Theorem imageval 36163

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 36161	. . 3 ⊢ Fun Image𝑅
2		funrel 6509	. . 3 ⊢ (Fun Image𝑅 → Rel Image𝑅)
3	1, 2	ax-mp 5	. 2 ⊢ Rel Image𝑅
4		mptrel 5775	. 2 ⊢ Rel (𝑥 ∈ V ↦ (𝑅 “ 𝑥))
5		vex 3436	. . . . 5 ⊢ 𝑦 ∈ V
6		vex 3436	. . . . 5 ⊢ 𝑧 ∈ V
7	5, 6	breldm 5857	. . . 4 ⊢ (𝑦Image𝑅𝑧 → 𝑦 ∈ dom Image𝑅)
8		fnimage 36162	. . . . 5 ⊢ Image𝑅 Fn {𝑥 ∣ (𝑅 “ 𝑥) ∈ V}
9	8	fndmi 6596	. . . 4 ⊢ dom Image𝑅 = {𝑥 ∣ (𝑅 “ 𝑥) ∈ V}
10	7, 9	eleqtrdi 2850	. . 3 ⊢ (𝑦Image𝑅𝑧 → 𝑦 ∈ {𝑥 ∣ (𝑅 “ 𝑥) ∈ V})
11	5, 6	breldm 5857	. . . 4 ⊢ (𝑦(𝑥 ∈ V ↦ (𝑅 “ 𝑥))𝑧 → 𝑦 ∈ dom (𝑥 ∈ V ↦ (𝑅 “ 𝑥)))
12		eqid 2740	. . . . . 6 ⊢ (𝑥 ∈ V ↦ (𝑅 “ 𝑥)) = (𝑥 ∈ V ↦ (𝑅 “ 𝑥))
13	12	dmmpt 6198	. . . . 5 ⊢ dom (𝑥 ∈ V ↦ (𝑅 “ 𝑥)) = {𝑥 ∈ V ∣ (𝑅 “ 𝑥) ∈ V}
14		rabab 3463	. . . . 5 ⊢ {𝑥 ∈ V ∣ (𝑅 “ 𝑥) ∈ V} = {𝑥 ∣ (𝑅 “ 𝑥) ∈ V}
15	13, 14	eqtri 2763	. . . 4 ⊢ dom (𝑥 ∈ V ↦ (𝑅 “ 𝑥)) = {𝑥 ∣ (𝑅 “ 𝑥) ∈ V}
16	11, 15	eleqtrdi 2850	. . 3 ⊢ (𝑦(𝑥 ∈ V ↦ (𝑅 “ 𝑥))𝑧 → 𝑦 ∈ {𝑥 ∣ (𝑅 “ 𝑥) ∈ V})
17		imaeq2 6015	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑅 “ 𝑥) = (𝑅 “ 𝑦))
18	17	eleq1d 2825	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑅 “ 𝑥) ∈ V ↔ (𝑅 “ 𝑦) ∈ V))
19	5, 18	elab 3624	. . . 4 ⊢ (𝑦 ∈ {𝑥 ∣ (𝑅 “ 𝑥) ∈ V} ↔ (𝑅 “ 𝑦) ∈ V)
20	5, 6	brimage 36159	. . . . 5 ⊢ (𝑦Image𝑅𝑧 ↔ 𝑧 = (𝑅 “ 𝑦))
21		eqcom 2747	. . . . . 6 ⊢ (𝑧 = (𝑅 “ 𝑦) ↔ (𝑅 “ 𝑦) = 𝑧)
22	17, 12	fvmptg 6940	. . . . . . . . 9 ⊢ ((𝑦 ∈ V ∧ (𝑅 “ 𝑦) ∈ V) → ((𝑥 ∈ V ↦ (𝑅 “ 𝑥))‘𝑦) = (𝑅 “ 𝑦))
23	5, 22	mpan 696	. . . . . . . 8 ⊢ ((𝑅 “ 𝑦) ∈ V → ((𝑥 ∈ V ↦ (𝑅 “ 𝑥))‘𝑦) = (𝑅 “ 𝑦))
24	23	eqeq1d 2742	. . . . . . 7 ⊢ ((𝑅 “ 𝑦) ∈ V → (((𝑥 ∈ V ↦ (𝑅 “ 𝑥))‘𝑦) = 𝑧 ↔ (𝑅 “ 𝑦) = 𝑧))
25		funmpt 6530	. . . . . . . . 9 ⊢ Fun (𝑥 ∈ V ↦ (𝑅 “ 𝑥))
26		df-fn 6495	. . . . . . . . 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 717	. . . . . . . 8 ⊢ (𝑥 ∈ V ↦ (𝑅 “ 𝑥)) Fn {𝑥 ∣ (𝑅 “ 𝑥) ∈ V}
28	19	biimpri 229	. . . . . . . 8 ⊢ ((𝑅 “ 𝑦) ∈ V → 𝑦 ∈ {𝑥 ∣ (𝑅 “ 𝑥) ∈ V})
29		fnbrfvb 6884	. . . . . . . 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 593	. . . . . . 7 ⊢ ((𝑅 “ 𝑦) ∈ V → (((𝑥 ∈ V ↦ (𝑅 “ 𝑥))‘𝑦) = 𝑧 ↔ 𝑦(𝑥 ∈ V ↦ (𝑅 “ 𝑥))𝑧))
31	24, 30	bitr3d 282	. . . . . 6 ⊢ ((𝑅 “ 𝑦) ∈ V → ((𝑅 “ 𝑦) = 𝑧 ↔ 𝑦(𝑥 ∈ V ↦ (𝑅 “ 𝑥))𝑧))
32	21, 31	bitrid 284	. . . . 5 ⊢ ((𝑅 “ 𝑦) ∈ V → (𝑧 = (𝑅 “ 𝑦) ↔ 𝑦(𝑥 ∈ V ↦ (𝑅 “ 𝑥))𝑧))
33	20, 32	bitrid 284	. . . 4 ⊢ ((𝑅 “ 𝑦) ∈ V → (𝑦Image𝑅𝑧 ↔ 𝑦(𝑥 ∈ V ↦ (𝑅 “ 𝑥))𝑧))
34	19, 33	sylbi 218	. . 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 5741	1 ⊢ Image𝑅 = (𝑥 ∈ V ↦ (𝑅 “ 𝑥))

Syntax hints:   ↔ wb 207   = wceq 1547   ∈ wcel 2119  {cab 2718  {crab 3392  Vcvv 3432   class class class wbr 5079   ↦ cmpt 5160  dom cdm 5625   “ cima 5628  Rel wrel 5630  Fun wfun 6486   Fn wfn 6487  ‘cfv 6492  Imagecimage 36073

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-symdif 4188  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-eprel 5525  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fo 6498  df-fv 6500  df-1st 7938  df-2nd 7939  df-txp 36087  df-image 36097

This theorem is referenced by:  fvimage  36164