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Theorem orvcval2 34305
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 34304 . 2 (𝜑 → (𝑋RV/𝑐𝑅𝐴) = (𝑋 “ {𝑦𝑦𝑅𝐴}))
5 funfn 6581 . . . 4 (Fun 𝑋𝑋 Fn dom 𝑋)
61, 5sylib 217 . . 3 (𝜑𝑋 Fn dom 𝑋)
7 fncnvima2 7066 . . 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 6906 . . . . 5 (𝑋𝑧) ∈ V
10 breq1 5148 . . . . 5 (𝑦 = (𝑋𝑧) → (𝑦𝑅𝐴 ↔ (𝑋𝑧)𝑅𝐴))
119, 10elab 3665 . . . 4 ((𝑋𝑧) ∈ {𝑦𝑦𝑅𝐴} ↔ (𝑋𝑧)𝑅𝐴)
1211rabbii 3425 . . 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 2770 1 (𝜑 → (𝑋RV/𝑐𝑅𝐴) = {𝑧 ∈ dom 𝑋 ∣ (𝑋𝑧)𝑅𝐴})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wcel 2099  {cab 2703  {crab 3419   class class class wbr 5145  ccnv 5673  dom cdm 5674  cima 5677  Fun wfun 6540   Fn wfn 6541  cfv 6546  (class class class)co 7416  RV/𝑐corvc 34302
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3776  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-br 5146  df-opab 5208  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-iota 6498  df-fun 6548  df-fn 6549  df-fv 6554  df-ov 7419  df-oprab 7420  df-mpo 7421  df-orvc 34303
This theorem is referenced by:  elorvc  34306
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