Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fnimage Structured version   Visualization version   GIF version

Theorem fnimage 32036
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 32035 . 2 Fun Image𝑅
2 vex 3203 . . . . . . . 8 𝑦 ∈ V
3 vex 3203 . . . . . . . 8 𝑥 ∈ V
42, 3brimage 32033 . . . . . . 7 (𝑦Image𝑅𝑥𝑥 = (𝑅𝑦))
5 eqvisset 3211 . . . . . . 7 (𝑥 = (𝑅𝑦) → (𝑅𝑦) ∈ V)
64, 5sylbi 207 . . . . . 6 (𝑦Image𝑅𝑥 → (𝑅𝑦) ∈ V)
76exlimiv 1858 . . . . 5 (∃𝑥 𝑦Image𝑅𝑥 → (𝑅𝑦) ∈ V)
8 eqid 2622 . . . . . . 7 (𝑅𝑦) = (𝑅𝑦)
9 brimageg 32034 . . . . . . . 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 4657 . . . . . . 7 (𝑥 = (𝑅𝑦) → (𝑦Image𝑅𝑥𝑦Image𝑅(𝑅𝑦)))
1312spcegv 3294 . . . . . 6 ((𝑅𝑦) ∈ V → (𝑦Image𝑅(𝑅𝑦) → ∃𝑥 𝑦Image𝑅𝑥))
1411, 13mpd 15 . . . . 5 ((𝑅𝑦) ∈ V → ∃𝑥 𝑦Image𝑅𝑥)
157, 14impbii 199 . . . 4 (∃𝑥 𝑦Image𝑅𝑥 ↔ (𝑅𝑦) ∈ V)
162eldm 5321 . . . 4 (𝑦 ∈ dom Image𝑅 ↔ ∃𝑥 𝑦Image𝑅𝑥)
17 imaeq2 5462 . . . . . 6 (𝑥 = 𝑦 → (𝑅𝑥) = (𝑅𝑦))
1817eleq1d 2686 . . . . 5 (𝑥 = 𝑦 → ((𝑅𝑥) ∈ V ↔ (𝑅𝑦) ∈ V))
192, 18elab 3350 . . . 4 (𝑦 ∈ {𝑥 ∣ (𝑅𝑥) ∈ V} ↔ (𝑅𝑦) ∈ V)
2015, 16, 193bitr4i 292 . . 3 (𝑦 ∈ dom Image𝑅𝑦 ∈ {𝑥 ∣ (𝑅𝑥) ∈ V})
2120eqriv 2619 . 2 dom Image𝑅 = {𝑥 ∣ (𝑅𝑥) ∈ V}
22 df-fn 5891 . 2 (Image𝑅 Fn {𝑥 ∣ (𝑅𝑥) ∈ V} ↔ (Fun Image𝑅 ∧ dom Image𝑅 = {𝑥 ∣ (𝑅𝑥) ∈ V}))
231, 21, 22mpbir2an 955 1 Image𝑅 Fn {𝑥 ∣ (𝑅𝑥) ∈ V}
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1483  wex 1704  wcel 1990  {cab 2608  Vcvv 3200   class class class wbr 4653  dom cdm 5114  cima 5117  Fun wfun 5882   Fn wfn 5883  Imagecimage 31947
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-symdif 3844  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-eprel 5029  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fo 5894  df-fv 5896  df-1st 7168  df-2nd 7169  df-txp 31961  df-image 31971
This theorem is referenced by:  imageval  32037
  Copyright terms: Public domain W3C validator