Theorem oveqprc 17288

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 33760. (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 → (𝐸‘𝑋) = (𝐸‘𝑍))

Proof of Theorem oveqprc

Step	Hyp	Ref	Expression
1		oveqprc.e	. . 3 ⊢ (𝐸‘∅) = ∅
2	1	eqcomi 2771	. 2 ⊢ ∅ = (𝐸‘∅)
3		fvprc 6874	. 2 ⊢ (¬ 𝑋 ∈ V → (𝐸‘𝑋) = ∅)
4		oveqprc.z	. . . 4 ⊢ 𝑍 = (𝑋𝑂𝑌)
5		oveqprc.r	. . . . 5 ⊢ Rel dom 𝑂
6	5	ovprc1 7455	. . . 4 ⊢ (¬ 𝑋 ∈ V → (𝑋𝑂𝑌) = ∅)
7	4, 6	eqtrid 2809	. . 3 ⊢ (¬ 𝑋 ∈ V → 𝑍 = ∅)
8	7	fveq2d 6886	. 2 ⊢ (¬ 𝑋 ∈ V → (𝐸‘𝑍) = (𝐸‘∅))
9	2, 3, 8	3eqtr4a 2823	1 ⊢ (¬ 𝑋 ∈ V → (𝐸‘𝑋) = (𝐸‘𝑍))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1570   ∈ wcel 2145  Vcvv 3453  ∅c0 4282  dom cdm 5659  Rel wrel 5664  ‘cfv 6537  (class class class)co 7416

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-xp 5665  df-rel 5666  df-dm 5669  df-iota 6493  df-fv 6545  df-ov 7419

This theorem is used by:  setsnid  17304  ressbas  17332  resseqnbas  17338  tnglem  24867  resvlem  33760