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Theorem imageval 32358
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 32356 . . 3 Fun Image𝑅
2 funrel 6118 . . 3 (Fun Image𝑅 → Rel Image𝑅)
31, 2ax-mp 5 . 2 Rel Image𝑅
4 mptrel 5450 . 2 Rel (𝑥 ∈ V ↦ (𝑅𝑥))
5 vex 3394 . . . . 5 𝑦 ∈ V
6 vex 3394 . . . . 5 𝑧 ∈ V
75, 6breldm 5530 . . . 4 (𝑦Image𝑅𝑧𝑦 ∈ dom Image𝑅)
8 fnimage 32357 . . . . 5 Image𝑅 Fn {𝑥 ∣ (𝑅𝑥) ∈ V}
9 fndm 6201 . . . . 5 (Image𝑅 Fn {𝑥 ∣ (𝑅𝑥) ∈ V} → dom Image𝑅 = {𝑥 ∣ (𝑅𝑥) ∈ V})
108, 9ax-mp 5 . . . 4 dom Image𝑅 = {𝑥 ∣ (𝑅𝑥) ∈ V}
117, 10syl6eleq 2895 . . 3 (𝑦Image𝑅𝑧𝑦 ∈ {𝑥 ∣ (𝑅𝑥) ∈ V})
125, 6breldm 5530 . . . 4 (𝑦(𝑥 ∈ V ↦ (𝑅𝑥))𝑧𝑦 ∈ dom (𝑥 ∈ V ↦ (𝑅𝑥)))
13 eqid 2806 . . . . . 6 (𝑥 ∈ V ↦ (𝑅𝑥)) = (𝑥 ∈ V ↦ (𝑅𝑥))
1413dmmpt 5844 . . . . 5 dom (𝑥 ∈ V ↦ (𝑅𝑥)) = {𝑥 ∈ V ∣ (𝑅𝑥) ∈ V}
15 rabab 3417 . . . . 5 {𝑥 ∈ V ∣ (𝑅𝑥) ∈ V} = {𝑥 ∣ (𝑅𝑥) ∈ V}
1614, 15eqtri 2828 . . . 4 dom (𝑥 ∈ V ↦ (𝑅𝑥)) = {𝑥 ∣ (𝑅𝑥) ∈ V}
1712, 16syl6eleq 2895 . . 3 (𝑦(𝑥 ∈ V ↦ (𝑅𝑥))𝑧𝑦 ∈ {𝑥 ∣ (𝑅𝑥) ∈ V})
18 imaeq2 5672 . . . . . 6 (𝑥 = 𝑦 → (𝑅𝑥) = (𝑅𝑦))
1918eleq1d 2870 . . . . 5 (𝑥 = 𝑦 → ((𝑅𝑥) ∈ V ↔ (𝑅𝑦) ∈ V))
205, 19elab 3545 . . . 4 (𝑦 ∈ {𝑥 ∣ (𝑅𝑥) ∈ V} ↔ (𝑅𝑦) ∈ V)
215, 6brimage 32354 . . . . 5 (𝑦Image𝑅𝑧𝑧 = (𝑅𝑦))
22 eqcom 2813 . . . . . 6 (𝑧 = (𝑅𝑦) ↔ (𝑅𝑦) = 𝑧)
2318, 13fvmptg 6501 . . . . . . . . 9 ((𝑦 ∈ V ∧ (𝑅𝑦) ∈ V) → ((𝑥 ∈ V ↦ (𝑅𝑥))‘𝑦) = (𝑅𝑦))
245, 23mpan 673 . . . . . . . 8 ((𝑅𝑦) ∈ V → ((𝑥 ∈ V ↦ (𝑅𝑥))‘𝑦) = (𝑅𝑦))
2524eqeq1d 2808 . . . . . . 7 ((𝑅𝑦) ∈ V → (((𝑥 ∈ V ↦ (𝑅𝑥))‘𝑦) = 𝑧 ↔ (𝑅𝑦) = 𝑧))
26 funmpt 6139 . . . . . . . . 9 Fun (𝑥 ∈ V ↦ (𝑅𝑥))
27 df-fn 6104 . . . . . . . . 9 ((𝑥 ∈ V ↦ (𝑅𝑥)) Fn {𝑥 ∣ (𝑅𝑥) ∈ V} ↔ (Fun (𝑥 ∈ V ↦ (𝑅𝑥)) ∧ dom (𝑥 ∈ V ↦ (𝑅𝑥)) = {𝑥 ∣ (𝑅𝑥) ∈ V}))
2826, 16, 27mpbir2an 693 . . . . . . . 8 (𝑥 ∈ V ↦ (𝑅𝑥)) Fn {𝑥 ∣ (𝑅𝑥) ∈ V}
2920biimpri 219 . . . . . . . 8 ((𝑅𝑦) ∈ V → 𝑦 ∈ {𝑥 ∣ (𝑅𝑥) ∈ V})
30 fnbrfvb 6456 . . . . . . . 8 (((𝑥 ∈ V ↦ (𝑅𝑥)) Fn {𝑥 ∣ (𝑅𝑥) ∈ V} ∧ 𝑦 ∈ {𝑥 ∣ (𝑅𝑥) ∈ V}) → (((𝑥 ∈ V ↦ (𝑅𝑥))‘𝑦) = 𝑧𝑦(𝑥 ∈ V ↦ (𝑅𝑥))𝑧))
3128, 29, 30sylancr 577 . . . . . . 7 ((𝑅𝑦) ∈ V → (((𝑥 ∈ V ↦ (𝑅𝑥))‘𝑦) = 𝑧𝑦(𝑥 ∈ V ↦ (𝑅𝑥))𝑧))
3225, 31bitr3d 272 . . . . . 6 ((𝑅𝑦) ∈ V → ((𝑅𝑦) = 𝑧𝑦(𝑥 ∈ V ↦ (𝑅𝑥))𝑧))
3322, 32syl5bb 274 . . . . 5 ((𝑅𝑦) ∈ V → (𝑧 = (𝑅𝑦) ↔ 𝑦(𝑥 ∈ V ↦ (𝑅𝑥))𝑧))
3421, 33syl5bb 274 . . . 4 ((𝑅𝑦) ∈ V → (𝑦Image𝑅𝑧𝑦(𝑥 ∈ V ↦ (𝑅𝑥))𝑧))
3520, 34sylbi 208 . . 3 (𝑦 ∈ {𝑥 ∣ (𝑅𝑥) ∈ V} → (𝑦Image𝑅𝑧𝑦(𝑥 ∈ V ↦ (𝑅𝑥))𝑧))
3611, 17, 35pm5.21nii 369 . 2 (𝑦Image𝑅𝑧𝑦(𝑥 ∈ V ↦ (𝑅𝑥))𝑧)
373, 4, 36eqbrriv 5417 1 Image𝑅 = (𝑥 ∈ V ↦ (𝑅𝑥))
Colors of variables: wff setvar class
Syntax hints:  wb 197   = wceq 1637  wcel 2156  {cab 2792  {crab 3100  Vcvv 3391   class class class wbr 4844  cmpt 4923  dom cdm 5311  cima 5314  Rel wrel 5316  Fun wfun 6095   Fn wfn 6096  cfv 6101  Imagecimage 32268
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-symdif 4042  df-nul 4117  df-if 4280  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5219  df-eprel 5224  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-fo 6107  df-fv 6109  df-1st 7398  df-2nd 7399  df-txp 32282  df-image 32292
This theorem is referenced by:  fvimage  32359
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