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Theorem imageval 31700
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 31698 . . 3 Fun Image𝑅
2 funrel 5866 . . 3 (Fun Image𝑅 → Rel Image𝑅)
31, 2ax-mp 5 . 2 Rel Image𝑅
4 mptrel 5210 . 2 Rel (𝑥 ∈ V ↦ (𝑅𝑥))
5 vex 3189 . . . . 5 𝑦 ∈ V
6 vex 3189 . . . . 5 𝑧 ∈ V
75, 6breldm 5291 . . . 4 (𝑦Image𝑅𝑧𝑦 ∈ dom Image𝑅)
8 fnimage 31699 . . . . 5 Image𝑅 Fn {𝑥 ∣ (𝑅𝑥) ∈ V}
9 fndm 5950 . . . . 5 (Image𝑅 Fn {𝑥 ∣ (𝑅𝑥) ∈ V} → dom Image𝑅 = {𝑥 ∣ (𝑅𝑥) ∈ V})
108, 9ax-mp 5 . . . 4 dom Image𝑅 = {𝑥 ∣ (𝑅𝑥) ∈ V}
117, 10syl6eleq 2708 . . 3 (𝑦Image𝑅𝑧𝑦 ∈ {𝑥 ∣ (𝑅𝑥) ∈ V})
125, 6breldm 5291 . . . 4 (𝑦(𝑥 ∈ V ↦ (𝑅𝑥))𝑧𝑦 ∈ dom (𝑥 ∈ V ↦ (𝑅𝑥)))
13 eqid 2621 . . . . . 6 (𝑥 ∈ V ↦ (𝑅𝑥)) = (𝑥 ∈ V ↦ (𝑅𝑥))
1413dmmpt 5591 . . . . 5 dom (𝑥 ∈ V ↦ (𝑅𝑥)) = {𝑥 ∈ V ∣ (𝑅𝑥) ∈ V}
15 rabab 3209 . . . . 5 {𝑥 ∈ V ∣ (𝑅𝑥) ∈ V} = {𝑥 ∣ (𝑅𝑥) ∈ V}
1614, 15eqtri 2643 . . . 4 dom (𝑥 ∈ V ↦ (𝑅𝑥)) = {𝑥 ∣ (𝑅𝑥) ∈ V}
1712, 16syl6eleq 2708 . . 3 (𝑦(𝑥 ∈ V ↦ (𝑅𝑥))𝑧𝑦 ∈ {𝑥 ∣ (𝑅𝑥) ∈ V})
18 imaeq2 5423 . . . . . 6 (𝑥 = 𝑦 → (𝑅𝑥) = (𝑅𝑦))
1918eleq1d 2683 . . . . 5 (𝑥 = 𝑦 → ((𝑅𝑥) ∈ V ↔ (𝑅𝑦) ∈ V))
205, 19elab 3334 . . . 4 (𝑦 ∈ {𝑥 ∣ (𝑅𝑥) ∈ V} ↔ (𝑅𝑦) ∈ V)
215, 6brimage 31696 . . . . 5 (𝑦Image𝑅𝑧𝑧 = (𝑅𝑦))
22 eqcom 2628 . . . . . 6 (𝑧 = (𝑅𝑦) ↔ (𝑅𝑦) = 𝑧)
2318, 13fvmptg 6239 . . . . . . . . 9 ((𝑦 ∈ V ∧ (𝑅𝑦) ∈ V) → ((𝑥 ∈ V ↦ (𝑅𝑥))‘𝑦) = (𝑅𝑦))
245, 23mpan 705 . . . . . . . 8 ((𝑅𝑦) ∈ V → ((𝑥 ∈ V ↦ (𝑅𝑥))‘𝑦) = (𝑅𝑦))
2524eqeq1d 2623 . . . . . . 7 ((𝑅𝑦) ∈ V → (((𝑥 ∈ V ↦ (𝑅𝑥))‘𝑦) = 𝑧 ↔ (𝑅𝑦) = 𝑧))
26 funmpt 5886 . . . . . . . . 9 Fun (𝑥 ∈ V ↦ (𝑅𝑥))
27 df-fn 5852 . . . . . . . . 9 ((𝑥 ∈ V ↦ (𝑅𝑥)) Fn {𝑥 ∣ (𝑅𝑥) ∈ V} ↔ (Fun (𝑥 ∈ V ↦ (𝑅𝑥)) ∧ dom (𝑥 ∈ V ↦ (𝑅𝑥)) = {𝑥 ∣ (𝑅𝑥) ∈ V}))
2826, 16, 27mpbir2an 954 . . . . . . . 8 (𝑥 ∈ V ↦ (𝑅𝑥)) Fn {𝑥 ∣ (𝑅𝑥) ∈ V}
2920biimpri 218 . . . . . . . 8 ((𝑅𝑦) ∈ V → 𝑦 ∈ {𝑥 ∣ (𝑅𝑥) ∈ V})
30 fnbrfvb 6195 . . . . . . . 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 270 . . . . . 6 ((𝑅𝑦) ∈ V → ((𝑅𝑦) = 𝑧𝑦(𝑥 ∈ V ↦ (𝑅𝑥))𝑧))
3322, 32syl5bb 272 . . . . 5 ((𝑅𝑦) ∈ V → (𝑧 = (𝑅𝑦) ↔ 𝑦(𝑥 ∈ V ↦ (𝑅𝑥))𝑧))
3421, 33syl5bb 272 . . . 4 ((𝑅𝑦) ∈ V → (𝑦Image𝑅𝑧𝑦(𝑥 ∈ V ↦ (𝑅𝑥))𝑧))
3520, 34sylbi 207 . . 3 (𝑦 ∈ {𝑥 ∣ (𝑅𝑥) ∈ V} → (𝑦Image𝑅𝑧𝑦(𝑥 ∈ V ↦ (𝑅𝑥))𝑧))
3611, 17, 35pm5.21nii 368 . 2 (𝑦Image𝑅𝑧𝑦(𝑥 ∈ V ↦ (𝑅𝑥))𝑧)
373, 4, 36eqbrriv 5178 1 Image𝑅 = (𝑥 ∈ V ↦ (𝑅𝑥))
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1480  wcel 1987  {cab 2607  {crab 2911  Vcvv 3186   class class class wbr 4615  cmpt 4675  dom cdm 5076  cima 5079  Rel wrel 5081  Fun wfun 5843   Fn wfn 5844  cfv 5849  Imagecimage 31609
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3419  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-symdif 3824  df-nul 3894  df-if 4061  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-br 4616  df-opab 4676  df-mpt 4677  df-eprel 4987  df-id 4991  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-fo 5855  df-fv 5857  df-1st 7116  df-2nd 7117  df-txp 31623  df-image 31633
This theorem is referenced by:  fvimage  31701
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