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Theorem fnimage 32625
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 32624 . 2 Fun Image𝑅
2 vex 3401 . . . . . . . 8 𝑦 ∈ V
3 vex 3401 . . . . . . . 8 𝑥 ∈ V
42, 3brimage 32622 . . . . . . 7 (𝑦Image𝑅𝑥𝑥 = (𝑅𝑦))
5 eqvisset 3413 . . . . . . 7 (𝑥 = (𝑅𝑦) → (𝑅𝑦) ∈ V)
64, 5sylbi 209 . . . . . 6 (𝑦Image𝑅𝑥 → (𝑅𝑦) ∈ V)
76exlimiv 1973 . . . . 5 (∃𝑥 𝑦Image𝑅𝑥 → (𝑅𝑦) ∈ V)
8 eqid 2778 . . . . . . 7 (𝑅𝑦) = (𝑅𝑦)
9 brimageg 32623 . . . . . . . 8 ((𝑦 ∈ V ∧ (𝑅𝑦) ∈ V) → (𝑦Image𝑅(𝑅𝑦) ↔ (𝑅𝑦) = (𝑅𝑦)))
102, 9mpan 680 . . . . . . 7 ((𝑅𝑦) ∈ V → (𝑦Image𝑅(𝑅𝑦) ↔ (𝑅𝑦) = (𝑅𝑦)))
118, 10mpbiri 250 . . . . . 6 ((𝑅𝑦) ∈ V → 𝑦Image𝑅(𝑅𝑦))
12 breq2 4890 . . . . . . 7 (𝑥 = (𝑅𝑦) → (𝑦Image𝑅𝑥𝑦Image𝑅(𝑅𝑦)))
1312spcegv 3496 . . . . . 6 ((𝑅𝑦) ∈ V → (𝑦Image𝑅(𝑅𝑦) → ∃𝑥 𝑦Image𝑅𝑥))
1411, 13mpd 15 . . . . 5 ((𝑅𝑦) ∈ V → ∃𝑥 𝑦Image𝑅𝑥)
157, 14impbii 201 . . . 4 (∃𝑥 𝑦Image𝑅𝑥 ↔ (𝑅𝑦) ∈ V)
162eldm 5566 . . . 4 (𝑦 ∈ dom Image𝑅 ↔ ∃𝑥 𝑦Image𝑅𝑥)
17 imaeq2 5716 . . . . . 6 (𝑥 = 𝑦 → (𝑅𝑥) = (𝑅𝑦))
1817eleq1d 2844 . . . . 5 (𝑥 = 𝑦 → ((𝑅𝑥) ∈ V ↔ (𝑅𝑦) ∈ V))
192, 18elab 3558 . . . 4 (𝑦 ∈ {𝑥 ∣ (𝑅𝑥) ∈ V} ↔ (𝑅𝑦) ∈ V)
2015, 16, 193bitr4i 295 . . 3 (𝑦 ∈ dom Image𝑅𝑦 ∈ {𝑥 ∣ (𝑅𝑥) ∈ V})
2120eqriv 2775 . 2 dom Image𝑅 = {𝑥 ∣ (𝑅𝑥) ∈ V}
22 df-fn 6138 . 2 (Image𝑅 Fn {𝑥 ∣ (𝑅𝑥) ∈ V} ↔ (Fun Image𝑅 ∧ dom Image𝑅 = {𝑥 ∣ (𝑅𝑥) ∈ V}))
231, 21, 22mpbir2an 701 1 Image𝑅 Fn {𝑥 ∣ (𝑅𝑥) ∈ V}
Colors of variables: wff setvar class
Syntax hints:  wb 198   = wceq 1601  wex 1823  wcel 2107  {cab 2763  Vcvv 3398   class class class wbr 4886  dom cdm 5355  cima 5358  Fun wfun 6129   Fn wfn 6130  Imagecimage 32536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-symdif 4067  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-eprel 5266  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-fo 6141  df-fv 6143  df-1st 7445  df-2nd 7446  df-txp 32550  df-image 32560
This theorem is referenced by:  imageval  32626
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