Mathbox for Thierry Arnoux < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  orvcval2 Structured version   Visualization version   GIF version

Theorem orvcval2 30854
 Description: Another way to express the value of the preimage mapping operator. (Contributed by Thierry Arnoux, 19-Jan-2017.)
Hypotheses
Ref Expression
orvcval.1 (𝜑 → Fun 𝑋)
orvcval.2 (𝜑𝑋𝑉)
orvcval.3 (𝜑𝐴𝑊)
Assertion
Ref Expression
orvcval2 (𝜑 → (𝑋RV/𝑐𝑅𝐴) = {𝑧 ∈ dom 𝑋 ∣ (𝑋𝑧)𝑅𝐴})
Distinct variable groups:   𝑧,𝐴   𝑧,𝑅   𝑧,𝑋
Allowed substitution hints:   𝜑(𝑧)   𝑉(𝑧)   𝑊(𝑧)

Proof of Theorem orvcval2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 orvcval.1 . . 3 (𝜑 → Fun 𝑋)
2 orvcval.2 . . 3 (𝜑𝑋𝑉)
3 orvcval.3 . . 3 (𝜑𝐴𝑊)
41, 2, 3orvcval 30853 . 2 (𝜑 → (𝑋RV/𝑐𝑅𝐴) = (𝑋 “ {𝑦𝑦𝑅𝐴}))
5 funfn 6061 . . . 4 (Fun 𝑋𝑋 Fn dom 𝑋)
61, 5sylib 208 . . 3 (𝜑𝑋 Fn dom 𝑋)
7 fncnvima2 6482 . . 3 (𝑋 Fn dom 𝑋 → (𝑋 “ {𝑦𝑦𝑅𝐴}) = {𝑧 ∈ dom 𝑋 ∣ (𝑋𝑧) ∈ {𝑦𝑦𝑅𝐴}})
86, 7syl 17 . 2 (𝜑 → (𝑋 “ {𝑦𝑦𝑅𝐴}) = {𝑧 ∈ dom 𝑋 ∣ (𝑋𝑧) ∈ {𝑦𝑦𝑅𝐴}})
9 fvex 6342 . . . . 5 (𝑋𝑧) ∈ V
10 breq1 4787 . . . . 5 (𝑦 = (𝑋𝑧) → (𝑦𝑅𝐴 ↔ (𝑋𝑧)𝑅𝐴))
119, 10elab 3499 . . . 4 ((𝑋𝑧) ∈ {𝑦𝑦𝑅𝐴} ↔ (𝑋𝑧)𝑅𝐴)
1211rabbii 3334 . . 3 {𝑧 ∈ dom 𝑋 ∣ (𝑋𝑧) ∈ {𝑦𝑦𝑅𝐴}} = {𝑧 ∈ dom 𝑋 ∣ (𝑋𝑧)𝑅𝐴}
1312a1i 11 . 2 (𝜑 → {𝑧 ∈ dom 𝑋 ∣ (𝑋𝑧) ∈ {𝑦𝑦𝑅𝐴}} = {𝑧 ∈ dom 𝑋 ∣ (𝑋𝑧)𝑅𝐴})
144, 8, 133eqtrd 2808 1 (𝜑 → (𝑋RV/𝑐𝑅𝐴) = {𝑧 ∈ dom 𝑋 ∣ (𝑋𝑧)𝑅𝐴})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1630   ∈ wcel 2144  {cab 2756  {crab 3064   class class class wbr 4784  ◡ccnv 5248  dom cdm 5249   “ cima 5252  Fun wfun 6025   Fn wfn 6026  ‘cfv 6031  (class class class)co 6792  ∘RV/𝑐corvc 30851 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-orvc 30852 This theorem is referenced by:  elorvc  30855
 Copyright terms: Public domain W3C validator