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Theorem upgr1wlkdlem2 27124
Description: Lemma 2 for upgr1wlkd 27125. (Contributed by AV, 22-Jan-2021.)
Hypotheses
Ref Expression
upgr1wlkd.p 𝑃 = ⟨“𝑋𝑌”⟩
upgr1wlkd.f 𝐹 = ⟨“𝐽”⟩
upgr1wlkd.x (𝜑𝑋 ∈ (Vtx‘𝐺))
upgr1wlkd.y (𝜑𝑌 ∈ (Vtx‘𝐺))
upgr1wlkd.j (𝜑 → ((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌})
Assertion
Ref Expression
upgr1wlkdlem2 ((𝜑𝑋𝑌) → {𝑋, 𝑌} ⊆ ((iEdg‘𝐺)‘𝐽))

Proof of Theorem upgr1wlkdlem2
StepHypRef Expression
1 upgr1wlkd.j . 2 (𝜑 → ((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌})
2 ssid 3657 . . 3 {𝑋, 𝑌} ⊆ {𝑋, 𝑌}
3 sseq2 3660 . . . 4 (((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌} → ({𝑋, 𝑌} ⊆ ((iEdg‘𝐺)‘𝐽) ↔ {𝑋, 𝑌} ⊆ {𝑋, 𝑌}))
43adantl 481 . . 3 (((𝜑𝑋𝑌) ∧ ((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌}) → ({𝑋, 𝑌} ⊆ ((iEdg‘𝐺)‘𝐽) ↔ {𝑋, 𝑌} ⊆ {𝑋, 𝑌}))
52, 4mpbiri 248 . 2 (((𝜑𝑋𝑌) ∧ ((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌}) → {𝑋, 𝑌} ⊆ ((iEdg‘𝐺)‘𝐽))
61, 5mpidan 705 1 ((𝜑𝑋𝑌) → {𝑋, 𝑌} ⊆ ((iEdg‘𝐺)‘𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wne 2823  wss 3607  {cpr 4212  cfv 5926  ⟨“cs1 13326  ⟨“cs2 13632  Vtxcvtx 25919  iEdgciedg 25920
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-in 3614  df-ss 3621
This theorem is referenced by:  upgr1wlkd  27125  upgr1trld  27126  upgr1pthd  27127  upgr1pthond  27128
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