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Theorem isubgr3stgrlem1 47868
Description: Lemma 1 for isubgr3stgr 47877. (Contributed by AV, 16-Sep-2025.)
Hypotheses
Ref Expression
isubgr3stgr.v 𝑉 = (Vtx‘𝐺)
isubgr3stgr.u 𝑈 = (𝐺 NeighbVtx 𝑋)
isubgr3stgr.c 𝐶 = (𝐺 ClNeighbVtx 𝑋)
isubgr3stgr.f 𝐹 = (𝐻 ∪ {⟨𝑋, 𝑌⟩})
Assertion
Ref Expression
isubgr3stgrlem1 ((𝐻:𝑈1-1-onto𝑅𝑋𝑉 ∧ (𝑌𝑊𝑌𝑅)) → 𝐹:𝐶1-1-onto→(𝑅 ∪ {𝑌}))

Proof of Theorem isubgr3stgrlem1
StepHypRef Expression
1 isubgr3stgr.u . . . . . 6 𝑈 = (𝐺 NeighbVtx 𝑋)
2 f1oeq2 6837 . . . . . 6 (𝑈 = (𝐺 NeighbVtx 𝑋) → (𝐻:𝑈1-1-onto𝑅𝐻:(𝐺 NeighbVtx 𝑋)–1-1-onto𝑅))
31, 2ax-mp 5 . . . . 5 (𝐻:𝑈1-1-onto𝑅𝐻:(𝐺 NeighbVtx 𝑋)–1-1-onto𝑅)
43biimpi 216 . . . 4 (𝐻:𝑈1-1-onto𝑅𝐻:(𝐺 NeighbVtx 𝑋)–1-1-onto𝑅)
543ad2ant1 1132 . . 3 ((𝐻:𝑈1-1-onto𝑅𝑋𝑉 ∧ (𝑌𝑊𝑌𝑅)) → 𝐻:(𝐺 NeighbVtx 𝑋)–1-1-onto𝑅)
6 simpl 482 . . . . 5 ((𝑌𝑊𝑌𝑅) → 𝑌𝑊)
76anim2i 617 . . . 4 ((𝑋𝑉 ∧ (𝑌𝑊𝑌𝑅)) → (𝑋𝑉𝑌𝑊))
873adant1 1129 . . 3 ((𝐻:𝑈1-1-onto𝑅𝑋𝑉 ∧ (𝑌𝑊𝑌𝑅)) → (𝑋𝑉𝑌𝑊))
9 nbgrnself2 29391 . . . 4 𝑋 ∉ (𝐺 NeighbVtx 𝑋)
109a1i 11 . . 3 ((𝐻:𝑈1-1-onto𝑅𝑋𝑉 ∧ (𝑌𝑊𝑌𝑅)) → 𝑋 ∉ (𝐺 NeighbVtx 𝑋))
11 simp3r 1201 . . 3 ((𝐻:𝑈1-1-onto𝑅𝑋𝑉 ∧ (𝑌𝑊𝑌𝑅)) → 𝑌𝑅)
12 isubgr3stgr.f . . . 4 𝐹 = (𝐻 ∪ {⟨𝑋, 𝑌⟩})
1312f1ounsn 7291 . . 3 ((𝐻:(𝐺 NeighbVtx 𝑋)–1-1-onto𝑅 ∧ (𝑋𝑉𝑌𝑊) ∧ (𝑋 ∉ (𝐺 NeighbVtx 𝑋) ∧ 𝑌𝑅)) → 𝐹:((𝐺 NeighbVtx 𝑋) ∪ {𝑋})–1-1-onto→(𝑅 ∪ {𝑌}))
145, 8, 10, 11, 13syl112anc 1373 . 2 ((𝐻:𝑈1-1-onto𝑅𝑋𝑉 ∧ (𝑌𝑊𝑌𝑅)) → 𝐹:((𝐺 NeighbVtx 𝑋) ∪ {𝑋})–1-1-onto→(𝑅 ∪ {𝑌}))
15 isubgr3stgr.c . . . 4 𝐶 = (𝐺 ClNeighbVtx 𝑋)
16 isubgr3stgr.v . . . . . . 7 𝑉 = (Vtx‘𝐺)
1716dfclnbgr4 47748 . . . . . 6 (𝑋𝑉 → (𝐺 ClNeighbVtx 𝑋) = ({𝑋} ∪ (𝐺 NeighbVtx 𝑋)))
18173ad2ant2 1133 . . . . 5 ((𝐻:𝑈1-1-onto𝑅𝑋𝑉 ∧ (𝑌𝑊𝑌𝑅)) → (𝐺 ClNeighbVtx 𝑋) = ({𝑋} ∪ (𝐺 NeighbVtx 𝑋)))
19 uncom 4167 . . . . 5 ({𝑋} ∪ (𝐺 NeighbVtx 𝑋)) = ((𝐺 NeighbVtx 𝑋) ∪ {𝑋})
2018, 19eqtrdi 2790 . . . 4 ((𝐻:𝑈1-1-onto𝑅𝑋𝑉 ∧ (𝑌𝑊𝑌𝑅)) → (𝐺 ClNeighbVtx 𝑋) = ((𝐺 NeighbVtx 𝑋) ∪ {𝑋}))
2115, 20eqtrid 2786 . . 3 ((𝐻:𝑈1-1-onto𝑅𝑋𝑉 ∧ (𝑌𝑊𝑌𝑅)) → 𝐶 = ((𝐺 NeighbVtx 𝑋) ∪ {𝑋}))
2221f1oeq2d 6844 . 2 ((𝐻:𝑈1-1-onto𝑅𝑋𝑉 ∧ (𝑌𝑊𝑌𝑅)) → (𝐹:𝐶1-1-onto→(𝑅 ∪ {𝑌}) ↔ 𝐹:((𝐺 NeighbVtx 𝑋) ∪ {𝑋})–1-1-onto→(𝑅 ∪ {𝑌})))
2314, 22mpbird 257 1 ((𝐻:𝑈1-1-onto𝑅𝑋𝑉 ∧ (𝑌𝑊𝑌𝑅)) → 𝐹:𝐶1-1-onto→(𝑅 ∪ {𝑌}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1536  wcel 2105  wnel 3043  cun 3960  {csn 4630  cop 4636  1-1-ontowf1o 6561  cfv 6562  (class class class)co 7430  Vtxcvtx 29027   NeighbVtx cnbgr 29363   ClNeighbVtx cclnbgr 47742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-1st 8012  df-2nd 8013  df-nbgr 29364  df-clnbgr 47743
This theorem is referenced by:  isubgr3stgrlem3  47870
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