Theorem fveqprc 17227

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 21568. (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 2771	. 2 ⊢ ∅ = (𝐸‘∅)
3		fvprc 6859	. 2 ⊢ (¬ 𝑋 ∈ V → (𝐸‘𝑋) = ∅)
4		fveqprc.y	. . . 4 ⊢ 𝑌 = (𝐹‘𝑋)
5		fvprc 6859	. . . 4 ⊢ (¬ 𝑋 ∈ V → (𝐹‘𝑋) = ∅)
6	4, 5	eqtrid 2809	. . 3 ⊢ (¬ 𝑋 ∈ V → 𝑌 = ∅)
7	6	fveq2d 6871	. 2 ⊢ (¬ 𝑋 ∈ V → (𝐸‘𝑌) = (𝐸‘∅))
8	2, 3, 7	3eqtr4a 2823	1 ⊢ (¬ 𝑋 ∈ V → (𝐸‘𝑋) = (𝐸‘𝑌))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1560   ∈ wcel 2142  Vcvv 3454  ∅c0 4285  ‘cfv 6521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-nul 5256  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-iota 6477  df-fv 6529

This theorem is referenced by:  oppcbas  17750  zlmlem  21568  ttglem  29076  mendsca  43762