Theorem isubgr3stgrlem1 48471

Description: Lemma 1 for isubgr3stgr 48480. (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 6760	. . . . . 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 218	. . . 4 ⊢ (𝐻:𝑈–1-1-onto→𝑅 → 𝐻:(𝐺 NeighbVtx 𝑋)–1-1-onto→𝑅)
5	4	3ad2ant1 1140	. . 3 ⊢ ((𝐻:𝑈–1-1-onto→𝑅 ∧ 𝑋 ∈ 𝑉 ∧ (𝑌 ∈ 𝑊 ∧ 𝑌 ∉ 𝑅)) → 𝐻:(𝐺 NeighbVtx 𝑋)–1-1-onto→𝑅)
6		simpl 484	. . . . 5 ⊢ ((𝑌 ∈ 𝑊 ∧ 𝑌 ∉ 𝑅) → 𝑌 ∈ 𝑊)
7	6	anim2i 624	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝑌 ∈ 𝑊 ∧ 𝑌 ∉ 𝑅)) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊))
8	7	3adant1 1137	. . 3 ⊢ ((𝐻:𝑈–1-1-onto→𝑅 ∧ 𝑋 ∈ 𝑉 ∧ (𝑌 ∈ 𝑊 ∧ 𝑌 ∉ 𝑅)) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊))
9		nbgrnself2 29451	. . . 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 1210	. . 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 1383	. 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 48329	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → (𝐺 ClNeighbVtx 𝑋) = ({𝑋} ∪ (𝐺 NeighbVtx 𝑋)))
18	17	3ad2ant2 1141	. . . . 5 ⊢ ((𝐻:𝑈–1-1-onto→𝑅 ∧ 𝑋 ∈ 𝑉 ∧ (𝑌 ∈ 𝑊 ∧ 𝑌 ∉ 𝑅)) → (𝐺 ClNeighbVtx 𝑋) = ({𝑋} ∪ (𝐺 NeighbVtx 𝑋)))
19		uncom 4091	. . . . 5 ⊢ ({𝑋} ∪ (𝐺 NeighbVtx 𝑋)) = ((𝐺 NeighbVtx 𝑋) ∪ {𝑋})
20	18, 19	eqtrdi 2792	. . . 4 ⊢ ((𝐻:𝑈–1-1-onto→𝑅 ∧ 𝑋 ∈ 𝑉 ∧ (𝑌 ∈ 𝑊 ∧ 𝑌 ∉ 𝑅)) → (𝐺 ClNeighbVtx 𝑋) = ((𝐺 NeighbVtx 𝑋) ∪ {𝑋}))
21	15, 20	eqtrid 2788	. . 3 ⊢ ((𝐻:𝑈–1-1-onto→𝑅 ∧ 𝑋 ∈ 𝑉 ∧ (𝑌 ∈ 𝑊 ∧ 𝑌 ∉ 𝑅)) → 𝐶 = ((𝐺 NeighbVtx 𝑋) ∪ {𝑋}))
22	21	f1oeq2d 6767	. 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 259	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 208   ∧ wa 397   ∧ w3a 1093   = wceq 1548   ∈ wcel 2121   ∉ wnel 3040   ∪ cun 3883  {csn 4558  ⟨cop 4564  –1-1-onto→wf1o 6488  ‘cfv 6489  (class class class)co 7360  Vtxcvtx 29087   NeighbVtx cnbgr 29423   ClNeighbVtx cclnbgr 48323

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5221  ax-nul 5231  ax-pr 5365  ax-un 7682

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-nel 3041  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-ov 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-nbgr 29424  df-clnbgr 48324

This theorem is referenced by:  isubgr3stgrlem3  48473