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

Theorem orvcval2 29847
 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 29846 . 2 (𝜑 → (𝑋RV/𝑐𝑅𝐴) = (𝑋 “ {𝑦𝑦𝑅𝐴}))
5 funfn 5833 . . . 4 (Fun 𝑋𝑋 Fn dom 𝑋)
61, 5sylib 207 . . 3 (𝜑𝑋 Fn dom 𝑋)
7 fncnvima2 6247 . . 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 6113 . . . . . 6 (𝑋𝑧) ∈ V
10 breq1 4586 . . . . . 6 (𝑦 = (𝑋𝑧) → (𝑦𝑅𝐴 ↔ (𝑋𝑧)𝑅𝐴))
119, 10elab 3319 . . . . 5 ((𝑋𝑧) ∈ {𝑦𝑦𝑅𝐴} ↔ (𝑋𝑧)𝑅𝐴)
1211a1i 11 . . . 4 (𝑧 ∈ dom 𝑋 → ((𝑋𝑧) ∈ {𝑦𝑦𝑅𝐴} ↔ (𝑋𝑧)𝑅𝐴))
1312rabbiia 3161 . . 3 {𝑧 ∈ dom 𝑋 ∣ (𝑋𝑧) ∈ {𝑦𝑦𝑅𝐴}} = {𝑧 ∈ dom 𝑋 ∣ (𝑋𝑧)𝑅𝐴}
1413a1i 11 . 2 (𝜑 → {𝑧 ∈ dom 𝑋 ∣ (𝑋𝑧) ∈ {𝑦𝑦𝑅𝐴}} = {𝑧 ∈ dom 𝑋 ∣ (𝑋𝑧)𝑅𝐴})
154, 8, 143eqtrd 2648 1 (𝜑 → (𝑋RV/𝑐𝑅𝐴) = {𝑧 ∈ dom 𝑋 ∣ (𝑋𝑧)𝑅𝐴})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475   ∈ wcel 1977  {cab 2596  {crab 2900   class class class wbr 4583  ◡ccnv 5037  dom cdm 5038   “ cima 5041  Fun wfun 5798   Fn wfn 5799  ‘cfv 5804  (class class class)co 6549  ∘RV/𝑐corvc 29844 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-orvc 29845 This theorem is referenced by:  elorvc  29848
 Copyright terms: Public domain W3C validator