Theorem imageval 32945

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 32943	. . 3 ⊢ Fun Image𝑅
2		funrel 6234	. . 3 ⊢ (Fun Image𝑅 → Rel Image𝑅)
3	1, 2	ax-mp 5	. 2 ⊢ Rel Image𝑅
4		mptrel 5575	. 2 ⊢ Rel (𝑥 ∈ V ↦ (𝑅 “ 𝑥))
5		vex 3435	. . . . 5 ⊢ 𝑦 ∈ V
6		vex 3435	. . . . 5 ⊢ 𝑧 ∈ V
7	5, 6	breldm 5655	. . . 4 ⊢ (𝑦Image𝑅𝑧 → 𝑦 ∈ dom Image𝑅)
8		fnimage 32944	. . . . 5 ⊢ Image𝑅 Fn {𝑥 ∣ (𝑅 “ 𝑥) ∈ V}
9		fndm 6317	. . . . 5 ⊢ (Image𝑅 Fn {𝑥 ∣ (𝑅 “ 𝑥) ∈ V} → dom Image𝑅 = {𝑥 ∣ (𝑅 “ 𝑥) ∈ V})
10	8, 9	ax-mp 5	. . . 4 ⊢ dom Image𝑅 = {𝑥 ∣ (𝑅 “ 𝑥) ∈ V}
11	7, 10	syl6eleq 2891	. . 3 ⊢ (𝑦Image𝑅𝑧 → 𝑦 ∈ {𝑥 ∣ (𝑅 “ 𝑥) ∈ V})
12	5, 6	breldm 5655	. . . 4 ⊢ (𝑦(𝑥 ∈ V ↦ (𝑅 “ 𝑥))𝑧 → 𝑦 ∈ dom (𝑥 ∈ V ↦ (𝑅 “ 𝑥)))
13		eqid 2793	. . . . . 6 ⊢ (𝑥 ∈ V ↦ (𝑅 “ 𝑥)) = (𝑥 ∈ V ↦ (𝑅 “ 𝑥))
14	13	dmmpt 5961	. . . . 5 ⊢ dom (𝑥 ∈ V ↦ (𝑅 “ 𝑥)) = {𝑥 ∈ V ∣ (𝑅 “ 𝑥) ∈ V}
15		rabab 3461	. . . . 5 ⊢ {𝑥 ∈ V ∣ (𝑅 “ 𝑥) ∈ V} = {𝑥 ∣ (𝑅 “ 𝑥) ∈ V}
16	14, 15	eqtri 2817	. . . 4 ⊢ dom (𝑥 ∈ V ↦ (𝑅 “ 𝑥)) = {𝑥 ∣ (𝑅 “ 𝑥) ∈ V}
17	12, 16	syl6eleq 2891	. . 3 ⊢ (𝑦(𝑥 ∈ V ↦ (𝑅 “ 𝑥))𝑧 → 𝑦 ∈ {𝑥 ∣ (𝑅 “ 𝑥) ∈ V})
18		imaeq2 5794	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑅 “ 𝑥) = (𝑅 “ 𝑦))
19	18	eleq1d 2865	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑅 “ 𝑥) ∈ V ↔ (𝑅 “ 𝑦) ∈ V))
20	5, 19	elab 3600	. . . 4 ⊢ (𝑦 ∈ {𝑥 ∣ (𝑅 “ 𝑥) ∈ V} ↔ (𝑅 “ 𝑦) ∈ V)
21	5, 6	brimage 32941	. . . . 5 ⊢ (𝑦Image𝑅𝑧 ↔ 𝑧 = (𝑅 “ 𝑦))
22		eqcom 2800	. . . . . 6 ⊢ (𝑧 = (𝑅 “ 𝑦) ↔ (𝑅 “ 𝑦) = 𝑧)
23	18, 13	fvmptg 6624	. . . . . . . . 9 ⊢ ((𝑦 ∈ V ∧ (𝑅 “ 𝑦) ∈ V) → ((𝑥 ∈ V ↦ (𝑅 “ 𝑥))‘𝑦) = (𝑅 “ 𝑦))
24	5, 23	mpan 686	. . . . . . . 8 ⊢ ((𝑅 “ 𝑦) ∈ V → ((𝑥 ∈ V ↦ (𝑅 “ 𝑥))‘𝑦) = (𝑅 “ 𝑦))
25	24	eqeq1d 2795	. . . . . . 7 ⊢ ((𝑅 “ 𝑦) ∈ V → (((𝑥 ∈ V ↦ (𝑅 “ 𝑥))‘𝑦) = 𝑧 ↔ (𝑅 “ 𝑦) = 𝑧))
26		funmpt 6255	. . . . . . . . 9 ⊢ Fun (𝑥 ∈ V ↦ (𝑅 “ 𝑥))
27		df-fn 6220	. . . . . . . . 9 ⊢ ((𝑥 ∈ V ↦ (𝑅 “ 𝑥)) Fn {𝑥 ∣ (𝑅 “ 𝑥) ∈ V} ↔ (Fun (𝑥 ∈ V ↦ (𝑅 “ 𝑥)) ∧ dom (𝑥 ∈ V ↦ (𝑅 “ 𝑥)) = {𝑥 ∣ (𝑅 “ 𝑥) ∈ V}))
28	26, 16, 27	mpbir2an 707	. . . . . . . 8 ⊢ (𝑥 ∈ V ↦ (𝑅 “ 𝑥)) Fn {𝑥 ∣ (𝑅 “ 𝑥) ∈ V}
29	20	biimpri 229	. . . . . . . 8 ⊢ ((𝑅 “ 𝑦) ∈ V → 𝑦 ∈ {𝑥 ∣ (𝑅 “ 𝑥) ∈ V})
30		fnbrfvb 6578	. . . . . . . 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))
31	28, 29, 30	sylancr 587	. . . . . . 7 ⊢ ((𝑅 “ 𝑦) ∈ V → (((𝑥 ∈ V ↦ (𝑅 “ 𝑥))‘𝑦) = 𝑧 ↔ 𝑦(𝑥 ∈ V ↦ (𝑅 “ 𝑥))𝑧))
32	25, 31	bitr3d 282	. . . . . 6 ⊢ ((𝑅 “ 𝑦) ∈ V → ((𝑅 “ 𝑦) = 𝑧 ↔ 𝑦(𝑥 ∈ V ↦ (𝑅 “ 𝑥))𝑧))
33	22, 32	syl5bb 284	. . . . 5 ⊢ ((𝑅 “ 𝑦) ∈ V → (𝑧 = (𝑅 “ 𝑦) ↔ 𝑦(𝑥 ∈ V ↦ (𝑅 “ 𝑥))𝑧))
34	21, 33	syl5bb 284	. . . 4 ⊢ ((𝑅 “ 𝑦) ∈ V → (𝑦Image𝑅𝑧 ↔ 𝑦(𝑥 ∈ V ↦ (𝑅 “ 𝑥))𝑧))
35	20, 34	sylbi 218	. . 3 ⊢ (𝑦 ∈ {𝑥 ∣ (𝑅 “ 𝑥) ∈ V} → (𝑦Image𝑅𝑧 ↔ 𝑦(𝑥 ∈ V ↦ (𝑅 “ 𝑥))𝑧))
36	11, 17, 35	pm5.21nii 380	. 2 ⊢ (𝑦Image𝑅𝑧 ↔ 𝑦(𝑥 ∈ V ↦ (𝑅 “ 𝑥))𝑧)
37	3, 4, 36	eqbrriv 5542	1 ⊢ Image𝑅 = (𝑥 ∈ V ↦ (𝑅 “ 𝑥))

Syntax hints:   ↔ wb 207   = wceq 1520   ∈ wcel 2079  {cab 2773  {crab 3107  Vcvv 3432   class class class wbr 4956   ↦ cmpt 5035  dom cdm 5435   “ cima 5438  Rel wrel 5440  Fun wfun 6211   Fn wfn 6212  ‘cfv 6217  Imagecimage 32855

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-13 2342  ax-ext 2767  ax-sep 5088  ax-nul 5095  ax-pow 5150  ax-pr 5214  ax-un 7310

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1080  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-mo 2574  df-eu 2610  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ne 2983  df-ral 3108  df-rex 3109  df-rab 3112  df-v 3434  df-sbc 3702  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-symdif 4134  df-nul 4207  df-if 4376  df-sn 4467  df-pr 4469  df-op 4473  df-uni 4740  df-br 4957  df-opab 5019  df-mpt 5036  df-id 5340  df-eprel 5345  df-xp 5441  df-rel 5442  df-cnv 5443  df-co 5444  df-dm 5445  df-rn 5446  df-res 5447  df-ima 5448  df-iota 6181  df-fun 6219  df-fn 6220  df-f 6221  df-fo 6223  df-fv 6225  df-1st 7536  df-2nd 7537  df-txp 32869  df-image 32879

This theorem is referenced by:  fvimage  32946