 “ 𝑥)) Fn {𝑥 ∣ (𝑅 “ 𝑥) ∈ V} ∧ 𝑦 ∈ {𝑥 ∣ (𝑅 “ 𝑥) ∈ V}) → (((𝑥 ∈ V ↦ (𝑅 “ 𝑥))‘𝑦) = 𝑧 ↔ 𝑦(𝑥 ∈ V ↦ (𝑅 “ 𝑥))𝑧))
30	27, 28, 29	sylancr 586	. . . . . . 7 ⊢ ((𝑅 “ 𝑦) ∈ V → (((𝑥 ∈ V ↦ (𝑅 “ 𝑥))‘𝑦) = 𝑧 ↔ 𝑦(𝑥 ∈ V ↦ (𝑅 “ 𝑥))𝑧))
31	24, 30	bitr3d 280	. . . . . 6 ⊢ ((𝑅 “ 𝑦) ∈ V → ((𝑅 “ 𝑦) = 𝑧 ↔ 𝑦(𝑥 ∈ V ↦ (𝑅 “ 𝑥))𝑧))
32	21, 31	syl5bb 282	. . . . 5 ⊢ ((𝑅 “ 𝑦) ∈ V → (𝑧 = (𝑅 “ 𝑦) ↔ 𝑦(𝑥 ∈ V ↦ (𝑅 “ 𝑥))𝑧))
33	20, 32	syl5bb 282	. . . 4 ⊢ ((𝑅 “ 𝑦) ∈ V → (𝑦Image𝑅𝑧 ↔ 𝑦(𝑥 ∈ V ↦ (𝑅 “ 𝑥))𝑧))
34	19, 33	sylbi 216	. . 3 ⊢ (𝑦 ∈ {𝑥 ∣ (𝑅 “ 𝑥) ∈ V} → (𝑦Image𝑅𝑧 ↔ 𝑦(𝑥 ∈ V ↦ (𝑅 “ 𝑥))𝑧))
35	10, 16, 34	pm5.21nii 379	. 2 ⊢ (𝑦Image𝑅𝑧 ↔ 𝑦(𝑥 ∈ V ↦ (𝑅 “ 𝑥))𝑧)
36	3, 4, 35	eqbrriv 5690	1 ⊢ Image𝑅 = (𝑥 ∈ V ↦ (𝑅 “ 𝑥))

Syntax hints:   ↔ wb 205   = wceq 1539   ∈ wcel 2108  {cab 2715  {crab 3067  Vcvv 3422   class class class wbr 5070   ↦ cmpt 5153  dom cdm 5580   “ cima 5583  Rel wrel 5585  Fun wfun 6412   Fn wfn 6413  ‘cfv 6418  Imagecimage 34069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-symdif 4173  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-eprel 5486  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-fo 6424  df-fv 6426  df-1st 7804  df-2nd 7805  df-txp 34083  df-image 34093

This theorem is referenced by:  fvimage  34160