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Theorem 2trllemF 26079
 Description: Lemma 5 for constr2trl 26129. (Contributed by Alexander van der Vekens, 31-Jan-2018.)
Assertion
Ref Expression
2trllemF (((𝐸𝐼) = {𝑋, 𝑌} ∧ 𝑌𝑉) → 𝐼 ∈ dom 𝐸)

Proof of Theorem 2trllemF
StepHypRef Expression
1 prid2g 4240 . . . 4 (𝑌𝑉𝑌 ∈ {𝑋, 𝑌})
2 eleq2 2677 . . . 4 ((𝐸𝐼) = {𝑋, 𝑌} → (𝑌 ∈ (𝐸𝐼) ↔ 𝑌 ∈ {𝑋, 𝑌}))
31, 2syl5ibr 235 . . 3 ((𝐸𝐼) = {𝑋, 𝑌} → (𝑌𝑉𝑌 ∈ (𝐸𝐼)))
43imp 444 . 2 (((𝐸𝐼) = {𝑋, 𝑌} ∧ 𝑌𝑉) → 𝑌 ∈ (𝐸𝐼))
5 elfvdm 6130 . 2 (𝑌 ∈ (𝐸𝐼) → 𝐼 ∈ dom 𝐸)
64, 5syl 17 1 (((𝐸𝐼) = {𝑋, 𝑌} ∧ 𝑌𝑉) → 𝐼 ∈ dom 𝐸)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cpr 4127  dom cdm 5038  ‘cfv 5804 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717  ax-pow 4769 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-dm 5048  df-iota 5768  df-fv 5812 This theorem is referenced by:  2trllemH  26082  2trllemE  26083
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