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Theorem fnimage 34503
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 34502 . 2 Fun Image𝑅
2 vex 3448 . . . . . . . 8 𝑦 ∈ V
3 vex 3448 . . . . . . . 8 𝑥 ∈ V
42, 3brimage 34500 . . . . . . 7 (𝑦Image𝑅𝑥𝑥 = (𝑅𝑦))
5 eqvisset 3461 . . . . . . 7 (𝑥 = (𝑅𝑦) → (𝑅𝑦) ∈ V)
64, 5sylbi 216 . . . . . 6 (𝑦Image𝑅𝑥 → (𝑅𝑦) ∈ V)
76exlimiv 1933 . . . . 5 (∃𝑥 𝑦Image𝑅𝑥 → (𝑅𝑦) ∈ V)
8 eqid 2736 . . . . . . 7 (𝑅𝑦) = (𝑅𝑦)
9 brimageg 34501 . . . . . . . 8 ((𝑦 ∈ V ∧ (𝑅𝑦) ∈ V) → (𝑦Image𝑅(𝑅𝑦) ↔ (𝑅𝑦) = (𝑅𝑦)))
102, 9mpan 688 . . . . . . 7 ((𝑅𝑦) ∈ V → (𝑦Image𝑅(𝑅𝑦) ↔ (𝑅𝑦) = (𝑅𝑦)))
118, 10mpbiri 257 . . . . . 6 ((𝑅𝑦) ∈ V → 𝑦Image𝑅(𝑅𝑦))
12 breq2 5108 . . . . . . 7 (𝑥 = (𝑅𝑦) → (𝑦Image𝑅𝑥𝑦Image𝑅(𝑅𝑦)))
1312spcegv 3555 . . . . . 6 ((𝑅𝑦) ∈ V → (𝑦Image𝑅(𝑅𝑦) → ∃𝑥 𝑦Image𝑅𝑥))
1411, 13mpd 15 . . . . 5 ((𝑅𝑦) ∈ V → ∃𝑥 𝑦Image𝑅𝑥)
157, 14impbii 208 . . . 4 (∃𝑥 𝑦Image𝑅𝑥 ↔ (𝑅𝑦) ∈ V)
162eldm 5855 . . . 4 (𝑦 ∈ dom Image𝑅 ↔ ∃𝑥 𝑦Image𝑅𝑥)
17 imaeq2 6008 . . . . . 6 (𝑥 = 𝑦 → (𝑅𝑥) = (𝑅𝑦))
1817eleq1d 2822 . . . . 5 (𝑥 = 𝑦 → ((𝑅𝑥) ∈ V ↔ (𝑅𝑦) ∈ V))
192, 18elab 3629 . . . 4 (𝑦 ∈ {𝑥 ∣ (𝑅𝑥) ∈ V} ↔ (𝑅𝑦) ∈ V)
2015, 16, 193bitr4i 302 . . 3 (𝑦 ∈ dom Image𝑅𝑦 ∈ {𝑥 ∣ (𝑅𝑥) ∈ V})
2120eqriv 2733 . 2 dom Image𝑅 = {𝑥 ∣ (𝑅𝑥) ∈ V}
22 df-fn 6497 . 2 (Image𝑅 Fn {𝑥 ∣ (𝑅𝑥) ∈ V} ↔ (Fun Image𝑅 ∧ dom Image𝑅 = {𝑥 ∣ (𝑅𝑥) ∈ V}))
231, 21, 22mpbir2an 709 1 Image𝑅 Fn {𝑥 ∣ (𝑅𝑥) ∈ V}
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1541  wex 1781  wcel 2106  {cab 2713  Vcvv 3444   class class class wbr 5104  dom cdm 5632  cima 5635  Fun wfun 6488   Fn wfn 6489  Imagecimage 34414
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5255  ax-nul 5262  ax-pr 5383  ax-un 7669
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2888  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-symdif 4201  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-br 5105  df-opab 5167  df-mpt 5188  df-id 5530  df-eprel 5536  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6446  df-fun 6496  df-fn 6497  df-f 6498  df-fo 6500  df-fv 6502  df-1st 7918  df-2nd 7919  df-txp 34428  df-image 34438
This theorem is referenced by:  imageval  34504
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