Theorem fveqprc 17215

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 21482. (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 2745	. 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 2783	. . 3 ⊢ (¬ 𝑋 ∈ V → 𝑌 = ∅)
7	6	fveq2d 6885	. 2 ⊢ (¬ 𝑋 ∈ V → (𝐸‘𝑌) = (𝐸‘∅))
8	2, 3, 7	3eqtr4a 2797	1 ⊢ (¬ 𝑋 ∈ V → (𝐸‘𝑋) = (𝐸‘𝑌))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1540   ∈ wcel 2109  Vcvv 3464  ∅c0 4313  ‘cfv 6536

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-nul 5281  ax-pr 5407

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 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-iota 6489  df-fv 6544

This theorem is referenced by:  oppcbas  17735  zlmlem  21482  ttglem  28860  mendsca  43176