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Theorem fveqprc 17225
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 21545. (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 2744 . 2 ∅ = (𝐸‘∅)
3 fvprc 6899 . 2 𝑋 ∈ V → (𝐸𝑋) = ∅)
4 fveqprc.y . . . 4 𝑌 = (𝐹𝑋)
5 fvprc 6899 . . . 4 𝑋 ∈ V → (𝐹𝑋) = ∅)
64, 5eqtrid 2787 . . 3 𝑋 ∈ V → 𝑌 = ∅)
76fveq2d 6911 . 2 𝑋 ∈ V → (𝐸𝑌) = (𝐸‘∅))
82, 3, 73eqtr4a 2801 1 𝑋 ∈ V → (𝐸𝑋) = (𝐸𝑌))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1537  wcel 2106  Vcvv 3478  c0 4339  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571
This theorem is referenced by:  oppcbas  17764  zlmlem  21545  ttglem  28900  mendsca  43174
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