Theorem 11wlkdlem2 41305

Description: Lemma 2 for 11wlkd 41308. (Contributed by AV, 22-Jan-2021.)

Hypotheses

Ref	Expression
11wlkd.p	⊢ 𝑃 = ⟨“𝑋𝑌”⟩
11wlkd.f	⊢ 𝐹 = ⟨“𝐽”⟩
11wlkd.x	⊢ (𝜑 → 𝑋 ∈ 𝑉)
11wlkd.y	⊢ (𝜑 → 𝑌 ∈ 𝑉)
11wlkd.l	⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝐼‘𝐽) = {𝑋})
11wlkd.j	⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → {𝑋, 𝑌} ⊆ (𝐼‘𝐽))

Assertion

Ref	Expression
11wlkdlem2	⊢ (𝜑 → 𝑋 ∈ (𝐼‘𝐽))

Proof of Theorem 11wlkdlem2

Step	Hyp	Ref	Expression
1		11wlkd.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
2		snidg 4153	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ {𝑋})
3	1, 2	syl 17	. . . 4 ⊢ (𝜑 → 𝑋 ∈ {𝑋})
4	3	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑋 ∈ {𝑋})
5		11wlkd.l	. . 3 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝐼‘𝐽) = {𝑋})
6	4, 5	eleqtrrd 2691	. 2 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑋 ∈ (𝐼‘𝐽))
7		11wlkd.j	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → {𝑋, 𝑌} ⊆ (𝐼‘𝐽))
8		11wlkd.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
9	8	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑌 ∈ 𝑉)
10		prssg 4290	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ((𝑋 ∈ (𝐼‘𝐽) ∧ 𝑌 ∈ (𝐼‘𝐽)) ↔ {𝑋, 𝑌} ⊆ (𝐼‘𝐽)))
11	1, 9, 10	syl2an2r 872	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → ((𝑋 ∈ (𝐼‘𝐽) ∧ 𝑌 ∈ (𝐼‘𝐽)) ↔ {𝑋, 𝑌} ⊆ (𝐼‘𝐽)))
12	7, 11	mpbird 246	. . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → (𝑋 ∈ (𝐼‘𝐽) ∧ 𝑌 ∈ (𝐼‘𝐽)))
13	12	simpld 474	. 2 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑋 ∈ (𝐼‘𝐽))
14	6, 13	pm2.61dane 2869	1 ⊢ (𝜑 → 𝑋 ∈ (𝐼‘𝐽))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780   ⊆ wss 3540  {csn 4125  {cpr 4127  ‘cfv 5804  ⟨“cs1 13149  ⟨“cs2 13437

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-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

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-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-v 3175  df-un 3545  df-in 3547  df-ss 3554  df-sn 4126  df-pr 4128

This theorem is referenced by:  11wlkdlem3  41306  11wlkdlem4  41307