Theorem oveqprc 17239

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 33322. (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 2749	. 2 ⊢ ∅ = (𝐸‘∅)
3		fvprc 6912	. 2 ⊢ (¬ 𝑋 ∈ V → (𝐸‘𝑋) = ∅)
4		oveqprc.z	. . . 4 ⊢ 𝑍 = (𝑋𝑂𝑌)
5		oveqprc.r	. . . . 5 ⊢ Rel dom 𝑂
6	5	ovprc1 7487	. . . 4 ⊢ (¬ 𝑋 ∈ V → (𝑋𝑂𝑌) = ∅)
7	4, 6	eqtrid 2792	. . 3 ⊢ (¬ 𝑋 ∈ V → 𝑍 = ∅)
8	7	fveq2d 6924	. 2 ⊢ (¬ 𝑋 ∈ V → (𝐸‘𝑍) = (𝐸‘∅))
9	2, 3, 8	3eqtr4a 2806	1 ⊢ (¬ 𝑋 ∈ V → (𝐸‘𝑋) = (𝐸‘𝑍))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1537   ∈ wcel 2108  Vcvv 3488  ∅c0 4352  dom cdm 5700  Rel wrel 5705  ‘cfv 6573  (class class class)co 7448

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-xp 5706  df-rel 5707  df-dm 5710  df-iota 6525  df-fv 6581  df-ov 7451

This theorem is referenced by:  setsnid  17256  ressbas  17293  resseqnbas  17300  tnglem  24674  resvlem  33322