Theorem isubgr3stgrlem1 48279

Description: Lemma 1 for isubgr3stgr 48288. (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:𝑈–1-1-onto→𝑅 ∧ 𝑋 ∈ 𝑉 ∧ (𝑌 ∈ 𝑊 ∧ 𝑌 ∉ 𝑅)) → 𝐹:𝐶–1-1-onto→(𝑅 ∪ {𝑌}))

Proof of Theorem isubgr3stgrlem1

Step	Hyp	Ref	Expression
1		isubgr3stgr.u	. . . . . 6 ⊢ 𝑈 = (𝐺 NeighbVtx 𝑋)
2		f1oeq2 6764	. . . . . 6 ⊢ (𝑈 = (𝐺 NeighbVtx 𝑋) → (𝐻:𝑈–1-1-onto→𝑅 ↔ 𝐻:(𝐺 NeighbVtx 𝑋)–1-1-onto→𝑅))
3	1, 2	ax-mp 5	. . . . 5 ⊢ (𝐻:𝑈–1-1-onto→𝑅 ↔ 𝐻:(𝐺 NeighbVtx 𝑋)–1-1-onto→𝑅)
4	3	biimpi 216	. . . 4 ⊢ (𝐻:𝑈–1-1-onto→𝑅 → 𝐻:(𝐺 NeighbVtx 𝑋)–1-1-onto→𝑅)
5	4	3ad2ant1 1134	. . 3 ⊢ ((𝐻:𝑈–1-1-onto→𝑅 ∧ 𝑋 ∈ 𝑉 ∧ (𝑌 ∈ 𝑊 ∧ 𝑌 ∉ 𝑅)) → 𝐻:(𝐺 NeighbVtx 𝑋)–1-1-onto→𝑅)
6		simpl 482	. . . . 5 ⊢ ((𝑌 ∈ 𝑊 ∧ 𝑌 ∉ 𝑅) → 𝑌 ∈ 𝑊)
7	6	anim2i 618	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝑌 ∈ 𝑊 ∧ 𝑌 ∉ 𝑅)) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊))
8	7	3adant1 1131	. . 3 ⊢ ((𝐻:𝑈–1-1-onto→𝑅 ∧ 𝑋 ∈ 𝑉 ∧ (𝑌 ∈ 𝑊 ∧ 𝑌 ∉ 𝑅)) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊))
9		nbgrnself2 29437	. . . 4 ⊢ 𝑋 ∉ (𝐺 NeighbVtx 𝑋)
10	9	a1i 11	. . 3 ⊢ ((𝐻:𝑈–1-1-onto→𝑅 ∧ 𝑋 ∈ 𝑉 ∧ (𝑌 ∈ 𝑊 ∧ 𝑌 ∉ 𝑅)) → 𝑋 ∉ (𝐺 NeighbVtx 𝑋))
11		simp3r 1204	. . 3 ⊢ ((𝐻:𝑈–1-1-onto→𝑅 ∧ 𝑋 ∈ 𝑉 ∧ (𝑌 ∈ 𝑊 ∧ 𝑌 ∉ 𝑅)) → 𝑌 ∉ 𝑅)
12		isubgr3stgr.f	. . . 4 ⊢ 𝐹 = (𝐻 ∪ {⟨𝑋, 𝑌⟩})
13	12	f1ounsn 7220	. . 3 ⊢ ((𝐻:(𝐺 NeighbVtx 𝑋)–1-1-onto→𝑅 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (𝑋 ∉ (𝐺 NeighbVtx 𝑋) ∧ 𝑌 ∉ 𝑅)) → 𝐹:((𝐺 NeighbVtx 𝑋) ∪ {𝑋})–1-1-onto→(𝑅 ∪ {𝑌}))
14	5, 8, 10, 11, 13	syl112anc 1377	. 2 ⊢ ((𝐻:𝑈–1-1-onto→𝑅 ∧ 𝑋 ∈ 𝑉 ∧ (𝑌 ∈ 𝑊 ∧ 𝑌 ∉ 𝑅)) → 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)
17	16	dfclnbgr4 48137	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → (𝐺 ClNeighbVtx 𝑋) = ({𝑋} ∪ (𝐺 NeighbVtx 𝑋)))
18	17	3ad2ant2 1135	. . . . 5 ⊢ ((𝐻:𝑈–1-1-onto→𝑅 ∧ 𝑋 ∈ 𝑉 ∧ (𝑌 ∈ 𝑊 ∧ 𝑌 ∉ 𝑅)) → (𝐺 ClNeighbVtx 𝑋) = ({𝑋} ∪ (𝐺 NeighbVtx 𝑋)))
19		uncom 4111	. . . . 5 ⊢ ({𝑋} ∪ (𝐺 NeighbVtx 𝑋)) = ((𝐺 NeighbVtx 𝑋) ∪ {𝑋})
20	18, 19	eqtrdi 2788	. . . 4 ⊢ ((𝐻:𝑈–1-1-onto→𝑅 ∧ 𝑋 ∈ 𝑉 ∧ (𝑌 ∈ 𝑊 ∧ 𝑌 ∉ 𝑅)) → (𝐺 ClNeighbVtx 𝑋) = ((𝐺 NeighbVtx 𝑋) ∪ {𝑋}))
21	15, 20	eqtrid 2784	. . 3 ⊢ ((𝐻:𝑈–1-1-onto→𝑅 ∧ 𝑋 ∈ 𝑉 ∧ (𝑌 ∈ 𝑊 ∧ 𝑌 ∉ 𝑅)) → 𝐶 = ((𝐺 NeighbVtx 𝑋) ∪ {𝑋}))
22	21	f1oeq2d 6771	. 2 ⊢ ((𝐻:𝑈–1-1-onto→𝑅 ∧ 𝑋 ∈ 𝑉 ∧ (𝑌 ∈ 𝑊 ∧ 𝑌 ∉ 𝑅)) → (𝐹:𝐶–1-1-onto→(𝑅 ∪ {𝑌}) ↔ 𝐹:((𝐺 NeighbVtx 𝑋) ∪ {𝑋})–1-1-onto→(𝑅 ∪ {𝑌})))
23	14, 22	mpbird 257	1 ⊢ ((𝐻:𝑈–1-1-onto→𝑅 ∧ 𝑋 ∈ 𝑉 ∧ (𝑌 ∈ 𝑊 ∧ 𝑌 ∉ 𝑅)) → 𝐹:𝐶–1-1-onto→(𝑅 ∪ {𝑌}))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ∉ wnel 3037   ∪ cun 3900  {csn 4581  ⟨cop 4587  –1-1-onto→wf1o 6492  ‘cfv 6493  (class class class)co 7360  Vtxcvtx 29073   NeighbVtx cnbgr 29409   ClNeighbVtx cclnbgr 48131

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-nbgr 29410  df-clnbgr 48132

This theorem is referenced by:  isubgr3stgrlem3  48281