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Theorem elorvc 31037
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 31036 . . . 4 (𝜑 → (𝑋RV/𝑐𝑅𝐴) = {𝑧 ∈ dom 𝑋 ∣ (𝑋𝑧)𝑅𝐴})
54eleq2d 2865 . . 3 (𝜑 → (𝑧 ∈ (𝑋RV/𝑐𝑅𝐴) ↔ 𝑧 ∈ {𝑧 ∈ dom 𝑋 ∣ (𝑋𝑧)𝑅𝐴}))
6 rabid 3298 . . 3 (𝑧 ∈ {𝑧 ∈ dom 𝑋 ∣ (𝑋𝑧)𝑅𝐴} ↔ (𝑧 ∈ dom 𝑋 ∧ (𝑋𝑧)𝑅𝐴))
75, 6syl6bb 279 . 2 (𝜑 → (𝑧 ∈ (𝑋RV/𝑐𝑅𝐴) ↔ (𝑧 ∈ dom 𝑋 ∧ (𝑋𝑧)𝑅𝐴)))
87baibd 536 1 ((𝜑𝑧 ∈ dom 𝑋) → (𝑧 ∈ (𝑋RV/𝑐𝑅𝐴) ↔ (𝑋𝑧)𝑅𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  wcel 2157  {crab 3094   class class class wbr 4844  dom cdm 5313  Fun wfun 6096  cfv 6102  (class class class)co 6879  RV/𝑐corvc 31033
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-sep 4976  ax-nul 4984  ax-pow 5036  ax-pr 5098  ax-un 7184
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ne 2973  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3388  df-sbc 3635  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-nul 4117  df-if 4279  df-pw 4352  df-sn 4370  df-pr 4372  df-op 4376  df-uni 4630  df-br 4845  df-opab 4907  df-id 5221  df-xp 5319  df-rel 5320  df-cnv 5321  df-co 5322  df-dm 5323  df-rn 5324  df-res 5325  df-ima 5326  df-iota 6065  df-fun 6104  df-fn 6105  df-fv 6110  df-ov 6882  df-oprab 6883  df-mpt2 6884  df-orvc 31034
This theorem is referenced by:  elorrvc  31041
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