Theorem oveqprc 17133

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 33432. (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 2746	. 2 ⊢ ∅ = (𝐸‘∅)
3		fvprc 6836	. 2 ⊢ (¬ 𝑋 ∈ V → (𝐸‘𝑋) = ∅)
4		oveqprc.z	. . . 4 ⊢ 𝑍 = (𝑋𝑂𝑌)
5		oveqprc.r	. . . . 5 ⊢ Rel dom 𝑂
6	5	ovprc1 7409	. . . 4 ⊢ (¬ 𝑋 ∈ V → (𝑋𝑂𝑌) = ∅)
7	4, 6	eqtrid 2784	. . 3 ⊢ (¬ 𝑋 ∈ V → 𝑍 = ∅)
8	7	fveq2d 6848	. 2 ⊢ (¬ 𝑋 ∈ V → (𝐸‘𝑍) = (𝐸‘∅))
9	2, 3, 8	3eqtr4a 2798	1 ⊢ (¬ 𝑋 ∈ V → (𝐸‘𝑋) = (𝐸‘𝑍))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1542   ∈ wcel 2114  Vcvv 3442  ∅c0 4287  dom cdm 5634  Rel wrel 5639  ‘cfv 6502  (class class class)co 7370

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5245  ax-nul 5255  ax-pr 5381

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-xp 5640  df-rel 5641  df-dm 5644  df-iota 6458  df-fv 6510  df-ov 7373

This theorem is referenced by:  setsnid  17149  ressbas  17177  resseqnbas  17183  tnglem  24601  resvlem  33432