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Theorem orvcval2 30301
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 30300 . 2 (𝜑 → (𝑋RV/𝑐𝑅𝐴) = (𝑋 “ {𝑦𝑦𝑅𝐴}))
5 funfn 5877 . . . 4 (Fun 𝑋𝑋 Fn dom 𝑋)
61, 5sylib 208 . . 3 (𝜑𝑋 Fn dom 𝑋)
7 fncnvima2 6295 . . 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 6158 . . . . . 6 (𝑋𝑧) ∈ V
10 breq1 4616 . . . . . 6 (𝑦 = (𝑋𝑧) → (𝑦𝑅𝐴 ↔ (𝑋𝑧)𝑅𝐴))
119, 10elab 3333 . . . . 5 ((𝑋𝑧) ∈ {𝑦𝑦𝑅𝐴} ↔ (𝑋𝑧)𝑅𝐴)
1211a1i 11 . . . 4 (𝑧 ∈ dom 𝑋 → ((𝑋𝑧) ∈ {𝑦𝑦𝑅𝐴} ↔ (𝑋𝑧)𝑅𝐴))
1312rabbiia 3173 . . 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 2659 1 (𝜑 → (𝑋RV/𝑐𝑅𝐴) = {𝑧 ∈ dom 𝑋 ∣ (𝑋𝑧)𝑅𝐴})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1480  wcel 1987  {cab 2607  {crab 2911   class class class wbr 4613  ccnv 5073  dom cdm 5074  cima 5077  Fun wfun 5841   Fn wfn 5842  cfv 5847  (class class class)co 6604  RV/𝑐corvc 30298
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-orvc 30299
This theorem is referenced by:  elorvc  30302
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