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Theorem oveqprc 17072
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 32176. (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 2742 . 2 ∅ = (𝐸‘∅)
3 fvprc 6838 . 2 𝑋 ∈ V → (𝐸𝑋) = ∅)
4 oveqprc.z . . . 4 𝑍 = (𝑋𝑂𝑌)
5 oveqprc.r . . . . 5 Rel dom 𝑂
65ovprc1 7400 . . . 4 𝑋 ∈ V → (𝑋𝑂𝑌) = ∅)
74, 6eqtrid 2785 . . 3 𝑋 ∈ V → 𝑍 = ∅)
87fveq2d 6850 . 2 𝑋 ∈ V → (𝐸𝑍) = (𝐸‘∅))
92, 3, 83eqtr4a 2799 1 𝑋 ∈ V → (𝐸𝑋) = (𝐸𝑍))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1542  wcel 2107  Vcvv 3447  c0 4286  dom cdm 5637  Rel wrel 5642  cfv 6500  (class class class)co 7361
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-xp 5643  df-rel 5644  df-dm 5647  df-iota 6452  df-fv 6508  df-ov 7364
This theorem is referenced by:  setsnid  17089  ressbas  17126  resseqnbas  17130  tnglem  24019  resvlem  32176
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