Theorem imageval 36398

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 36396	. . 3 ⊢ Fun Image𝑅
2		funrel 6555	. . 3 ⊢ (Fun Image𝑅 → Rel Image𝑅)
3	1, 2	ax-mp 5	. 2 ⊢ Rel Image𝑅
4		mptrel 5814	. 2 ⊢ Rel (𝑥 ∈ V ↦ (𝑅 “ 𝑥))
5		vex 3459	. . . . 5 ⊢ 𝑦 ∈ V
6		vex 3459	. . . . 5 ⊢ 𝑧 ∈ V
7	5, 6	breldm 5900	. . . 4 ⊢ (𝑦Image𝑅𝑧 → 𝑦 ∈ dom Image𝑅)
8		fnimage 36397	. . . . 5 ⊢ Image𝑅 Fn {𝑥 ∣ (𝑅 “ 𝑥) ∈ V}
9	8	fndmi 6641	. . . 4 ⊢ dom Image𝑅 = {𝑥 ∣ (𝑅 “ 𝑥) ∈ V}
10	7, 9	eleqtrdi 2873	. . 3 ⊢ (𝑦Image𝑅𝑧 → 𝑦 ∈ {𝑥 ∣ (𝑅 “ 𝑥) ∈ V})
11	5, 6	breldm 5900	. . . 4 ⊢ (𝑦(𝑥 ∈ V ↦ (𝑅 “ 𝑥))𝑧 → 𝑦 ∈ dom (𝑥 ∈ V ↦ (𝑅 “ 𝑥)))
12		eqid 2763	. . . . . 6 ⊢ (𝑥 ∈ V ↦ (𝑅 “ 𝑥)) = (𝑥 ∈ V ↦ (𝑅 “ 𝑥))
13	12	dmmpt 6243	. . . . 5 ⊢ dom (𝑥 ∈ V ↦ (𝑅 “ 𝑥)) = {𝑥 ∈ V ∣ (𝑅 “ 𝑥) ∈ V}
14		rabab 3485	. . . . 5 ⊢ {𝑥 ∈ V ∣ (𝑅 “ 𝑥) ∈ V} = {𝑥 ∣ (𝑅 “ 𝑥) ∈ V}
15	13, 14	eqtri 2786	. . . 4 ⊢ dom (𝑥 ∈ V ↦ (𝑅 “ 𝑥)) = {𝑥 ∣ (𝑅 “ 𝑥) ∈ V}
16	11, 15	eleqtrdi 2873	. . 3 ⊢ (𝑦(𝑥 ∈ V ↦ (𝑅 “ 𝑥))𝑧 → 𝑦 ∈ {𝑥 ∣ (𝑅 “ 𝑥) ∈ V})
17		imaeq2 6060	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑅 “ 𝑥) = (𝑅 “ 𝑦))
18	17	eleq1d 2848	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑅 “ 𝑥) ∈ V ↔ (𝑅 “ 𝑦) ∈ V))
19	5, 18	elab 3639	. . . 4 ⊢ (𝑦 ∈ {𝑥 ∣ (𝑅 “ 𝑥) ∈ V} ↔ (𝑅 “ 𝑦) ∈ V)
20	5, 6	brimage 36394	. . . . 5 ⊢ (𝑦Image𝑅𝑧 ↔ 𝑧 = (𝑅 “ 𝑦))
21		eqcom 2770	. . . . . 6 ⊢ (𝑧 = (𝑅 “ 𝑦) ↔ (𝑅 “ 𝑦) = 𝑧)
22	17, 12	fvmptg 6989	. . . . . . . . 9 ⊢ ((𝑦 ∈ V ∧ (𝑅 “ 𝑦) ∈ V) → ((𝑥 ∈ V ↦ (𝑅 “ 𝑥))‘𝑦) = (𝑅 “ 𝑦))
23	5, 22	mpan 702	. . . . . . . 8 ⊢ ((𝑅 “ 𝑦) ∈ V → ((𝑥 ∈ V ↦ (𝑅 “ 𝑥))‘𝑦) = (𝑅 “ 𝑦))
24	23	eqeq1d 2765	. . . . . . 7 ⊢ ((𝑅 “ 𝑦) ∈ V → (((𝑥 ∈ V ↦ (𝑅 “ 𝑥))‘𝑦) = 𝑧 ↔ (𝑅 “ 𝑦) = 𝑧))
25		funmpt 6576	. . . . . . . . 9 ⊢ Fun (𝑥 ∈ V ↦ (𝑅 “ 𝑥))
26		df-fn 6541	. . . . . . . . 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 723	. . . . . . . 8 ⊢ (𝑥 ∈ V ↦ (𝑅 “ 𝑥)) Fn {𝑥 ∣ (𝑅 “ 𝑥) ∈ V}
28	19	biimpri 231	. . . . . . . 8 ⊢ ((𝑅 “ 𝑦) ∈ V → 𝑦 ∈ {𝑥 ∣ (𝑅 “ 𝑥) ∈ V})
29		fnbrfvb 6933	. . . . . . . 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 598	. . . . . . 7 ⊢ ((𝑅 “ 𝑦) ∈ V → (((𝑥 ∈ V ↦ (𝑅 “ 𝑥))‘𝑦) = 𝑧 ↔ 𝑦(𝑥 ∈ V ↦ (𝑅 “ 𝑥))𝑧))
31	24, 30	bitr3d 284	. . . . . 6 ⊢ ((𝑅 “ 𝑦) ∈ V → ((𝑅 “ 𝑦) = 𝑧 ↔ 𝑦(𝑥 ∈ V ↦ (𝑅 “ 𝑥))𝑧))
32	21, 31	bitrid 286	. . . . 5 ⊢ ((𝑅 “ 𝑦) ∈ V → (𝑧 = (𝑅 “ 𝑦) ↔ 𝑦(𝑥 ∈ V ↦ (𝑅 “ 𝑥))𝑧))
33	20, 32	bitrid 286	. . . 4 ⊢ ((𝑅 “ 𝑦) ∈ V → (𝑦Image𝑅𝑧 ↔ 𝑦(𝑥 ∈ V ↦ (𝑅 “ 𝑥))𝑧))
34	19, 33	sylbi 220	. . 3 ⊢ (𝑦 ∈ {𝑥 ∣ (𝑅 “ 𝑥) ∈ V} → (𝑦Image𝑅𝑧 ↔ 𝑦(𝑥 ∈ V ↦ (𝑅 “ 𝑥))𝑧))
35	10, 16, 34	pm5.21nii 381	. 2 ⊢ (𝑦Image𝑅𝑧 ↔ 𝑦(𝑥 ∈ V ↦ (𝑅 “ 𝑥))𝑧)
36	3, 4, 35	eqbrriv 5779	1 ⊢ Image𝑅 = (𝑥 ∈ V ↦ (𝑅 “ 𝑥))

Syntax hints:   ↔ wb 209   = wceq 1570   ∈ wcel 2143  {cab 2741  {crab 3416  Vcvv 3455   class class class wbr 5110   ↦ cmpt 5193  dom cdm 5663   “ cima 5666  Rel wrel 5668  Fun wfun 6532   Fn wfn 6533  ‘cfv 6538  Imagecimage 36308

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-symdif 4207  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-eprel 5563  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fo 6544  df-fv 6546  df-1st 7987  df-2nd 7988  df-txp 36322  df-image 36332

This theorem is referenced by:  fvimage  36399