Theorem imageval 35903

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 35901	. . 3 ⊢ Fun Image𝑅
2		funrel 6503	. . 3 ⊢ (Fun Image𝑅 → Rel Image𝑅)
3	1, 2	ax-mp 5	. 2 ⊢ Rel Image𝑅
4		mptrel 5772	. 2 ⊢ Rel (𝑥 ∈ V ↦ (𝑅 “ 𝑥))
5		vex 3442	. . . . 5 ⊢ 𝑦 ∈ V
6		vex 3442	. . . . 5 ⊢ 𝑧 ∈ V
7	5, 6	breldm 5855	. . . 4 ⊢ (𝑦Image𝑅𝑧 → 𝑦 ∈ dom Image𝑅)
8		fnimage 35902	. . . . 5 ⊢ Image𝑅 Fn {𝑥 ∣ (𝑅 “ 𝑥) ∈ V}
9	8	fndmi 6590	. . . 4 ⊢ dom Image𝑅 = {𝑥 ∣ (𝑅 “ 𝑥) ∈ V}
10	7, 9	eleqtrdi 2838	. . 3 ⊢ (𝑦Image𝑅𝑧 → 𝑦 ∈ {𝑥 ∣ (𝑅 “ 𝑥) ∈ V})
11	5, 6	breldm 5855	. . . 4 ⊢ (𝑦(𝑥 ∈ V ↦ (𝑅 “ 𝑥))𝑧 → 𝑦 ∈ dom (𝑥 ∈ V ↦ (𝑅 “ 𝑥)))
12		eqid 2729	. . . . . 6 ⊢ (𝑥 ∈ V ↦ (𝑅 “ 𝑥)) = (𝑥 ∈ V ↦ (𝑅 “ 𝑥))
13	12	dmmpt 6193	. . . . 5 ⊢ dom (𝑥 ∈ V ↦ (𝑅 “ 𝑥)) = {𝑥 ∈ V ∣ (𝑅 “ 𝑥) ∈ V}
14		rabab 3469	. . . . 5 ⊢ {𝑥 ∈ V ∣ (𝑅 “ 𝑥) ∈ V} = {𝑥 ∣ (𝑅 “ 𝑥) ∈ V}
15	13, 14	eqtri 2752	. . . 4 ⊢ dom (𝑥 ∈ V ↦ (𝑅 “ 𝑥)) = {𝑥 ∣ (𝑅 “ 𝑥) ∈ V}
16	11, 15	eleqtrdi 2838	. . 3 ⊢ (𝑦(𝑥 ∈ V ↦ (𝑅 “ 𝑥))𝑧 → 𝑦 ∈ {𝑥 ∣ (𝑅 “ 𝑥) ∈ V})
17		imaeq2 6011	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑅 “ 𝑥) = (𝑅 “ 𝑦))
18	17	eleq1d 2813	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑅 “ 𝑥) ∈ V ↔ (𝑅 “ 𝑦) ∈ V))
19	5, 18	elab 3637	. . . 4 ⊢ (𝑦 ∈ {𝑥 ∣ (𝑅 “ 𝑥) ∈ V} ↔ (𝑅 “ 𝑦) ∈ V)
20	5, 6	brimage 35899	. . . . 5 ⊢ (𝑦Image𝑅𝑧 ↔ 𝑧 = (𝑅 “ 𝑦))
21		eqcom 2736	. . . . . 6 ⊢ (𝑧 = (𝑅 “ 𝑦) ↔ (𝑅 “ 𝑦) = 𝑧)
22	17, 12	fvmptg 6932	. . . . . . . . 9 ⊢ ((𝑦 ∈ V ∧ (𝑅 “ 𝑦) ∈ V) → ((𝑥 ∈ V ↦ (𝑅 “ 𝑥))‘𝑦) = (𝑅 “ 𝑦))
23	5, 22	mpan 690	. . . . . . . 8 ⊢ ((𝑅 “ 𝑦) ∈ V → ((𝑥 ∈ V ↦ (𝑅 “ 𝑥))‘𝑦) = (𝑅 “ 𝑦))
24	23	eqeq1d 2731	. . . . . . 7 ⊢ ((𝑅 “ 𝑦) ∈ V → (((𝑥 ∈ V ↦ (𝑅 “ 𝑥))‘𝑦) = 𝑧 ↔ (𝑅 “ 𝑦) = 𝑧))
25		funmpt 6524	. . . . . . . . 9 ⊢ Fun (𝑥 ∈ V ↦ (𝑅 “ 𝑥))
26		df-fn 6489	. . . . . . . . 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 711	. . . . . . . 8 ⊢ (𝑥 ∈ V ↦ (𝑅 “ 𝑥)) Fn {𝑥 ∣ (𝑅 “ 𝑥) ∈ V}
28	19	biimpri 228	. . . . . . . 8 ⊢ ((𝑅 “ 𝑦) ∈ V → 𝑦 ∈ {𝑥 ∣ (𝑅 “ 𝑥) ∈ V})
29		fnbrfvb 6877	. . . . . . . 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 587	. . . . . . 7 ⊢ ((𝑅 “ 𝑦) ∈ V → (((𝑥 ∈ V ↦ (𝑅 “ 𝑥))‘𝑦) = 𝑧 ↔ 𝑦(𝑥 ∈ V ↦ (𝑅 “ 𝑥))𝑧))
31	24, 30	bitr3d 281	. . . . . 6 ⊢ ((𝑅 “ 𝑦) ∈ V → ((𝑅 “ 𝑦) = 𝑧 ↔ 𝑦(𝑥 ∈ V ↦ (𝑅 “ 𝑥))𝑧))
32	21, 31	bitrid 283	. . . . 5 ⊢ ((𝑅 “ 𝑦) ∈ V → (𝑧 = (𝑅 “ 𝑦) ↔ 𝑦(𝑥 ∈ V ↦ (𝑅 “ 𝑥))𝑧))
33	20, 32	bitrid 283	. . . 4 ⊢ ((𝑅 “ 𝑦) ∈ V → (𝑦Image𝑅𝑧 ↔ 𝑦(𝑥 ∈ V ↦ (𝑅 “ 𝑥))𝑧))
34	19, 33	sylbi 217	. . 3 ⊢ (𝑦 ∈ {𝑥 ∣ (𝑅 “ 𝑥) ∈ V} → (𝑦Image𝑅𝑧 ↔ 𝑦(𝑥 ∈ V ↦ (𝑅 “ 𝑥))𝑧))
35	10, 16, 34	pm5.21nii 378	. 2 ⊢ (𝑦Image𝑅𝑧 ↔ 𝑦(𝑥 ∈ V ↦ (𝑅 “ 𝑥))𝑧)
36	3, 4, 35	eqbrriv 5738	1 ⊢ Image𝑅 = (𝑥 ∈ V ↦ (𝑅 “ 𝑥))

Syntax hints:   ↔ wb 206   = wceq 1540   ∈ wcel 2109  {cab 2707  {crab 3396  Vcvv 3438   class class class wbr 5095   ↦ cmpt 5176  dom cdm 5623   “ cima 5626  Rel wrel 5628  Fun wfun 6480   Fn wfn 6481  ‘cfv 6486  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:  fvimage  35904