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Theorem orvcval2 30845
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 30844 . 2 (𝜑 → (𝑋RV/𝑐𝑅𝐴) = (𝑋 “ {𝑦𝑦𝑅𝐴}))
5 funfn 6131 . . . 4 (Fun 𝑋𝑋 Fn dom 𝑋)
61, 5sylib 209 . . 3 (𝜑𝑋 Fn dom 𝑋)
7 fncnvima2 6561 . . 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 6421 . . . . 5 (𝑋𝑧) ∈ V
10 breq1 4847 . . . . 5 (𝑦 = (𝑋𝑧) → (𝑦𝑅𝐴 ↔ (𝑋𝑧)𝑅𝐴))
119, 10elab 3545 . . . 4 ((𝑋𝑧) ∈ {𝑦𝑦𝑅𝐴} ↔ (𝑋𝑧)𝑅𝐴)
1211rabbii 3375 . . 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 2844 1 (𝜑 → (𝑋RV/𝑐𝑅𝐴) = {𝑧 ∈ dom 𝑋 ∣ (𝑋𝑧)𝑅𝐴})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1637  wcel 2156  {cab 2792  {crab 3100   class class class wbr 4844  ccnv 5310  dom cdm 5311  cima 5314  Fun wfun 6095   Fn wfn 6096  cfv 6101  (class class class)co 6874  RV/𝑐corvc 30842
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-br 4845  df-opab 4907  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-iota 6064  df-fun 6103  df-fn 6104  df-fv 6109  df-ov 6877  df-oprab 6878  df-mpt2 6879  df-orvc 30843
This theorem is referenced by:  elorvc  30846
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