Theorem isubgr3stgrlem4 47929

Description: Lemma 4 for isubgr3stgr 47935. (Contributed by AV, 24-Sep-2025.)

Hypotheses

Ref	Expression
isubgr3stgr.v	⊢ 𝑉 = (Vtx‘𝐺)
isubgr3stgr.u	⊢ 𝑈 = (𝐺 NeighbVtx 𝑋)
isubgr3stgr.c	⊢ 𝐶 = (𝐺 ClNeighbVtx 𝑋)
isubgr3stgr.n	⊢ 𝑁 ∈ ℕ0
isubgr3stgr.s	⊢ 𝑆 = (StarGr‘𝑁)
isubgr3stgr.w	⊢ 𝑊 = (Vtx‘𝑆)
isubgr3stgr.e	⊢ 𝐸 = (Edg‘𝐺)

Assertion

Ref	Expression
isubgr3stgrlem4	⊢ ((𝐴 = 𝑋 ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶)) → ∃𝑧 ∈ (1...𝑁)(𝐹 “ {𝐴, 𝐵}) = {0, 𝑧})

Distinct variable groups:   𝑧,𝐴   𝑧,𝐵   𝑧,𝐶   𝑧,𝐹   𝑧,𝑁   𝑧,𝑊   𝑧,𝑋

Allowed substitution hints:   𝑆(𝑧)   𝑈(𝑧)   𝐸(𝑧)   𝐺(𝑧)   𝑉(𝑧)

Proof of Theorem isubgr3stgrlem4

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		preq2 4710	. . . . . 6 ⊢ (𝑧 = (𝐹‘𝐵) → {0, 𝑧} = {0, (𝐹‘𝐵)})
2	1	eqeq2d 2746	. . . . 5 ⊢ (𝑧 = (𝐹‘𝐵) → ((𝐹 “ {𝑋, 𝐵}) = {0, 𝑧} ↔ (𝐹 “ {𝑋, 𝐵}) = {0, (𝐹‘𝐵)}))
3		f1of 6817	. . . . . . . . 9 ⊢ (𝐹:𝐶–1-1-onto→𝑊 → 𝐹:𝐶⟶𝑊)
4	3	adantr 480	. . . . . . . 8 ⊢ ((𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0) → 𝐹:𝐶⟶𝑊)
5	4	adantr 480	. . . . . . 7 ⊢ (((𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0) ∧ (𝑋 ≠ 𝐵 ∧ 𝑋 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶)) → 𝐹:𝐶⟶𝑊)
6		simpr3 1197	. . . . . . 7 ⊢ (((𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0) ∧ (𝑋 ≠ 𝐵 ∧ 𝑋 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶)) → 𝐵 ∈ 𝐶)
7	5, 6	ffvelcdmd 7074	. . . . . 6 ⊢ (((𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0) ∧ (𝑋 ≠ 𝐵 ∧ 𝑋 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶)) → (𝐹‘𝐵) ∈ 𝑊)
8		isubgr3stgr.w	. . . . . . . . . 10 ⊢ 𝑊 = (Vtx‘𝑆)
9		isubgr3stgr.s	. . . . . . . . . . 11 ⊢ 𝑆 = (StarGr‘𝑁)
10	9	fveq2i 6878	. . . . . . . . . 10 ⊢ (Vtx‘𝑆) = (Vtx‘(StarGr‘𝑁))
11		isubgr3stgr.n	. . . . . . . . . . 11 ⊢ 𝑁 ∈ ℕ0
12		stgrvtx 47914	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ0 → (Vtx‘(StarGr‘𝑁)) = (0...𝑁))
13	11, 12	ax-mp 5	. . . . . . . . . 10 ⊢ (Vtx‘(StarGr‘𝑁)) = (0...𝑁)
14	8, 10, 13	3eqtri 2762	. . . . . . . . 9 ⊢ 𝑊 = (0...𝑁)
15	14	eleq2i 2826	. . . . . . . 8 ⊢ ((𝐹‘𝐵) ∈ 𝑊 ↔ (𝐹‘𝐵) ∈ (0...𝑁))
16		fz0sn0fz1 13660	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ0 → (0...𝑁) = ({0} ∪ (1...𝑁)))
17	11, 16	ax-mp 5	. . . . . . . . 9 ⊢ (0...𝑁) = ({0} ∪ (1...𝑁))
18	17	eleq2i 2826	. . . . . . . 8 ⊢ ((𝐹‘𝐵) ∈ (0...𝑁) ↔ (𝐹‘𝐵) ∈ ({0} ∪ (1...𝑁)))
19		elun 4128	. . . . . . . . 9 ⊢ ((𝐹‘𝐵) ∈ ({0} ∪ (1...𝑁)) ↔ ((𝐹‘𝐵) ∈ {0} ∨ (𝐹‘𝐵) ∈ (1...𝑁)))
20		fvex 6888	. . . . . . . . . . 11 ⊢ (𝐹‘𝐵) ∈ V
21	20	elsn 4616	. . . . . . . . . 10 ⊢ ((𝐹‘𝐵) ∈ {0} ↔ (𝐹‘𝐵) = 0)
22	21	orbi1i 913	. . . . . . . . 9 ⊢ (((𝐹‘𝐵) ∈ {0} ∨ (𝐹‘𝐵) ∈ (1...𝑁)) ↔ ((𝐹‘𝐵) = 0 ∨ (𝐹‘𝐵) ∈ (1...𝑁)))
23	19, 22	bitri 275	. . . . . . . 8 ⊢ ((𝐹‘𝐵) ∈ ({0} ∪ (1...𝑁)) ↔ ((𝐹‘𝐵) = 0 ∨ (𝐹‘𝐵) ∈ (1...𝑁)))
24	15, 18, 23	3bitri 297	. . . . . . 7 ⊢ ((𝐹‘𝐵) ∈ 𝑊 ↔ ((𝐹‘𝐵) = 0 ∨ (𝐹‘𝐵) ∈ (1...𝑁)))
25		eqeq2 2747	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑋) = 0 → ((𝐹‘𝐵) = (𝐹‘𝑋) ↔ (𝐹‘𝐵) = 0))
26	25	adantl 481	. . . . . . . . . 10 ⊢ ((𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0) → ((𝐹‘𝐵) = (𝐹‘𝑋) ↔ (𝐹‘𝐵) = 0))
27	26	adantr 480	. . . . . . . . 9 ⊢ (((𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0) ∧ (𝑋 ≠ 𝐵 ∧ 𝑋 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶)) → ((𝐹‘𝐵) = (𝐹‘𝑋) ↔ (𝐹‘𝐵) = 0))
28		f1of1 6816	. . . . . . . . . . . 12 ⊢ (𝐹:𝐶–1-1-onto→𝑊 → 𝐹:𝐶–1-1→𝑊)
29		dff14a 7262	. . . . . . . . . . . . 13 ⊢ (𝐹:𝐶–1-1→𝑊 ↔ (𝐹:𝐶⟶𝑊 ∧ ∀𝑎 ∈ 𝐶 ∀𝑏 ∈ 𝐶 (𝑎 ≠ 𝑏 → (𝐹‘𝑎) ≠ (𝐹‘𝑏))))
30		simpl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑎 = 𝑋 ∧ 𝑏 = 𝐵) → 𝑎 = 𝑋)
31		simpr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑎 = 𝑋 ∧ 𝑏 = 𝐵) → 𝑏 = 𝐵)
32	30, 31	neeq12d 2993	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑎 = 𝑋 ∧ 𝑏 = 𝐵) → (𝑎 ≠ 𝑏 ↔ 𝑋 ≠ 𝐵))
33		fveq2 6875	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝑋 → (𝐹‘𝑎) = (𝐹‘𝑋))
34	33	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑎 = 𝑋 ∧ 𝑏 = 𝐵) → (𝐹‘𝑎) = (𝐹‘𝑋))
35		fveq2 6875	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑏 = 𝐵 → (𝐹‘𝑏) = (𝐹‘𝐵))
36	35	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑎 = 𝑋 ∧ 𝑏 = 𝐵) → (𝐹‘𝑏) = (𝐹‘𝐵))
37	34, 36	neeq12d 2993	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑎 = 𝑋 ∧ 𝑏 = 𝐵) → ((𝐹‘𝑎) ≠ (𝐹‘𝑏) ↔ (𝐹‘𝑋) ≠ (𝐹‘𝐵)))
38	32, 37	imbi12d 344	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎 = 𝑋 ∧ 𝑏 = 𝐵) → ((𝑎 ≠ 𝑏 → (𝐹‘𝑎) ≠ (𝐹‘𝑏)) ↔ (𝑋 ≠ 𝐵 → (𝐹‘𝑋) ≠ (𝐹‘𝐵))))
39	38	rspc2gv 3611	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → (∀𝑎 ∈ 𝐶 ∀𝑏 ∈ 𝐶 (𝑎 ≠ 𝑏 → (𝐹‘𝑎) ≠ (𝐹‘𝑏)) → (𝑋 ≠ 𝐵 → (𝐹‘𝑋) ≠ (𝐹‘𝐵))))
40	39	3adant1 1130	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 ≠ 𝐵 ∧ 𝑋 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → (∀𝑎 ∈ 𝐶 ∀𝑏 ∈ 𝐶 (𝑎 ≠ 𝑏 → (𝐹‘𝑎) ≠ (𝐹‘𝑏)) → (𝑋 ≠ 𝐵 → (𝐹‘𝑋) ≠ (𝐹‘𝐵))))
41		id 22	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑋 ≠ 𝐵 → (𝐹‘𝑋) ≠ (𝐹‘𝐵)) → (𝑋 ≠ 𝐵 → (𝐹‘𝑋) ≠ (𝐹‘𝐵)))
42		eqneqall 2943	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹‘𝑋) = (𝐹‘𝐵) → ((𝐹‘𝑋) ≠ (𝐹‘𝐵) → (𝐹‘𝐵) ∈ (1...𝑁)))
43	42	eqcoms 2743	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹‘𝐵) = (𝐹‘𝑋) → ((𝐹‘𝑋) ≠ (𝐹‘𝐵) → (𝐹‘𝐵) ∈ (1...𝑁)))
44	43	com12 32	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹‘𝑋) ≠ (𝐹‘𝐵) → ((𝐹‘𝐵) = (𝐹‘𝑋) → (𝐹‘𝐵) ∈ (1...𝑁)))
45	41, 44	syl6com 37	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 ≠ 𝐵 → ((𝑋 ≠ 𝐵 → (𝐹‘𝑋) ≠ (𝐹‘𝐵)) → ((𝐹‘𝐵) = (𝐹‘𝑋) → (𝐹‘𝐵) ∈ (1...𝑁))))
46	45	3ad2ant1 1133	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 ≠ 𝐵 ∧ 𝑋 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → ((𝑋 ≠ 𝐵 → (𝐹‘𝑋) ≠ (𝐹‘𝐵)) → ((𝐹‘𝐵) = (𝐹‘𝑋) → (𝐹‘𝐵) ∈ (1...𝑁))))
47	40, 46	syld 47	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ≠ 𝐵 ∧ 𝑋 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → (∀𝑎 ∈ 𝐶 ∀𝑏 ∈ 𝐶 (𝑎 ≠ 𝑏 → (𝐹‘𝑎) ≠ (𝐹‘𝑏)) → ((𝐹‘𝐵) = (𝐹‘𝑋) → (𝐹‘𝐵) ∈ (1...𝑁))))
48	47	adantld 490	. . . . . . . . . . . . 13 ⊢ ((𝑋 ≠ 𝐵 ∧ 𝑋 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → ((𝐹:𝐶⟶𝑊 ∧ ∀𝑎 ∈ 𝐶 ∀𝑏 ∈ 𝐶 (𝑎 ≠ 𝑏 → (𝐹‘𝑎) ≠ (𝐹‘𝑏))) → ((𝐹‘𝐵) = (𝐹‘𝑋) → (𝐹‘𝐵) ∈ (1...𝑁))))
49	29, 48	biimtrid 242	. . . . . . . . . . . 12 ⊢ ((𝑋 ≠ 𝐵 ∧ 𝑋 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → (𝐹:𝐶–1-1→𝑊 → ((𝐹‘𝐵) = (𝐹‘𝑋) → (𝐹‘𝐵) ∈ (1...𝑁))))
50	28, 49	syl5com 31	. . . . . . . . . . 11 ⊢ (𝐹:𝐶–1-1-onto→𝑊 → ((𝑋 ≠ 𝐵 ∧ 𝑋 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → ((𝐹‘𝐵) = (𝐹‘𝑋) → (𝐹‘𝐵) ∈ (1...𝑁))))
51	50	adantr 480	. . . . . . . . . 10 ⊢ ((𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0) → ((𝑋 ≠ 𝐵 ∧ 𝑋 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → ((𝐹‘𝐵) = (𝐹‘𝑋) → (𝐹‘𝐵) ∈ (1...𝑁))))
52	51	imp 406	. . . . . . . . 9 ⊢ (((𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0) ∧ (𝑋 ≠ 𝐵 ∧ 𝑋 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶)) → ((𝐹‘𝐵) = (𝐹‘𝑋) → (𝐹‘𝐵) ∈ (1...𝑁)))
53	27, 52	sylbird 260	. . . . . . . 8 ⊢ (((𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0) ∧ (𝑋 ≠ 𝐵 ∧ 𝑋 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶)) → ((𝐹‘𝐵) = 0 → (𝐹‘𝐵) ∈ (1...𝑁)))
54		idd 24	. . . . . . . 8 ⊢ (((𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0) ∧ (𝑋 ≠ 𝐵 ∧ 𝑋 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶)) → ((𝐹‘𝐵) ∈ (1...𝑁) → (𝐹‘𝐵) ∈ (1...𝑁)))
55	53, 54	jaod 859	. . . . . . 7 ⊢ (((𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0) ∧ (𝑋 ≠ 𝐵 ∧ 𝑋 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶)) → (((𝐹‘𝐵) = 0 ∨ (𝐹‘𝐵) ∈ (1...𝑁)) → (𝐹‘𝐵) ∈ (1...𝑁)))
56	24, 55	biimtrid 242	. . . . . 6 ⊢ (((𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0) ∧ (𝑋 ≠ 𝐵 ∧ 𝑋 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶)) → ((𝐹‘𝐵) ∈ 𝑊 → (𝐹‘𝐵) ∈ (1...𝑁)))
57	7, 56	mpd 15	. . . . 5 ⊢ (((𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0) ∧ (𝑋 ≠ 𝐵 ∧ 𝑋 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶)) → (𝐹‘𝐵) ∈ (1...𝑁))
58		f1ofn 6818	. . . . . . . . . 10 ⊢ (𝐹:𝐶–1-1-onto→𝑊 → 𝐹 Fn 𝐶)
59	58	adantr 480	. . . . . . . . 9 ⊢ ((𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0) → 𝐹 Fn 𝐶)
60		3simpc 1150	. . . . . . . . 9 ⊢ ((𝑋 ≠ 𝐵 ∧ 𝑋 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → (𝑋 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶))
61	59, 60	anim12i 613	. . . . . . . 8 ⊢ (((𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0) ∧ (𝑋 ≠ 𝐵 ∧ 𝑋 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶)) → (𝐹 Fn 𝐶 ∧ (𝑋 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶)))
62		3anass 1094	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐶 ∧ 𝑋 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ↔ (𝐹 Fn 𝐶 ∧ (𝑋 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶)))
63	61, 62	sylibr 234	. . . . . . 7 ⊢ (((𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0) ∧ (𝑋 ≠ 𝐵 ∧ 𝑋 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶)) → (𝐹 Fn 𝐶 ∧ 𝑋 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶))
64		fnimapr 6961	. . . . . . 7 ⊢ ((𝐹 Fn 𝐶 ∧ 𝑋 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → (𝐹 “ {𝑋, 𝐵}) = {(𝐹‘𝑋), (𝐹‘𝐵)})
65	63, 64	syl 17	. . . . . 6 ⊢ (((𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0) ∧ (𝑋 ≠ 𝐵 ∧ 𝑋 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶)) → (𝐹 “ {𝑋, 𝐵}) = {(𝐹‘𝑋), (𝐹‘𝐵)})
66		simpr 484	. . . . . . . 8 ⊢ ((𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0) → (𝐹‘𝑋) = 0)
67	66	adantr 480	. . . . . . 7 ⊢ (((𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0) ∧ (𝑋 ≠ 𝐵 ∧ 𝑋 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶)) → (𝐹‘𝑋) = 0)
68	67	preq1d 4715	. . . . . 6 ⊢ (((𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0) ∧ (𝑋 ≠ 𝐵 ∧ 𝑋 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶)) → {(𝐹‘𝑋), (𝐹‘𝐵)} = {0, (𝐹‘𝐵)})
69	65, 68	eqtrd 2770	. . . . 5 ⊢ (((𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0) ∧ (𝑋 ≠ 𝐵 ∧ 𝑋 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶)) → (𝐹 “ {𝑋, 𝐵}) = {0, (𝐹‘𝐵)})
70	2, 57, 69	rspcedvdw 3604	. . . 4 ⊢ (((𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0) ∧ (𝑋 ≠ 𝐵 ∧ 𝑋 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶)) → ∃𝑧 ∈ (1...𝑁)(𝐹 “ {𝑋, 𝐵}) = {0, 𝑧})
71	70	ex 412	. . 3 ⊢ ((𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0) → ((𝑋 ≠ 𝐵 ∧ 𝑋 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → ∃𝑧 ∈ (1...𝑁)(𝐹 “ {𝑋, 𝐵}) = {0, 𝑧}))
72		neeq1 2994	. . . . 5 ⊢ (𝐴 = 𝑋 → (𝐴 ≠ 𝐵 ↔ 𝑋 ≠ 𝐵))
73		eleq1 2822	. . . . 5 ⊢ (𝐴 = 𝑋 → (𝐴 ∈ 𝐶 ↔ 𝑋 ∈ 𝐶))
74	72, 73	3anbi12d 1439	. . . 4 ⊢ (𝐴 = 𝑋 → ((𝐴 ≠ 𝐵 ∧ 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ↔ (𝑋 ≠ 𝐵 ∧ 𝑋 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶)))
75		preq1 4709	. . . . . . 7 ⊢ (𝐴 = 𝑋 → {𝐴, 𝐵} = {𝑋, 𝐵})
76	75	imaeq2d 6047	. . . . . 6 ⊢ (𝐴 = 𝑋 → (𝐹 “ {𝐴, 𝐵}) = (𝐹 “ {𝑋, 𝐵}))
77	76	eqeq1d 2737	. . . . 5 ⊢ (𝐴 = 𝑋 → ((𝐹 “ {𝐴, 𝐵}) = {0, 𝑧} ↔ (𝐹 “ {𝑋, 𝐵}) = {0, 𝑧}))
78	77	rexbidv 3164	. . . 4 ⊢ (𝐴 = 𝑋 → (∃𝑧 ∈ (1...𝑁)(𝐹 “ {𝐴, 𝐵}) = {0, 𝑧} ↔ ∃𝑧 ∈ (1...𝑁)(𝐹 “ {𝑋, 𝐵}) = {0, 𝑧}))
79	74, 78	imbi12d 344	. . 3 ⊢ (𝐴 = 𝑋 → (((𝐴 ≠ 𝐵 ∧ 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → ∃𝑧 ∈ (1...𝑁)(𝐹 “ {𝐴, 𝐵}) = {0, 𝑧}) ↔ ((𝑋 ≠ 𝐵 ∧ 𝑋 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → ∃𝑧 ∈ (1...𝑁)(𝐹 “ {𝑋, 𝐵}) = {0, 𝑧})))
80	71, 79	imbitrrid 246	. 2 ⊢ (𝐴 = 𝑋 → ((𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0) → ((𝐴 ≠ 𝐵 ∧ 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → ∃𝑧 ∈ (1...𝑁)(𝐹 “ {𝐴, 𝐵}) = {0, 𝑧})))
81	80	3imp 1110	1 ⊢ ((𝐴 = 𝑋 ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶)) → ∃𝑧 ∈ (1...𝑁)(𝐹 “ {𝐴, 𝐵}) = {0, 𝑧})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1540   ∈ wcel 2108   ≠ wne 2932  ∀wral 3051  ∃wrex 3060   ∪ cun 3924  {csn 4601  {cpr 4603   “ cima 5657   Fn wfn 6525  ⟶wf 6526  –1-1→wf1 6527  –1-1-onto→wf1o 6529  ‘cfv 6530  (class class class)co 7403  0cc0 11127  1c1 11128  ℕ₀cn0 12499  ...cfz 13522  Vtxcvtx 28921  Edgcedg 28972   NeighbVtx cnbgr 29257   ClNeighbVtx cclnbgr 47780  StarGrcstgr 47911

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 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7727  ax-cnex 11183  ax-resscn 11184  ax-1cn 11185  ax-icn 11186  ax-addcl 11187  ax-addrcl 11188  ax-mulcl 11189  ax-mulrcl 11190  ax-mulcom 11191  ax-addass 11192  ax-mulass 11193  ax-distr 11194  ax-i2m1 11195  ax-1ne0 11196  ax-1rid 11197  ax-rnegex 11198  ax-rrecex 11199  ax-cnre 11200  ax-pre-lttri 11201  ax-pre-lttrn 11202  ax-pre-ltadd 11203  ax-pre-mulgt0 11204

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6483  df-fun 6532  df-fn 6533  df-f 6534  df-f1 6535  df-fo 6536  df-f1o 6537  df-fv 6538  df-riota 7360  df-ov 7406  df-oprab 7407  df-mpo 7408  df-om 7860  df-1st 7986  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-rdg 8422  df-1o 8478  df-oadd 8482  df-er 8717  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-dju 9913  df-card 9951  df-pnf 11269  df-mnf 11270  df-xr 11271  df-ltxr 11272  df-le 11273  df-sub 11466  df-neg 11467  df-nn 12239  df-2 12301  df-3 12302  df-4 12303  df-5 12304  df-6 12305  df-7 12306  df-8 12307  df-9 12308  df-n0 12500  df-xnn0 12573  df-z 12587  df-dec 12707  df-uz 12851  df-fz 13523  df-hash 14347  df-struct 17164  df-slot 17199  df-ndx 17211  df-base 17227  df-edgf 28914  df-vtx 28923  df-stgr 47912

This theorem is referenced by:  isubgr3stgrlem6  47931