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Theorem isubgr3stgrlem1 47933
Description: Lemma 1 for isubgr3stgr 47942. (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 1134 . . 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 1131 . . 3 ((𝐻:𝑈1-1-onto𝑅𝑋𝑉 ∧ (𝑌𝑊𝑌𝑅)) → (𝑋𝑉𝑌𝑊))
9 nbgrnself2 29377 . . . 4 𝑋 ∉ (𝐺 NeighbVtx 𝑋)
109a1i 11 . . 3 ((𝐻:𝑈1-1-onto𝑅𝑋𝑉 ∧ (𝑌𝑊𝑌𝑅)) → 𝑋 ∉ (𝐺 NeighbVtx 𝑋))
11 simp3r 1203 . . 3 ((𝐻:𝑈1-1-onto𝑅𝑋𝑉 ∧ (𝑌𝑊𝑌𝑅)) → 𝑌𝑅)
12 isubgr3stgr.f . . . 4 𝐹 = (𝐻 ∪ {⟨𝑋, 𝑌⟩})
1312f1ounsn 7292 . . 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 1376 . 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 47811 . . . . . 6 (𝑋𝑉 → (𝐺 ClNeighbVtx 𝑋) = ({𝑋} ∪ (𝐺 NeighbVtx 𝑋)))
18173ad2ant2 1135 . . . . 5 ((𝐻:𝑈1-1-onto𝑅𝑋𝑉 ∧ (𝑌𝑊𝑌𝑅)) → (𝐺 ClNeighbVtx 𝑋) = ({𝑋} ∪ (𝐺 NeighbVtx 𝑋)))
19 uncom 4158 . . . . 5 ({𝑋} ∪ (𝐺 NeighbVtx 𝑋)) = ((𝐺 NeighbVtx 𝑋) ∪ {𝑋})
2018, 19eqtrdi 2793 . . . 4 ((𝐻:𝑈1-1-onto𝑅𝑋𝑉 ∧ (𝑌𝑊𝑌𝑅)) → (𝐺 ClNeighbVtx 𝑋) = ((𝐺 NeighbVtx 𝑋) ∪ {𝑋}))
2115, 20eqtrid 2789 . . 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 1087   = wceq 1540  wcel 2108  wnel 3046  cun 3949  {csn 4626  cop 4632  1-1-ontowf1o 6560  cfv 6561  (class class class)co 7431  Vtxcvtx 29013   NeighbVtx cnbgr 29349   ClNeighbVtx cclnbgr 47805
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-nbgr 29350  df-clnbgr 47806
This theorem is referenced by:  isubgr3stgrlem3  47935
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