Theorem fveqprc 17246

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 21666. (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 6873	. 2 ⊢ (¬ 𝑋 ∈ V → (𝐸‘𝑋) = ∅)
4		fveqprc.y	. . . 4 ⊢ 𝑌 = (𝐹‘𝑋)
5		fvprc 6873	. . . 4 ⊢ (¬ 𝑋 ∈ V → (𝐹‘𝑋) = ∅)
6	4, 5	eqtrid 2810	. . 3 ⊢ (¬ 𝑋 ∈ V → 𝑌 = ∅)
7	6	fveq2d 6885	. 2 ⊢ (¬ 𝑋 ∈ V → (𝐸‘𝑌) = (𝐸‘∅))
8	2, 3, 7	3eqtr4a 2824	1 ⊢ (¬ 𝑋 ∈ V → (𝐸‘𝑋) = (𝐸‘𝑌))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1570   ∈ wcel 2143  Vcvv 3455  ∅c0 4286  ‘cfv 6536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-nul 5269  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-iota 6492  df-fv 6544

This theorem is referenced by:  oppcbas  17769  zlmlem  21666  ttglem  29225  mendsca  43932