Theorem fveqprc 17384

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 zlmlem 21837. (Contributed by AV, 31-Oct-2024.)

Hypotheses

Ref	Expression
fveqprc.e	⊢ (𝐸‘∅) = ∅
fveqprc.y	⊢ 𝑌 = (𝐹‘𝑋)

Assertion

Ref	Expression
fveqprc	⊢ (¬ 𝑋 ∈ V → (𝐸‘𝑋) = (𝐸‘𝑌))

Proof of Theorem fveqprc

Step	Hyp	Ref	Expression
1		fveqprc.e	. . 3 ⊢ (𝐸‘∅) = ∅
2	1	eqcomi 2772	. 2 ⊢ ∅ = (𝐸‘∅)
3		fvprc 6888	. 2 ⊢ (¬ 𝑋 ∈ V → (𝐸‘𝑋) = ∅)
4		fveqprc.y	. . . 4 ⊢ 𝑌 = (𝐹‘𝑋)
5		fvprc 6888	. . . 4 ⊢ (¬ 𝑋 ∈ V → (𝐹‘𝑋) = ∅)
6	4, 5	eqtrid 2810	. . 3 ⊢ (¬ 𝑋 ∈ V → 𝑌 = ∅)
7	6	fveq2d 6900	. 2 ⊢ (¬ 𝑋 ∈ V → (𝐸‘𝑌) = (𝐸‘∅))
8	2, 3, 7	3eqtr4a 2824	1 ⊢ (¬ 𝑋 ∈ V → (𝐸‘𝑋) = (𝐸‘𝑌))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1569   ∈ wcel 2146  Vcvv 3454  ∅c0 4281  ‘cfv 6547

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735  ax-nul 5263  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-iota 6501  df-fv 6555

This theorem is used by:  oppcbas  17907  zlmlem  21837  ttglem  29468  mendsca  44333