Theorem isubgr3stgrlem1 48759

Description: Lemma 1 for isubgr3stgr 48768. (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 6809	. . . . . 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 219	. . . 4 ⊢ (𝐻:𝑈–1-1-onto→𝑅 → 𝐻:(𝐺 NeighbVtx 𝑋)–1-1-onto→𝑅)
5	4	3ad2ant1 1151	. . 3 ⊢ ((𝐻:𝑈–1-1-onto→𝑅 ∧ 𝑋 ∈ 𝑉 ∧ (𝑌 ∈ 𝑊 ∧ 𝑌 ∉ 𝑅)) → 𝐻:(𝐺 NeighbVtx 𝑋)–1-1-onto→𝑅)
6		simpl 487	. . . . 5 ⊢ ((𝑌 ∈ 𝑊 ∧ 𝑌 ∉ 𝑅) → 𝑌 ∈ 𝑊)
7	6	anim2i 628	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝑌 ∈ 𝑊 ∧ 𝑌 ∉ 𝑅)) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊))
8	7	3adant1 1148	. . 3 ⊢ ((𝐻:𝑈–1-1-onto→𝑅 ∧ 𝑋 ∈ 𝑉 ∧ (𝑌 ∈ 𝑊 ∧ 𝑌 ∉ 𝑅)) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊))
9		nbgrnself2 29719	. . . 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 1221	. . 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 7270	. . 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 1401	. 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 48617	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → (𝐺 ClNeighbVtx 𝑋) = ({𝑋} ∪ (𝐺 NeighbVtx 𝑋)))
18	17	3ad2ant2 1152	. . . . 5 ⊢ ((𝐻:𝑈–1-1-onto→𝑅 ∧ 𝑋 ∈ 𝑉 ∧ (𝑌 ∈ 𝑊 ∧ 𝑌 ∉ 𝑅)) → (𝐺 ClNeighbVtx 𝑋) = ({𝑋} ∪ (𝐺 NeighbVtx 𝑋)))
19		uncom 4112	. . . . 5 ⊢ ({𝑋} ∪ (𝐺 NeighbVtx 𝑋)) = ((𝐺 NeighbVtx 𝑋) ∪ {𝑋})
20	18, 19	eqtrdi 2814	. . . 4 ⊢ ((𝐻:𝑈–1-1-onto→𝑅 ∧ 𝑋 ∈ 𝑉 ∧ (𝑌 ∈ 𝑊 ∧ 𝑌 ∉ 𝑅)) → (𝐺 ClNeighbVtx 𝑋) = ((𝐺 NeighbVtx 𝑋) ∪ {𝑋}))
21	15, 20	eqtrid 2810	. . 3 ⊢ ((𝐻:𝑈–1-1-onto→𝑅 ∧ 𝑋 ∈ 𝑉 ∧ (𝑌 ∈ 𝑊 ∧ 𝑌 ∉ 𝑅)) → 𝐶 = ((𝐺 NeighbVtx 𝑋) ∪ {𝑋}))
22	21	f1oeq2d 6816	. 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 260	1 ⊢ ((𝐻:𝑈–1-1-onto→𝑅 ∧ 𝑋 ∈ 𝑉 ∧ (𝑌 ∈ 𝑊 ∧ 𝑌 ∉ 𝑅)) → 𝐹:𝐶–1-1-onto→(𝑅 ∪ {𝑌}))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143   ∉ wnel 3064   ∪ cun 3903  {csn 4589  ⟨cop 4595  –1-1-onto→wf1o 6535  ‘cfv 6536  (class class class)co 7410  Vtxcvtx 29355   NeighbVtx cnbgr 29691   ClNeighbVtx cclnbgr 48611

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404  ax-un 7732

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7982  df-2nd 7983  df-nbgr 29692  df-clnbgr 48612

This theorem is used by:  isubgr3stgrlem3  48761