Theorem isubgr3stgrlem1 48442

Description: Lemma 1 for isubgr3stgr 48451. (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 6769	. . . . . 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 29429	. . . 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 7227	. . 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 48300	. . . . . 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 4098	. . . . 5 ⊢ ({𝑋} ∪ (𝐺 NeighbVtx 𝑋)) = ((𝐺 NeighbVtx 𝑋) ∪ {𝑋})
20	18, 19	eqtrdi 2787	. . . 4 ⊢ ((𝐻:𝑈–1-1-onto→𝑅 ∧ 𝑋 ∈ 𝑉 ∧ (𝑌 ∈ 𝑊 ∧ 𝑌 ∉ 𝑅)) → (𝐺 ClNeighbVtx 𝑋) = ((𝐺 NeighbVtx 𝑋) ∪ {𝑋}))
21	15, 20	eqtrid 2783	. . 3 ⊢ ((𝐻:𝑈–1-1-onto→𝑅 ∧ 𝑋 ∈ 𝑉 ∧ (𝑌 ∈ 𝑊 ∧ 𝑌 ∉ 𝑅)) → 𝐶 = ((𝐺 NeighbVtx 𝑋) ∪ {𝑋}))
22	21	f1oeq2d 6776	. 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 3036   ∪ cun 3887  {csn 4567  ⟨cop 4573  –1-1-onto→wf1o 6497  ‘cfv 6498  (class class class)co 7367  Vtxcvtx 29065   NeighbVtx cnbgr 29401   ClNeighbVtx cclnbgr 48294

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 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375  ax-un 7689

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-nbgr 29402  df-clnbgr 48295

This theorem is referenced by:  isubgr3stgrlem3  48444