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Theorem elorvc 34079
Description: Elementhood of a preimage. (Contributed by Thierry Arnoux, 21-Jan-2017.)
Hypotheses
Ref Expression
orvcval.1 (𝜑 → Fun 𝑋)
orvcval.2 (𝜑𝑋𝑉)
orvcval.3 (𝜑𝐴𝑊)
Assertion
Ref Expression
elorvc ((𝜑𝑧 ∈ dom 𝑋) → (𝑧 ∈ (𝑋RV/𝑐𝑅𝐴) ↔ (𝑋𝑧)𝑅𝐴))
Distinct variable groups:   𝑧,𝐴   𝑧,𝑅   𝑧,𝑋
Allowed substitution hints:   𝜑(𝑧)   𝑉(𝑧)   𝑊(𝑧)

Proof of Theorem elorvc
StepHypRef Expression
1 orvcval.1 . . . . 5 (𝜑 → Fun 𝑋)
2 orvcval.2 . . . . 5 (𝜑𝑋𝑉)
3 orvcval.3 . . . . 5 (𝜑𝐴𝑊)
41, 2, 3orvcval2 34078 . . . 4 (𝜑 → (𝑋RV/𝑐𝑅𝐴) = {𝑧 ∈ dom 𝑋 ∣ (𝑋𝑧)𝑅𝐴})
54eleq2d 2815 . . 3 (𝜑 → (𝑧 ∈ (𝑋RV/𝑐𝑅𝐴) ↔ 𝑧 ∈ {𝑧 ∈ dom 𝑋 ∣ (𝑋𝑧)𝑅𝐴}))
6 rabid 3449 . . 3 (𝑧 ∈ {𝑧 ∈ dom 𝑋 ∣ (𝑋𝑧)𝑅𝐴} ↔ (𝑧 ∈ dom 𝑋 ∧ (𝑋𝑧)𝑅𝐴))
75, 6bitrdi 287 . 2 (𝜑 → (𝑧 ∈ (𝑋RV/𝑐𝑅𝐴) ↔ (𝑧 ∈ dom 𝑋 ∧ (𝑋𝑧)𝑅𝐴)))
87baibd 539 1 ((𝜑𝑧 ∈ dom 𝑋) → (𝑧 ∈ (𝑋RV/𝑐𝑅𝐴) ↔ (𝑋𝑧)𝑅𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  wcel 2099  {crab 3429   class class class wbr 5148  dom cdm 5678  Fun wfun 6542  cfv 6548  (class class class)co 7420  RV/𝑐corvc 34075
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 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-orvc 34076
This theorem is referenced by:  elorrvc  34083
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