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: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 6837	. . . . . 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 617	. . . 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 29377	. . . 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 1203	. . 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 7292	. . 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 1376	. 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 47811	. . . . . 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 4158	. . . . 5 ⊢ ({𝑋} ∪ (𝐺 NeighbVtx 𝑋)) = ((𝐺 NeighbVtx 𝑋) ∪ {𝑋})
20	18, 19	eqtrdi 2793	. . . 4 ⊢ ((𝐻:𝑈–1-1-onto→𝑅 ∧ 𝑋 ∈ 𝑉 ∧ (𝑌 ∈ 𝑊 ∧ 𝑌 ∉ 𝑅)) → (𝐺 ClNeighbVtx 𝑋) = ((𝐺 NeighbVtx 𝑋) ∪ {𝑋}))
21	15, 20	eqtrid 2789	. . 3 ⊢ ((𝐻:𝑈–1-1-onto→𝑅 ∧ 𝑋 ∈ 𝑉 ∧ (𝑌 ∈ 𝑊 ∧ 𝑌 ∉ 𝑅)) → 𝐶 = ((𝐺 NeighbVtx 𝑋) ∪ {𝑋}))
22	21	f1oeq2d 6844	. 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 1540   ∈ wcel 2108   ∉ wnel 3046   ∪ cun 3949  {csn 4626  ⟨cop 4632  –1-1-onto→wf1o 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