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Theorem fnimage 32161
Description: Image𝑅 is a function over the set-like portion of 𝑅. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
fnimage Image𝑅 Fn {𝑥 ∣ (𝑅𝑥) ∈ V}
Distinct variable group:   𝑥,𝑅

Proof of Theorem fnimage
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 funimage 32160 . 2 Fun Image𝑅
2 vex 3234 . . . . . . . 8 𝑦 ∈ V
3 vex 3234 . . . . . . . 8 𝑥 ∈ V
42, 3brimage 32158 . . . . . . 7 (𝑦Image𝑅𝑥𝑥 = (𝑅𝑦))
5 eqvisset 3242 . . . . . . 7 (𝑥 = (𝑅𝑦) → (𝑅𝑦) ∈ V)
64, 5sylbi 207 . . . . . 6 (𝑦Image𝑅𝑥 → (𝑅𝑦) ∈ V)
76exlimiv 1898 . . . . 5 (∃𝑥 𝑦Image𝑅𝑥 → (𝑅𝑦) ∈ V)
8 eqid 2651 . . . . . . 7 (𝑅𝑦) = (𝑅𝑦)
9 brimageg 32159 . . . . . . . 8 ((𝑦 ∈ V ∧ (𝑅𝑦) ∈ V) → (𝑦Image𝑅(𝑅𝑦) ↔ (𝑅𝑦) = (𝑅𝑦)))
102, 9mpan 706 . . . . . . 7 ((𝑅𝑦) ∈ V → (𝑦Image𝑅(𝑅𝑦) ↔ (𝑅𝑦) = (𝑅𝑦)))
118, 10mpbiri 248 . . . . . 6 ((𝑅𝑦) ∈ V → 𝑦Image𝑅(𝑅𝑦))
12 breq2 4689 . . . . . . 7 (𝑥 = (𝑅𝑦) → (𝑦Image𝑅𝑥𝑦Image𝑅(𝑅𝑦)))
1312spcegv 3325 . . . . . 6 ((𝑅𝑦) ∈ V → (𝑦Image𝑅(𝑅𝑦) → ∃𝑥 𝑦Image𝑅𝑥))
1411, 13mpd 15 . . . . 5 ((𝑅𝑦) ∈ V → ∃𝑥 𝑦Image𝑅𝑥)
157, 14impbii 199 . . . 4 (∃𝑥 𝑦Image𝑅𝑥 ↔ (𝑅𝑦) ∈ V)
162eldm 5353 . . . 4 (𝑦 ∈ dom Image𝑅 ↔ ∃𝑥 𝑦Image𝑅𝑥)
17 imaeq2 5497 . . . . . 6 (𝑥 = 𝑦 → (𝑅𝑥) = (𝑅𝑦))
1817eleq1d 2715 . . . . 5 (𝑥 = 𝑦 → ((𝑅𝑥) ∈ V ↔ (𝑅𝑦) ∈ V))
192, 18elab 3382 . . . 4 (𝑦 ∈ {𝑥 ∣ (𝑅𝑥) ∈ V} ↔ (𝑅𝑦) ∈ V)
2015, 16, 193bitr4i 292 . . 3 (𝑦 ∈ dom Image𝑅𝑦 ∈ {𝑥 ∣ (𝑅𝑥) ∈ V})
2120eqriv 2648 . 2 dom Image𝑅 = {𝑥 ∣ (𝑅𝑥) ∈ V}
22 df-fn 5929 . 2 (Image𝑅 Fn {𝑥 ∣ (𝑅𝑥) ∈ V} ↔ (Fun Image𝑅 ∧ dom Image𝑅 = {𝑥 ∣ (𝑅𝑥) ∈ V}))
231, 21, 22mpbir2an 975 1 Image𝑅 Fn {𝑥 ∣ (𝑅𝑥) ∈ V}
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1523  wex 1744  wcel 2030  {cab 2637  Vcvv 3231   class class class wbr 4685  dom cdm 5143  cima 5146  Fun wfun 5920   Fn wfn 5921  Imagecimage 32072
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-symdif 3877  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-eprel 5058  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fo 5932  df-fv 5934  df-1st 7210  df-2nd 7211  df-txp 32086  df-image 32096
This theorem is referenced by:  imageval  32162
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