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Theorem fveqprc 17228
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 21527. (Contributed by AV, 31-Oct-2024.)
Hypotheses
Ref Expression
fveqprc.e (𝐸‘∅) = ∅
fveqprc.y 𝑌 = (𝐹𝑋)
Assertion
Ref Expression
fveqprc 𝑋 ∈ V → (𝐸𝑋) = (𝐸𝑌))

Proof of Theorem fveqprc
StepHypRef Expression
1 fveqprc.e . . 3 (𝐸‘∅) = ∅
21eqcomi 2746 . 2 ∅ = (𝐸‘∅)
3 fvprc 6898 . 2 𝑋 ∈ V → (𝐸𝑋) = ∅)
4 fveqprc.y . . . 4 𝑌 = (𝐹𝑋)
5 fvprc 6898 . . . 4 𝑋 ∈ V → (𝐹𝑋) = ∅)
64, 5eqtrid 2789 . . 3 𝑋 ∈ V → 𝑌 = ∅)
76fveq2d 6910 . 2 𝑋 ∈ V → (𝐸𝑌) = (𝐸‘∅))
82, 3, 73eqtr4a 2803 1 𝑋 ∈ V → (𝐸𝑋) = (𝐸𝑌))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1540  wcel 2108  Vcvv 3480  c0 4333  cfv 6561
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569
This theorem is referenced by:  oppcbas  17761  zlmlem  21527  ttglem  28885  mendsca  43197
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