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Theorem oveqprc 17229
Description: Lemma for showing the equality of values for functions like slot extractors 𝐸 at a proper class. Extracted from several former proofs of lemmas like resvlem 33357. (Contributed by AV, 31-Oct-2024.)
Hypotheses
Ref Expression
oveqprc.e (𝐸‘∅) = ∅
oveqprc.z 𝑍 = (𝑋𝑂𝑌)
oveqprc.r Rel dom 𝑂
Assertion
Ref Expression
oveqprc 𝑋 ∈ V → (𝐸𝑋) = (𝐸𝑍))

Proof of Theorem oveqprc
StepHypRef Expression
1 oveqprc.e . . 3 (𝐸‘∅) = ∅
21eqcomi 2746 . 2 ∅ = (𝐸‘∅)
3 fvprc 6898 . 2 𝑋 ∈ V → (𝐸𝑋) = ∅)
4 oveqprc.z . . . 4 𝑍 = (𝑋𝑂𝑌)
5 oveqprc.r . . . . 5 Rel dom 𝑂
65ovprc1 7470 . . . 4 𝑋 ∈ V → (𝑋𝑂𝑌) = ∅)
74, 6eqtrid 2789 . . 3 𝑋 ∈ V → 𝑍 = ∅)
87fveq2d 6910 . 2 𝑋 ∈ V → (𝐸𝑍) = (𝐸‘∅))
92, 3, 83eqtr4a 2803 1 𝑋 ∈ V → (𝐸𝑋) = (𝐸𝑍))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1540  wcel 2108  Vcvv 3480  c0 4333  dom cdm 5685  Rel wrel 5690  cfv 6561  (class class class)co 7431
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-dm 5695  df-iota 6514  df-fv 6569  df-ov 7434
This theorem is referenced by:  setsnid  17245  ressbas  17280  resseqnbas  17287  tnglem  24653  resvlem  33357
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