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Theorem imageval 35207
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 35205 . . 3 Fun Image𝑅
2 funrel 6565 . . 3 (Fun Image𝑅 → Rel Image𝑅)
31, 2ax-mp 5 . 2 Rel Image𝑅
4 mptrel 5825 . 2 Rel (𝑥 ∈ V ↦ (𝑅𝑥))
5 vex 3477 . . . . 5 𝑦 ∈ V
6 vex 3477 . . . . 5 𝑧 ∈ V
75, 6breldm 5908 . . . 4 (𝑦Image𝑅𝑧𝑦 ∈ dom Image𝑅)
8 fnimage 35206 . . . . 5 Image𝑅 Fn {𝑥 ∣ (𝑅𝑥) ∈ V}
98fndmi 6653 . . . 4 dom Image𝑅 = {𝑥 ∣ (𝑅𝑥) ∈ V}
107, 9eleqtrdi 2842 . . 3 (𝑦Image𝑅𝑧𝑦 ∈ {𝑥 ∣ (𝑅𝑥) ∈ V})
115, 6breldm 5908 . . . 4 (𝑦(𝑥 ∈ V ↦ (𝑅𝑥))𝑧𝑦 ∈ dom (𝑥 ∈ V ↦ (𝑅𝑥)))
12 eqid 2731 . . . . . 6 (𝑥 ∈ V ↦ (𝑅𝑥)) = (𝑥 ∈ V ↦ (𝑅𝑥))
1312dmmpt 6239 . . . . 5 dom (𝑥 ∈ V ↦ (𝑅𝑥)) = {𝑥 ∈ V ∣ (𝑅𝑥) ∈ V}
14 rabab 3502 . . . . 5 {𝑥 ∈ V ∣ (𝑅𝑥) ∈ V} = {𝑥 ∣ (𝑅𝑥) ∈ V}
1513, 14eqtri 2759 . . . 4 dom (𝑥 ∈ V ↦ (𝑅𝑥)) = {𝑥 ∣ (𝑅𝑥) ∈ V}
1611, 15eleqtrdi 2842 . . 3 (𝑦(𝑥 ∈ V ↦ (𝑅𝑥))𝑧𝑦 ∈ {𝑥 ∣ (𝑅𝑥) ∈ V})
17 imaeq2 6055 . . . . . 6 (𝑥 = 𝑦 → (𝑅𝑥) = (𝑅𝑦))
1817eleq1d 2817 . . . . 5 (𝑥 = 𝑦 → ((𝑅𝑥) ∈ V ↔ (𝑅𝑦) ∈ V))
195, 18elab 3668 . . . 4 (𝑦 ∈ {𝑥 ∣ (𝑅𝑥) ∈ V} ↔ (𝑅𝑦) ∈ V)
205, 6brimage 35203 . . . . 5 (𝑦Image𝑅𝑧𝑧 = (𝑅𝑦))
21 eqcom 2738 . . . . . 6 (𝑧 = (𝑅𝑦) ↔ (𝑅𝑦) = 𝑧)
2217, 12fvmptg 6996 . . . . . . . . 9 ((𝑦 ∈ V ∧ (𝑅𝑦) ∈ V) → ((𝑥 ∈ V ↦ (𝑅𝑥))‘𝑦) = (𝑅𝑦))
235, 22mpan 687 . . . . . . . 8 ((𝑅𝑦) ∈ V → ((𝑥 ∈ V ↦ (𝑅𝑥))‘𝑦) = (𝑅𝑦))
2423eqeq1d 2733 . . . . . . 7 ((𝑅𝑦) ∈ V → (((𝑥 ∈ V ↦ (𝑅𝑥))‘𝑦) = 𝑧 ↔ (𝑅𝑦) = 𝑧))
25 funmpt 6586 . . . . . . . . 9 Fun (𝑥 ∈ V ↦ (𝑅𝑥))
26 df-fn 6546 . . . . . . . . 9 ((𝑥 ∈ V ↦ (𝑅𝑥)) Fn {𝑥 ∣ (𝑅𝑥) ∈ V} ↔ (Fun (𝑥 ∈ V ↦ (𝑅𝑥)) ∧ dom (𝑥 ∈ V ↦ (𝑅𝑥)) = {𝑥 ∣ (𝑅𝑥) ∈ V}))
2725, 15, 26mpbir2an 708 . . . . . . . 8 (𝑥 ∈ V ↦ (𝑅𝑥)) Fn {𝑥 ∣ (𝑅𝑥) ∈ V}
2819biimpri 227 . . . . . . . 8 ((𝑅𝑦) ∈ V → 𝑦 ∈ {𝑥 ∣ (𝑅𝑥) ∈ V})
29 fnbrfvb 6944 . . . . . . . 8 (((𝑥 ∈ V ↦ (𝑅𝑥)) Fn {𝑥 ∣ (𝑅𝑥) ∈ V} ∧ 𝑦 ∈ {𝑥 ∣ (𝑅𝑥) ∈ V}) → (((𝑥 ∈ V ↦ (𝑅𝑥))‘𝑦) = 𝑧𝑦(𝑥 ∈ V ↦ (𝑅𝑥))𝑧))
3027, 28, 29sylancr 586 . . . . . . 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 216 . . 3 (𝑦 ∈ {𝑥 ∣ (𝑅𝑥) ∈ V} → (𝑦Image𝑅𝑧𝑦(𝑥 ∈ V ↦ (𝑅𝑥))𝑧))
3510, 16, 34pm5.21nii 378 . 2 (𝑦Image𝑅𝑧𝑦(𝑥 ∈ V ↦ (𝑅𝑥))𝑧)
363, 4, 35eqbrriv 5791 1 Image𝑅 = (𝑥 ∈ V ↦ (𝑅𝑥))
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1540  wcel 2105  {cab 2708  {crab 3431  Vcvv 3473   class class class wbr 5148  cmpt 5231  dom cdm 5676  cima 5679  Rel wrel 5681  Fun wfun 6537   Fn wfn 6538  cfv 6543  Imagecimage 35117
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7728
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-symdif 4242  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-eprel 5580  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fo 6549  df-fv 6551  df-1st 7978  df-2nd 7979  df-txp 35131  df-image 35141
This theorem is referenced by:  fvimage  35208
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