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Theorem elorvc 30314
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 30313 . . . 4 (𝜑 → (𝑋RV/𝑐𝑅𝐴) = {𝑧 ∈ dom 𝑋 ∣ (𝑋𝑧)𝑅𝐴})
54eleq2d 2684 . . 3 (𝜑 → (𝑧 ∈ (𝑋RV/𝑐𝑅𝐴) ↔ 𝑧 ∈ {𝑧 ∈ dom 𝑋 ∣ (𝑋𝑧)𝑅𝐴}))
6 rabid 3106 . . 3 (𝑧 ∈ {𝑧 ∈ dom 𝑋 ∣ (𝑋𝑧)𝑅𝐴} ↔ (𝑧 ∈ dom 𝑋 ∧ (𝑋𝑧)𝑅𝐴))
75, 6syl6bb 276 . 2 (𝜑 → (𝑧 ∈ (𝑋RV/𝑐𝑅𝐴) ↔ (𝑧 ∈ dom 𝑋 ∧ (𝑋𝑧)𝑅𝐴)))
87baibd 947 1 ((𝜑𝑧 ∈ dom 𝑋) → (𝑧 ∈ (𝑋RV/𝑐𝑅𝐴) ↔ (𝑋𝑧)𝑅𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wcel 1987  {crab 2911   class class class wbr 4615  dom cdm 5076  Fun wfun 5843  cfv 5849  (class class class)co 6607  RV/𝑐corvc 30310
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 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905
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 3419  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-br 4616  df-opab 4676  df-id 4991  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-iota 5812  df-fun 5851  df-fn 5852  df-fv 5857  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-orvc 30311
This theorem is referenced by:  elorrvc  30318
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