Theorem 2trllemF 26079

Description: Lemma 5 for constr2trl 26129. (Contributed by Alexander van der Vekens, 31-Jan-2018.)

Assertion

Ref	Expression
2trllemF	⊢ (((𝐸‘𝐼) = {𝑋, 𝑌} ∧ 𝑌 ∈ 𝑉) → 𝐼 ∈ dom 𝐸)

Proof of Theorem 2trllemF

Step	Hyp	Ref	Expression
1		prid2g 4240	. . . 4 ⊢ (𝑌 ∈ 𝑉 → 𝑌 ∈ {𝑋, 𝑌})
2		eleq2 2677	. . . 4 ⊢ ((𝐸‘𝐼) = {𝑋, 𝑌} → (𝑌 ∈ (𝐸‘𝐼) ↔ 𝑌 ∈ {𝑋, 𝑌}))
3	1, 2	syl5ibr 235	. . 3 ⊢ ((𝐸‘𝐼) = {𝑋, 𝑌} → (𝑌 ∈ 𝑉 → 𝑌 ∈ (𝐸‘𝐼)))
4	3	imp 444	. 2 ⊢ (((𝐸‘𝐼) = {𝑋, 𝑌} ∧ 𝑌 ∈ 𝑉) → 𝑌 ∈ (𝐸‘𝐼))
5		elfvdm 6130	. 2 ⊢ (𝑌 ∈ (𝐸‘𝐼) → 𝐼 ∈ dom 𝐸)
6	4, 5	syl 17	1 ⊢ (((𝐸‘𝐼) = {𝑋, 𝑌} ∧ 𝑌 ∈ 𝑉) → 𝐼 ∈ dom 𝐸)

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