Mathbox for Scott Fenton < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  imageval Structured version   Visualization version   GIF version

Theorem imageval 31207
 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 ↦ (𝑅𝑥))
Distinct variable group:   𝑥,𝑅

Proof of Theorem imageval
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 funimage 31205 . . 3 Fun Image𝑅
2 funrel 5821 . . 3 (Fun Image𝑅 → Rel Image𝑅)
31, 2ax-mp 5 . 2 Rel Image𝑅
4 mptrel 5170 . 2 Rel (𝑥 ∈ V ↦ (𝑅𝑥))
5 vex 3176 . . . . 5 𝑦 ∈ V
6 vex 3176 . . . . 5 𝑧 ∈ V
75, 6breldm 5251 . . . 4 (𝑦Image𝑅𝑧𝑦 ∈ dom Image𝑅)
8 fnimage 31206 . . . . 5 Image𝑅 Fn {𝑥 ∣ (𝑅𝑥) ∈ V}
9 fndm 5904 . . . . 5 (Image𝑅 Fn {𝑥 ∣ (𝑅𝑥) ∈ V} → dom Image𝑅 = {𝑥 ∣ (𝑅𝑥) ∈ V})
108, 9ax-mp 5 . . . 4 dom Image𝑅 = {𝑥 ∣ (𝑅𝑥) ∈ V}
117, 10syl6eleq 2698 . . 3 (𝑦Image𝑅𝑧𝑦 ∈ {𝑥 ∣ (𝑅𝑥) ∈ V})
125, 6breldm 5251 . . . 4 (𝑦(𝑥 ∈ V ↦ (𝑅𝑥))𝑧𝑦 ∈ dom (𝑥 ∈ V ↦ (𝑅𝑥)))
13 eqid 2610 . . . . . 6 (𝑥 ∈ V ↦ (𝑅𝑥)) = (𝑥 ∈ V ↦ (𝑅𝑥))
1413dmmpt 5547 . . . . 5 dom (𝑥 ∈ V ↦ (𝑅𝑥)) = {𝑥 ∈ V ∣ (𝑅𝑥) ∈ V}
15 rabab 3196 . . . . 5 {𝑥 ∈ V ∣ (𝑅𝑥) ∈ V} = {𝑥 ∣ (𝑅𝑥) ∈ V}
1614, 15eqtri 2632 . . . 4 dom (𝑥 ∈ V ↦ (𝑅𝑥)) = {𝑥 ∣ (𝑅𝑥) ∈ V}
1712, 16syl6eleq 2698 . . 3 (𝑦(𝑥 ∈ V ↦ (𝑅𝑥))𝑧𝑦 ∈ {𝑥 ∣ (𝑅𝑥) ∈ V})
18 imaeq2 5381 . . . . . 6 (𝑥 = 𝑦 → (𝑅𝑥) = (𝑅𝑦))
1918eleq1d 2672 . . . . 5 (𝑥 = 𝑦 → ((𝑅𝑥) ∈ V ↔ (𝑅𝑦) ∈ V))
205, 19elab 3319 . . . 4 (𝑦 ∈ {𝑥 ∣ (𝑅𝑥) ∈ V} ↔ (𝑅𝑦) ∈ V)
215, 6brimage 31203 . . . . 5 (𝑦Image𝑅𝑧𝑧 = (𝑅𝑦))
22 eqcom 2617 . . . . . 6 (𝑧 = (𝑅𝑦) ↔ (𝑅𝑦) = 𝑧)
2318, 13fvmptg 6189 . . . . . . . . 9 ((𝑦 ∈ V ∧ (𝑅𝑦) ∈ V) → ((𝑥 ∈ V ↦ (𝑅𝑥))‘𝑦) = (𝑅𝑦))
245, 23mpan 702 . . . . . . . 8 ((𝑅𝑦) ∈ V → ((𝑥 ∈ V ↦ (𝑅𝑥))‘𝑦) = (𝑅𝑦))
2524eqeq1d 2612 . . . . . . 7 ((𝑅𝑦) ∈ V → (((𝑥 ∈ V ↦ (𝑅𝑥))‘𝑦) = 𝑧 ↔ (𝑅𝑦) = 𝑧))
26 funmpt 5840 . . . . . . . . 9 Fun (𝑥 ∈ V ↦ (𝑅𝑥))
27 df-fn 5807 . . . . . . . . 9 ((𝑥 ∈ V ↦ (𝑅𝑥)) Fn {𝑥 ∣ (𝑅𝑥) ∈ V} ↔ (Fun (𝑥 ∈ V ↦ (𝑅𝑥)) ∧ dom (𝑥 ∈ V ↦ (𝑅𝑥)) = {𝑥 ∣ (𝑅𝑥) ∈ V}))
2826, 16, 27mpbir2an 957 . . . . . . . 8 (𝑥 ∈ V ↦ (𝑅𝑥)) Fn {𝑥 ∣ (𝑅𝑥) ∈ V}
2920biimpri 217 . . . . . . . 8 ((𝑅𝑦) ∈ V → 𝑦 ∈ {𝑥 ∣ (𝑅𝑥) ∈ V})
30 fnbrfvb 6146 . . . . . . . 8 (((𝑥 ∈ V ↦ (𝑅𝑥)) Fn {𝑥 ∣ (𝑅𝑥) ∈ V} ∧ 𝑦 ∈ {𝑥 ∣ (𝑅𝑥) ∈ V}) → (((𝑥 ∈ V ↦ (𝑅𝑥))‘𝑦) = 𝑧𝑦(𝑥 ∈ V ↦ (𝑅𝑥))𝑧))
3128, 29, 30sylancr 694 . . . . . . 7 ((𝑅𝑦) ∈ V → (((𝑥 ∈ V ↦ (𝑅𝑥))‘𝑦) = 𝑧𝑦(𝑥 ∈ V ↦ (𝑅𝑥))𝑧))
3225, 31bitr3d 269 . . . . . 6 ((𝑅𝑦) ∈ V → ((𝑅𝑦) = 𝑧𝑦(𝑥 ∈ V ↦ (𝑅𝑥))𝑧))
3322, 32syl5bb 271 . . . . 5 ((𝑅𝑦) ∈ V → (𝑧 = (𝑅𝑦) ↔ 𝑦(𝑥 ∈ V ↦ (𝑅𝑥))𝑧))
3421, 33syl5bb 271 . . . 4 ((𝑅𝑦) ∈ V → (𝑦Image𝑅𝑧𝑦(𝑥 ∈ V ↦ (𝑅𝑥))𝑧))
3520, 34sylbi 206 . . 3 (𝑦 ∈ {𝑥 ∣ (𝑅𝑥) ∈ V} → (𝑦Image𝑅𝑧𝑦(𝑥 ∈ V ↦ (𝑅𝑥))𝑧))
3611, 17, 35pm5.21nii 367 . 2 (𝑦Image𝑅𝑧𝑦(𝑥 ∈ V ↦ (𝑅𝑥))𝑧)
373, 4, 36eqbrriv 5138 1 Image𝑅 = (𝑥 ∈ V ↦ (𝑅𝑥))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   = wceq 1475   ∈ wcel 1977  {cab 2596  {crab 2900  Vcvv 3173   class class class wbr 4583   ↦ cmpt 4643  dom cdm 5038   “ cima 5041  Rel wrel 5043  Fun wfun 5798   Fn wfn 5799  ‘cfv 5804  Imagecimage 31116 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-symdif 3806  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-eprel 4949  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fo 5810  df-fv 5812  df-1st 7059  df-2nd 7060  df-txp 31130  df-image 31140 This theorem is referenced by:  fvimage  31208
 Copyright terms: Public domain W3C validator