Theorem fveqprc 17118

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 21471. (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 6826	. 2 ⊢ (¬ 𝑋 ∈ V → (𝐸‘𝑋) = ∅)
4		fveqprc.y	. . . 4 ⊢ 𝑌 = (𝐹‘𝑋)
5		fvprc 6826	. . . 4 ⊢ (¬ 𝑋 ∈ V → (𝐹‘𝑋) = ∅)
6	4, 5	eqtrid 2783	. . 3 ⊢ (¬ 𝑋 ∈ V → 𝑌 = ∅)
7	6	fveq2d 6838	. 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 1541   ∈ wcel 2113  Vcvv 3440  ∅c0 4285  ‘cfv 6492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-iota 6448  df-fv 6500

This theorem is referenced by:  oppcbas  17641  zlmlem  21471  ttglem  28948  mendsca  43427