Theorem oveqprc 17258

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 33662. (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 6873	. 2 ⊢ (¬ 𝑋 ∈ V → (𝐸‘𝑋) = ∅)
4		oveqprc.z	. . . 4 ⊢ 𝑍 = (𝑋𝑂𝑌)
5		oveqprc.r	. . . . 5 ⊢ Rel dom 𝑂
6	5	ovprc1 7451	. . . 4 ⊢ (¬ 𝑋 ∈ V → (𝑋𝑂𝑌) = ∅)
7	4, 6	eqtrid 2809	. . 3 ⊢ (¬ 𝑋 ∈ V → 𝑍 = ∅)
8	7	fveq2d 6885	. 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 1569   ∈ wcel 2142  Vcvv 3454  ∅c0 4285  dom cdm 5660  Rel wrel 5665  ‘cfv 6536  (class class class)co 7412

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  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 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-xp 5666  df-rel 5667  df-dm 5670  df-iota 6492  df-fv 6544  df-ov 7415

This theorem is used by:  setsnid  17274  ressbas  17302  resseqnbas  17308  tnglem  24808  resvlem  33662