Theorem oveqprc 17211

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 33349. (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 2744	. 2 ⊢ ∅ = (𝐸‘∅)
3		fvprc 6868	. 2 ⊢ (¬ 𝑋 ∈ V → (𝐸‘𝑋) = ∅)
4		oveqprc.z	. . . 4 ⊢ 𝑍 = (𝑋𝑂𝑌)
5		oveqprc.r	. . . . 5 ⊢ Rel dom 𝑂
6	5	ovprc1 7444	. . . 4 ⊢ (¬ 𝑋 ∈ V → (𝑋𝑂𝑌) = ∅)
7	4, 6	eqtrid 2782	. . 3 ⊢ (¬ 𝑋 ∈ V → 𝑍 = ∅)
8	7	fveq2d 6880	. 2 ⊢ (¬ 𝑋 ∈ V → (𝐸‘𝑍) = (𝐸‘∅))
9	2, 3, 8	3eqtr4a 2796	1 ⊢ (¬ 𝑋 ∈ V → (𝐸‘𝑋) = (𝐸‘𝑍))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1540   ∈ wcel 2108  Vcvv 3459  ∅c0 4308  dom cdm 5654  Rel wrel 5659  ‘cfv 6531  (class class class)co 7405

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-xp 5660  df-rel 5661  df-dm 5664  df-iota 6484  df-fv 6539  df-ov 7408

This theorem is referenced by:  setsnid  17227  ressbas  17257  resseqnbas  17263  tnglem  24579  resvlem  33349