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Theorem upgr1wlkdlem2 27612
Description: Lemma 2 for upgr1wlkd 27613. (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 3910 . . 3 {𝑋, 𝑌} ⊆ {𝑋, 𝑌}
3 sseq2 3914 . . . 4 (((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌} → ({𝑋, 𝑌} ⊆ ((iEdg‘𝐺)‘𝐽) ↔ {𝑋, 𝑌} ⊆ {𝑋, 𝑌}))
43adantl 482 . . 3 (((𝜑𝑋𝑌) ∧ ((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌}) → ({𝑋, 𝑌} ⊆ ((iEdg‘𝐺)‘𝐽) ↔ {𝑋, 𝑌} ⊆ {𝑋, 𝑌}))
52, 4mpbiri 259 . 2 (((𝜑𝑋𝑌) ∧ ((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌}) → {𝑋, 𝑌} ⊆ ((iEdg‘𝐺)‘𝐽))
61, 5mpidan 685 1 ((𝜑𝑋𝑌) → {𝑋, 𝑌} ⊆ ((iEdg‘𝐺)‘𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1522  wcel 2081  wne 2984  wss 3859  {cpr 4474  cfv 6225  ⟨“cs1 13793  ⟨“cs2 14039  Vtxcvtx 26464  iEdgciedg 26465
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-ext 2769
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-in 3866  df-ss 3874
This theorem is referenced by:  upgr1wlkd  27613  upgr1trld  27614  upgr1pthd  27615  upgr1pthond  27616
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