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Theorem upgr1wlkdlem2 29399
Description: Lemma 2 for upgr1wlkd 29400. (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 4005 . . 3 {𝑋, 𝑌} ⊆ {𝑋, 𝑌}
3 sseq2 4009 . . . 4 (((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌} → ({𝑋, 𝑌} ⊆ ((iEdg‘𝐺)‘𝐽) ↔ {𝑋, 𝑌} ⊆ {𝑋, 𝑌}))
43adantl 483 . . 3 (((𝜑𝑋𝑌) ∧ ((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌}) → ({𝑋, 𝑌} ⊆ ((iEdg‘𝐺)‘𝐽) ↔ {𝑋, 𝑌} ⊆ {𝑋, 𝑌}))
52, 4mpbiri 258 . 2 (((𝜑𝑋𝑌) ∧ ((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌}) → {𝑋, 𝑌} ⊆ ((iEdg‘𝐺)‘𝐽))
61, 5mpidan 688 1 ((𝜑𝑋𝑌) → {𝑋, 𝑌} ⊆ ((iEdg‘𝐺)‘𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wne 2941  wss 3949  {cpr 4631  cfv 6544  ⟨“cs1 14545  ⟨“cs2 14792  Vtxcvtx 28256  iEdgciedg 28257
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-in 3956  df-ss 3966
This theorem is referenced by:  upgr1wlkd  29400  upgr1trld  29401  upgr1pthd  29402  upgr1pthond  29403
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