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Theorem imageval 35912
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 35910 . . 3 Fun Image𝑅
2 funrel 6585 . . 3 (Fun Image𝑅 → Rel Image𝑅)
31, 2ax-mp 5 . 2 Rel Image𝑅
4 mptrel 5838 . 2 Rel (𝑥 ∈ V ↦ (𝑅𝑥))
5 vex 3482 . . . . 5 𝑦 ∈ V
6 vex 3482 . . . . 5 𝑧 ∈ V
75, 6breldm 5922 . . . 4 (𝑦Image𝑅𝑧𝑦 ∈ dom Image𝑅)
8 fnimage 35911 . . . . 5 Image𝑅 Fn {𝑥 ∣ (𝑅𝑥) ∈ V}
98fndmi 6673 . . . 4 dom Image𝑅 = {𝑥 ∣ (𝑅𝑥) ∈ V}
107, 9eleqtrdi 2849 . . 3 (𝑦Image𝑅𝑧𝑦 ∈ {𝑥 ∣ (𝑅𝑥) ∈ V})
115, 6breldm 5922 . . . 4 (𝑦(𝑥 ∈ V ↦ (𝑅𝑥))𝑧𝑦 ∈ dom (𝑥 ∈ V ↦ (𝑅𝑥)))
12 eqid 2735 . . . . . 6 (𝑥 ∈ V ↦ (𝑅𝑥)) = (𝑥 ∈ V ↦ (𝑅𝑥))
1312dmmpt 6262 . . . . 5 dom (𝑥 ∈ V ↦ (𝑅𝑥)) = {𝑥 ∈ V ∣ (𝑅𝑥) ∈ V}
14 rabab 3510 . . . . 5 {𝑥 ∈ V ∣ (𝑅𝑥) ∈ V} = {𝑥 ∣ (𝑅𝑥) ∈ V}
1513, 14eqtri 2763 . . . 4 dom (𝑥 ∈ V ↦ (𝑅𝑥)) = {𝑥 ∣ (𝑅𝑥) ∈ V}
1611, 15eleqtrdi 2849 . . 3 (𝑦(𝑥 ∈ V ↦ (𝑅𝑥))𝑧𝑦 ∈ {𝑥 ∣ (𝑅𝑥) ∈ V})
17 imaeq2 6076 . . . . . 6 (𝑥 = 𝑦 → (𝑅𝑥) = (𝑅𝑦))
1817eleq1d 2824 . . . . 5 (𝑥 = 𝑦 → ((𝑅𝑥) ∈ V ↔ (𝑅𝑦) ∈ V))
195, 18elab 3681 . . . 4 (𝑦 ∈ {𝑥 ∣ (𝑅𝑥) ∈ V} ↔ (𝑅𝑦) ∈ V)
205, 6brimage 35908 . . . . 5 (𝑦Image𝑅𝑧𝑧 = (𝑅𝑦))
21 eqcom 2742 . . . . . 6 (𝑧 = (𝑅𝑦) ↔ (𝑅𝑦) = 𝑧)
2217, 12fvmptg 7014 . . . . . . . . 9 ((𝑦 ∈ V ∧ (𝑅𝑦) ∈ V) → ((𝑥 ∈ V ↦ (𝑅𝑥))‘𝑦) = (𝑅𝑦))
235, 22mpan 690 . . . . . . . 8 ((𝑅𝑦) ∈ V → ((𝑥 ∈ V ↦ (𝑅𝑥))‘𝑦) = (𝑅𝑦))
2423eqeq1d 2737 . . . . . . 7 ((𝑅𝑦) ∈ V → (((𝑥 ∈ V ↦ (𝑅𝑥))‘𝑦) = 𝑧 ↔ (𝑅𝑦) = 𝑧))
25 funmpt 6606 . . . . . . . . 9 Fun (𝑥 ∈ V ↦ (𝑅𝑥))
26 df-fn 6566 . . . . . . . . 9 ((𝑥 ∈ V ↦ (𝑅𝑥)) Fn {𝑥 ∣ (𝑅𝑥) ∈ V} ↔ (Fun (𝑥 ∈ V ↦ (𝑅𝑥)) ∧ dom (𝑥 ∈ V ↦ (𝑅𝑥)) = {𝑥 ∣ (𝑅𝑥) ∈ V}))
2725, 15, 26mpbir2an 711 . . . . . . . 8 (𝑥 ∈ V ↦ (𝑅𝑥)) Fn {𝑥 ∣ (𝑅𝑥) ∈ V}
2819biimpri 228 . . . . . . . 8 ((𝑅𝑦) ∈ V → 𝑦 ∈ {𝑥 ∣ (𝑅𝑥) ∈ V})
29 fnbrfvb 6960 . . . . . . . 8 (((𝑥 ∈ V ↦ (𝑅𝑥)) Fn {𝑥 ∣ (𝑅𝑥) ∈ V} ∧ 𝑦 ∈ {𝑥 ∣ (𝑅𝑥) ∈ V}) → (((𝑥 ∈ V ↦ (𝑅𝑥))‘𝑦) = 𝑧𝑦(𝑥 ∈ V ↦ (𝑅𝑥))𝑧))
3027, 28, 29sylancr 587 . . . . . . 7 ((𝑅𝑦) ∈ V → (((𝑥 ∈ V ↦ (𝑅𝑥))‘𝑦) = 𝑧𝑦(𝑥 ∈ V ↦ (𝑅𝑥))𝑧))
3124, 30bitr3d 281 . . . . . 6 ((𝑅𝑦) ∈ V → ((𝑅𝑦) = 𝑧𝑦(𝑥 ∈ V ↦ (𝑅𝑥))𝑧))
3221, 31bitrid 283 . . . . 5 ((𝑅𝑦) ∈ V → (𝑧 = (𝑅𝑦) ↔ 𝑦(𝑥 ∈ V ↦ (𝑅𝑥))𝑧))
3320, 32bitrid 283 . . . 4 ((𝑅𝑦) ∈ V → (𝑦Image𝑅𝑧𝑦(𝑥 ∈ V ↦ (𝑅𝑥))𝑧))
3419, 33sylbi 217 . . 3 (𝑦 ∈ {𝑥 ∣ (𝑅𝑥) ∈ V} → (𝑦Image𝑅𝑧𝑦(𝑥 ∈ V ↦ (𝑅𝑥))𝑧))
3510, 16, 34pm5.21nii 378 . 2 (𝑦Image𝑅𝑧𝑦(𝑥 ∈ V ↦ (𝑅𝑥))𝑧)
363, 4, 35eqbrriv 5804 1 Image𝑅 = (𝑥 ∈ V ↦ (𝑅𝑥))
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1537  wcel 2106  {cab 2712  {crab 3433  Vcvv 3478   class class class wbr 5148  cmpt 5231  dom cdm 5689  cima 5692  Rel wrel 5694  Fun wfun 6557   Fn wfn 6558  cfv 6563  Imagecimage 35822
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-symdif 4259  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-eprel 5589  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fo 6569  df-fv 6571  df-1st 8013  df-2nd 8014  df-txp 35836  df-image 35846
This theorem is referenced by:  fvimage  35913
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