Theorem isomuspgrlem2d 45171

Description: Lemma 4 for isomuspgrlem2 45173. (Contributed by AV, 1-Dec-2022.)

Hypotheses

Ref	Expression
isomushgr.v	⊢ 𝑉 = (Vtx‘𝐴)
isomushgr.w	⊢ 𝑊 = (Vtx‘𝐵)
isomushgr.e	⊢ 𝐸 = (Edg‘𝐴)
isomushgr.k	⊢ 𝐾 = (Edg‘𝐵)
isomuspgrlem2.g	⊢ 𝐺 = (𝑥 ∈ 𝐸 ↦ (𝐹 “ 𝑥))
isomuspgrlem2.a	⊢ (𝜑 → 𝐴 ∈ USPGraph)
isomuspgrlem2.f	⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝑊)
isomuspgrlem2.i	⊢ (𝜑 → ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ({𝑎, 𝑏} ∈ 𝐸 ↔ {(𝐹‘𝑎), (𝐹‘𝑏)} ∈ 𝐾))
isomuspgrlem2.x	⊢ (𝜑 → 𝐹 ∈ 𝑋)
isomuspgrlem2.b	⊢ (𝜑 → 𝐵 ∈ USPGraph)

Assertion

Ref	Expression
isomuspgrlem2d	⊢ (𝜑 → 𝐺:𝐸–onto→𝐾)

Distinct variable groups:   𝑎,𝑏,𝑥   𝑥,𝐴   𝑥,𝐵   𝑥,𝐸   𝑥,𝐾   𝑥,𝑉   𝑥,𝑊   𝑥,𝐹   𝑥,𝑋   𝐸,𝑎,𝑏   𝐹,𝑎,𝑏   𝐾,𝑎,𝑏   𝑉,𝑎,𝑏   𝜑,𝑥

Allowed substitution hints:   𝜑(𝑎,𝑏)   𝐴(𝑎,𝑏)   𝐵(𝑎,𝑏)   𝐺(𝑥,𝑎,𝑏)   𝑊(𝑎,𝑏)   𝑋(𝑎,𝑏)

Proof of Theorem isomuspgrlem2d

Dummy variables 𝑒 𝑦 𝑐 𝑑 𝑚 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isomushgr.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐴)
2		isomushgr.w	. . 3 ⊢ 𝑊 = (Vtx‘𝐵)
3		isomushgr.e	. . 3 ⊢ 𝐸 = (Edg‘𝐴)
4		isomushgr.k	. . 3 ⊢ 𝐾 = (Edg‘𝐵)
5		isomuspgrlem2.g	. . 3 ⊢ 𝐺 = (𝑥 ∈ 𝐸 ↦ (𝐹 “ 𝑥))
6		isomuspgrlem2.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ USPGraph)
7		isomuspgrlem2.f	. . 3 ⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝑊)
8		isomuspgrlem2.i	. . 3 ⊢ (𝜑 → ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ({𝑎, 𝑏} ∈ 𝐸 ↔ {(𝐹‘𝑎), (𝐹‘𝑏)} ∈ 𝐾))
9	1, 2, 3, 4, 5, 6, 7, 8	isomuspgrlem2b 45169	. 2 ⊢ (𝜑 → 𝐺:𝐸⟶𝐾)
10		isomuspgrlem2.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ USPGraph)
11		uspgrupgr 27449	. . . . . 6 ⊢ (𝐵 ∈ USPGraph → 𝐵 ∈ UPGraph)
12	10, 11	syl 17	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ UPGraph)
13	2, 4	upgredg 27410	. . . . 5 ⊢ ((𝐵 ∈ UPGraph ∧ 𝑦 ∈ 𝐾) → ∃𝑐 ∈ 𝑊 ∃𝑑 ∈ 𝑊 𝑦 = {𝑐, 𝑑})
14	12, 13	sylan 579	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐾) → ∃𝑐 ∈ 𝑊 ∃𝑑 ∈ 𝑊 𝑦 = {𝑐, 𝑑})
15		eleq1 2826	. . . . . . . . . . 11 ⊢ (𝑦 = {𝑐, 𝑑} → (𝑦 ∈ 𝐾 ↔ {𝑐, 𝑑} ∈ 𝐾))
16	15	anbi2d 628	. . . . . . . . . 10 ⊢ (𝑦 = {𝑐, 𝑑} → ((𝜑 ∧ 𝑦 ∈ 𝐾) ↔ (𝜑 ∧ {𝑐, 𝑑} ∈ 𝐾)))
17		f1ofo 6707	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹:𝑉–1-1-onto→𝑊 → 𝐹:𝑉–onto→𝑊)
18	7, 17	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐹:𝑉–onto→𝑊)
19		foelrn 6964	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹:𝑉–onto→𝑊 ∧ 𝑐 ∈ 𝑊) → ∃𝑚 ∈ 𝑉 𝑐 = (𝐹‘𝑚))
20	18, 19	sylan 579	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝑊) → ∃𝑚 ∈ 𝑉 𝑐 = (𝐹‘𝑚))
21	20	ex 412	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑐 ∈ 𝑊 → ∃𝑚 ∈ 𝑉 𝑐 = (𝐹‘𝑚)))
22		foelrn 6964	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹:𝑉–onto→𝑊 ∧ 𝑑 ∈ 𝑊) → ∃𝑛 ∈ 𝑉 𝑑 = (𝐹‘𝑛))
23	18, 22	sylan 579	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝑊) → ∃𝑛 ∈ 𝑉 𝑑 = (𝐹‘𝑛))
24	23	ex 412	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑑 ∈ 𝑊 → ∃𝑛 ∈ 𝑉 𝑑 = (𝐹‘𝑛)))
25	21, 24	anim12d 608	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑐 ∈ 𝑊 ∧ 𝑑 ∈ 𝑊) → (∃𝑚 ∈ 𝑉 𝑐 = (𝐹‘𝑚) ∧ ∃𝑛 ∈ 𝑉 𝑑 = (𝐹‘𝑛))))
26	25	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ {𝑐, 𝑑} ∈ 𝐾) → ((𝑐 ∈ 𝑊 ∧ 𝑑 ∈ 𝑊) → (∃𝑚 ∈ 𝑉 𝑐 = (𝐹‘𝑚) ∧ ∃𝑛 ∈ 𝑉 𝑑 = (𝐹‘𝑛))))
27	26	imp 406	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ {𝑐, 𝑑} ∈ 𝐾) ∧ (𝑐 ∈ 𝑊 ∧ 𝑑 ∈ 𝑊)) → (∃𝑚 ∈ 𝑉 𝑐 = (𝐹‘𝑚) ∧ ∃𝑛 ∈ 𝑉 𝑑 = (𝐹‘𝑛)))
28		preq12 4668	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑐 = (𝐹‘𝑚) ∧ 𝑑 = (𝐹‘𝑛)) → {𝑐, 𝑑} = {(𝐹‘𝑚), (𝐹‘𝑛)})
29	28	ancoms 458	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑑 = (𝐹‘𝑛) ∧ 𝑐 = (𝐹‘𝑚)) → {𝑐, 𝑑} = {(𝐹‘𝑚), (𝐹‘𝑛)})
30	29	eleq1d 2823	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑑 = (𝐹‘𝑛) ∧ 𝑐 = (𝐹‘𝑚)) → ({𝑐, 𝑑} ∈ 𝐾 ↔ {(𝐹‘𝑚), (𝐹‘𝑛)} ∈ 𝐾))
31		preq1 4666	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑎 = 𝑚 → {𝑎, 𝑏} = {𝑚, 𝑏})
32	31	eleq1d 2823	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑎 = 𝑚 → ({𝑎, 𝑏} ∈ 𝐸 ↔ {𝑚, 𝑏} ∈ 𝐸))
33		fveq2 6756	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑎 = 𝑚 → (𝐹‘𝑎) = (𝐹‘𝑚))
34	33	preq1d 4672	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑎 = 𝑚 → {(𝐹‘𝑎), (𝐹‘𝑏)} = {(𝐹‘𝑚), (𝐹‘𝑏)})
35	34	eleq1d 2823	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑎 = 𝑚 → ({(𝐹‘𝑎), (𝐹‘𝑏)} ∈ 𝐾 ↔ {(𝐹‘𝑚), (𝐹‘𝑏)} ∈ 𝐾))
36	32, 35	bibi12d 345	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑎 = 𝑚 → (({𝑎, 𝑏} ∈ 𝐸 ↔ {(𝐹‘𝑎), (𝐹‘𝑏)} ∈ 𝐾) ↔ ({𝑚, 𝑏} ∈ 𝐸 ↔ {(𝐹‘𝑚), (𝐹‘𝑏)} ∈ 𝐾)))
37		preq2 4667	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑏 = 𝑛 → {𝑚, 𝑏} = {𝑚, 𝑛})
38	37	eleq1d 2823	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑏 = 𝑛 → ({𝑚, 𝑏} ∈ 𝐸 ↔ {𝑚, 𝑛} ∈ 𝐸))
39		fveq2 6756	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑏 = 𝑛 → (𝐹‘𝑏) = (𝐹‘𝑛))
40	39	preq2d 4673	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑏 = 𝑛 → {(𝐹‘𝑚), (𝐹‘𝑏)} = {(𝐹‘𝑚), (𝐹‘𝑛)})
41	40	eleq1d 2823	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑏 = 𝑛 → ({(𝐹‘𝑚), (𝐹‘𝑏)} ∈ 𝐾 ↔ {(𝐹‘𝑚), (𝐹‘𝑛)} ∈ 𝐾))
42	38, 41	bibi12d 345	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑏 = 𝑛 → (({𝑚, 𝑏} ∈ 𝐸 ↔ {(𝐹‘𝑚), (𝐹‘𝑏)} ∈ 𝐾) ↔ ({𝑚, 𝑛} ∈ 𝐸 ↔ {(𝐹‘𝑚), (𝐹‘𝑛)} ∈ 𝐾)))
43	36, 42	rspc2va 3563	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉) ∧ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ({𝑎, 𝑏} ∈ 𝐸 ↔ {(𝐹‘𝑎), (𝐹‘𝑏)} ∈ 𝐾)) → ({𝑚, 𝑛} ∈ 𝐸 ↔ {(𝐹‘𝑚), (𝐹‘𝑛)} ∈ 𝐾))
44	43	bicomd 222	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉) ∧ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ({𝑎, 𝑏} ∈ 𝐸 ↔ {(𝐹‘𝑎), (𝐹‘𝑏)} ∈ 𝐾)) → ({(𝐹‘𝑚), (𝐹‘𝑛)} ∈ 𝐾 ↔ {𝑚, 𝑛} ∈ 𝐸))
45	44	ancoms 458	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ({𝑎, 𝑏} ∈ 𝐸 ↔ {(𝐹‘𝑎), (𝐹‘𝑏)} ∈ 𝐾) ∧ (𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉)) → ({(𝐹‘𝑚), (𝐹‘𝑛)} ∈ 𝐾 ↔ {𝑚, 𝑛} ∈ 𝐸))
46	45	biimpd 228	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ({𝑎, 𝑏} ∈ 𝐸 ↔ {(𝐹‘𝑎), (𝐹‘𝑏)} ∈ 𝐾) ∧ (𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉)) → ({(𝐹‘𝑚), (𝐹‘𝑛)} ∈ 𝐾 → {𝑚, 𝑛} ∈ 𝐸))
47	46	ex 412	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ({𝑎, 𝑏} ∈ 𝐸 ↔ {(𝐹‘𝑎), (𝐹‘𝑏)} ∈ 𝐾) → ((𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉) → ({(𝐹‘𝑚), (𝐹‘𝑛)} ∈ 𝐾 → {𝑚, 𝑛} ∈ 𝐸)))
48	8, 47	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → ((𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉) → ({(𝐹‘𝑚), (𝐹‘𝑛)} ∈ 𝐾 → {𝑚, 𝑛} ∈ 𝐸)))
49	48	com13 88	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ({(𝐹‘𝑚), (𝐹‘𝑛)} ∈ 𝐾 → ((𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉) → (𝜑 → {𝑚, 𝑛} ∈ 𝐸)))
50	30, 49	syl6bi 252	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑑 = (𝐹‘𝑛) ∧ 𝑐 = (𝐹‘𝑚)) → ({𝑐, 𝑑} ∈ 𝐾 → ((𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉) → (𝜑 → {𝑚, 𝑛} ∈ 𝐸))))
51	50	com14 96	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ({𝑐, 𝑑} ∈ 𝐾 → ((𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉) → ((𝑑 = (𝐹‘𝑛) ∧ 𝑐 = (𝐹‘𝑚)) → {𝑚, 𝑛} ∈ 𝐸))))
52	51	imp 406	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ {𝑐, 𝑑} ∈ 𝐾) → ((𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉) → ((𝑑 = (𝐹‘𝑛) ∧ 𝑐 = (𝐹‘𝑚)) → {𝑚, 𝑛} ∈ 𝐸)))
53	52	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ {𝑐, 𝑑} ∈ 𝐾) ∧ (𝑐 ∈ 𝑊 ∧ 𝑑 ∈ 𝑊)) → ((𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉) → ((𝑑 = (𝐹‘𝑛) ∧ 𝑐 = (𝐹‘𝑚)) → {𝑚, 𝑛} ∈ 𝐸)))
54	53	imp31 417	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ {𝑐, 𝑑} ∈ 𝐾) ∧ (𝑐 ∈ 𝑊 ∧ 𝑑 ∈ 𝑊)) ∧ (𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉)) ∧ (𝑑 = (𝐹‘𝑛) ∧ 𝑐 = (𝐹‘𝑚))) → {𝑚, 𝑛} ∈ 𝐸)
55		imaeq2 5954	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑒 = {𝑚, 𝑛} → (𝐹 “ 𝑒) = (𝐹 “ {𝑚, 𝑛}))
56		f1ofn 6701	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝐹:𝑉–1-1-onto→𝑊 → 𝐹 Fn 𝑉)
57	7, 56	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → 𝐹 Fn 𝑉)
58	57	ad3antrrr 726	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ {𝑐, 𝑑} ∈ 𝐾) ∧ (𝑐 ∈ 𝑊 ∧ 𝑑 ∈ 𝑊)) ∧ (𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉)) → 𝐹 Fn 𝑉)
59		simprl 767	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ {𝑐, 𝑑} ∈ 𝐾) ∧ (𝑐 ∈ 𝑊 ∧ 𝑑 ∈ 𝑊)) ∧ (𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉)) → 𝑚 ∈ 𝑉)
60		simpr 484	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉) → 𝑛 ∈ 𝑉)
61	60	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ {𝑐, 𝑑} ∈ 𝐾) ∧ (𝑐 ∈ 𝑊 ∧ 𝑑 ∈ 𝑊)) ∧ (𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉)) → 𝑛 ∈ 𝑉)
62	58, 59, 61	3jca 1126	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ {𝑐, 𝑑} ∈ 𝐾) ∧ (𝑐 ∈ 𝑊 ∧ 𝑑 ∈ 𝑊)) ∧ (𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉)) → (𝐹 Fn 𝑉 ∧ 𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉))
63	62	adantr 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝜑 ∧ {𝑐, 𝑑} ∈ 𝐾) ∧ (𝑐 ∈ 𝑊 ∧ 𝑑 ∈ 𝑊)) ∧ (𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉)) ∧ (𝑑 = (𝐹‘𝑛) ∧ 𝑐 = (𝐹‘𝑚))) → (𝐹 Fn 𝑉 ∧ 𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉))
64		fnimapr 6834	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐹 Fn 𝑉 ∧ 𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉) → (𝐹 “ {𝑚, 𝑛}) = {(𝐹‘𝑚), (𝐹‘𝑛)})
65	63, 64	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝜑 ∧ {𝑐, 𝑑} ∈ 𝐾) ∧ (𝑐 ∈ 𝑊 ∧ 𝑑 ∈ 𝑊)) ∧ (𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉)) ∧ (𝑑 = (𝐹‘𝑛) ∧ 𝑐 = (𝐹‘𝑚))) → (𝐹 “ {𝑚, 𝑛}) = {(𝐹‘𝑚), (𝐹‘𝑛)})
66	55, 65	sylan9eqr 2801	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝜑 ∧ {𝑐, 𝑑} ∈ 𝐾) ∧ (𝑐 ∈ 𝑊 ∧ 𝑑 ∈ 𝑊)) ∧ (𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉)) ∧ (𝑑 = (𝐹‘𝑛) ∧ 𝑐 = (𝐹‘𝑚))) ∧ 𝑒 = {𝑚, 𝑛}) → (𝐹 “ 𝑒) = {(𝐹‘𝑚), (𝐹‘𝑛)})
67	66	eqeq2d 2749	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝜑 ∧ {𝑐, 𝑑} ∈ 𝐾) ∧ (𝑐 ∈ 𝑊 ∧ 𝑑 ∈ 𝑊)) ∧ (𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉)) ∧ (𝑑 = (𝐹‘𝑛) ∧ 𝑐 = (𝐹‘𝑚))) ∧ 𝑒 = {𝑚, 𝑛}) → ({𝑐, 𝑑} = (𝐹 “ 𝑒) ↔ {𝑐, 𝑑} = {(𝐹‘𝑚), (𝐹‘𝑛)}))
68	29	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ {𝑐, 𝑑} ∈ 𝐾) ∧ (𝑐 ∈ 𝑊 ∧ 𝑑 ∈ 𝑊)) ∧ (𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉)) ∧ (𝑑 = (𝐹‘𝑛) ∧ 𝑐 = (𝐹‘𝑚))) → {𝑐, 𝑑} = {(𝐹‘𝑚), (𝐹‘𝑛)})
69	54, 67, 68	rspcedvd 3555	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ {𝑐, 𝑑} ∈ 𝐾) ∧ (𝑐 ∈ 𝑊 ∧ 𝑑 ∈ 𝑊)) ∧ (𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉)) ∧ (𝑑 = (𝐹‘𝑛) ∧ 𝑐 = (𝐹‘𝑚))) → ∃𝑒 ∈ 𝐸 {𝑐, 𝑑} = (𝐹 “ 𝑒))
70	69	ex 412	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ {𝑐, 𝑑} ∈ 𝐾) ∧ (𝑐 ∈ 𝑊 ∧ 𝑑 ∈ 𝑊)) ∧ (𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉)) → ((𝑑 = (𝐹‘𝑛) ∧ 𝑐 = (𝐹‘𝑚)) → ∃𝑒 ∈ 𝐸 {𝑐, 𝑑} = (𝐹 “ 𝑒)))
71	70	anassrs 467	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ {𝑐, 𝑑} ∈ 𝐾) ∧ (𝑐 ∈ 𝑊 ∧ 𝑑 ∈ 𝑊)) ∧ 𝑚 ∈ 𝑉) ∧ 𝑛 ∈ 𝑉) → ((𝑑 = (𝐹‘𝑛) ∧ 𝑐 = (𝐹‘𝑚)) → ∃𝑒 ∈ 𝐸 {𝑐, 𝑑} = (𝐹 “ 𝑒)))
72	71	expd 415	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ {𝑐, 𝑑} ∈ 𝐾) ∧ (𝑐 ∈ 𝑊 ∧ 𝑑 ∈ 𝑊)) ∧ 𝑚 ∈ 𝑉) ∧ 𝑛 ∈ 𝑉) → (𝑑 = (𝐹‘𝑛) → (𝑐 = (𝐹‘𝑚) → ∃𝑒 ∈ 𝐸 {𝑐, 𝑑} = (𝐹 “ 𝑒))))
73	72	rexlimdva 3212	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ {𝑐, 𝑑} ∈ 𝐾) ∧ (𝑐 ∈ 𝑊 ∧ 𝑑 ∈ 𝑊)) ∧ 𝑚 ∈ 𝑉) → (∃𝑛 ∈ 𝑉 𝑑 = (𝐹‘𝑛) → (𝑐 = (𝐹‘𝑚) → ∃𝑒 ∈ 𝐸 {𝑐, 𝑑} = (𝐹 “ 𝑒))))
74	73	com23 86	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ {𝑐, 𝑑} ∈ 𝐾) ∧ (𝑐 ∈ 𝑊 ∧ 𝑑 ∈ 𝑊)) ∧ 𝑚 ∈ 𝑉) → (𝑐 = (𝐹‘𝑚) → (∃𝑛 ∈ 𝑉 𝑑 = (𝐹‘𝑛) → ∃𝑒 ∈ 𝐸 {𝑐, 𝑑} = (𝐹 “ 𝑒))))
75	74	rexlimdva 3212	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ {𝑐, 𝑑} ∈ 𝐾) ∧ (𝑐 ∈ 𝑊 ∧ 𝑑 ∈ 𝑊)) → (∃𝑚 ∈ 𝑉 𝑐 = (𝐹‘𝑚) → (∃𝑛 ∈ 𝑉 𝑑 = (𝐹‘𝑛) → ∃𝑒 ∈ 𝐸 {𝑐, 𝑑} = (𝐹 “ 𝑒))))
76	75	impd 410	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ {𝑐, 𝑑} ∈ 𝐾) ∧ (𝑐 ∈ 𝑊 ∧ 𝑑 ∈ 𝑊)) → ((∃𝑚 ∈ 𝑉 𝑐 = (𝐹‘𝑚) ∧ ∃𝑛 ∈ 𝑉 𝑑 = (𝐹‘𝑛)) → ∃𝑒 ∈ 𝐸 {𝑐, 𝑑} = (𝐹 “ 𝑒)))
77	27, 76	mpd 15	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ {𝑐, 𝑑} ∈ 𝐾) ∧ (𝑐 ∈ 𝑊 ∧ 𝑑 ∈ 𝑊)) → ∃𝑒 ∈ 𝐸 {𝑐, 𝑑} = (𝐹 “ 𝑒))
78	77	ex 412	. . . . . . . . . 10 ⊢ ((𝜑 ∧ {𝑐, 𝑑} ∈ 𝐾) → ((𝑐 ∈ 𝑊 ∧ 𝑑 ∈ 𝑊) → ∃𝑒 ∈ 𝐸 {𝑐, 𝑑} = (𝐹 “ 𝑒)))
79	16, 78	syl6bi 252	. . . . . . . . 9 ⊢ (𝑦 = {𝑐, 𝑑} → ((𝜑 ∧ 𝑦 ∈ 𝐾) → ((𝑐 ∈ 𝑊 ∧ 𝑑 ∈ 𝑊) → ∃𝑒 ∈ 𝐸 {𝑐, 𝑑} = (𝐹 “ 𝑒))))
80	79	impd 410	. . . . . . . 8 ⊢ (𝑦 = {𝑐, 𝑑} → (((𝜑 ∧ 𝑦 ∈ 𝐾) ∧ (𝑐 ∈ 𝑊 ∧ 𝑑 ∈ 𝑊)) → ∃𝑒 ∈ 𝐸 {𝑐, 𝑑} = (𝐹 “ 𝑒)))
81	80	impcom 407	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐾) ∧ (𝑐 ∈ 𝑊 ∧ 𝑑 ∈ 𝑊)) ∧ 𝑦 = {𝑐, 𝑑}) → ∃𝑒 ∈ 𝐸 {𝑐, 𝑑} = (𝐹 “ 𝑒))
82		simpr 484	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐾) ∧ (𝑐 ∈ 𝑊 ∧ 𝑑 ∈ 𝑊)) ∧ 𝑦 = {𝑐, 𝑑}) → 𝑦 = {𝑐, 𝑑})
83	82	adantr 480	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑦 ∈ 𝐾) ∧ (𝑐 ∈ 𝑊 ∧ 𝑑 ∈ 𝑊)) ∧ 𝑦 = {𝑐, 𝑑}) ∧ 𝑒 ∈ 𝐸) → 𝑦 = {𝑐, 𝑑})
84		isomuspgrlem2.x	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 ∈ 𝑋)
85	84	ad4antr 728	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑦 ∈ 𝐾) ∧ (𝑐 ∈ 𝑊 ∧ 𝑑 ∈ 𝑊)) ∧ 𝑦 = {𝑐, 𝑑}) ∧ 𝑒 ∈ 𝐸) → 𝐹 ∈ 𝑋)
86	1, 2, 3, 4, 5	isomuspgrlem2a 45168	. . . . . . . . . . 11 ⊢ (𝐹 ∈ 𝑋 → ∀𝑦 ∈ 𝐸 (𝐹 “ 𝑦) = (𝐺‘𝑦))
87	85, 86	syl 17	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑦 ∈ 𝐾) ∧ (𝑐 ∈ 𝑊 ∧ 𝑑 ∈ 𝑊)) ∧ 𝑦 = {𝑐, 𝑑}) ∧ 𝑒 ∈ 𝐸) → ∀𝑦 ∈ 𝐸 (𝐹 “ 𝑦) = (𝐺‘𝑦))
88		imaeq2 5954	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑒 → (𝐹 “ 𝑦) = (𝐹 “ 𝑒))
89		fveq2 6756	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑒 → (𝐺‘𝑦) = (𝐺‘𝑒))
90	88, 89	eqeq12d 2754	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑒 → ((𝐹 “ 𝑦) = (𝐺‘𝑦) ↔ (𝐹 “ 𝑒) = (𝐺‘𝑒)))
91	90	rspcv 3547	. . . . . . . . . . . 12 ⊢ (𝑒 ∈ 𝐸 → (∀𝑦 ∈ 𝐸 (𝐹 “ 𝑦) = (𝐺‘𝑦) → (𝐹 “ 𝑒) = (𝐺‘𝑒)))
92		eqcom 2745	. . . . . . . . . . . 12 ⊢ ((𝐺‘𝑒) = (𝐹 “ 𝑒) ↔ (𝐹 “ 𝑒) = (𝐺‘𝑒))
93	91, 92	syl6ibr 251	. . . . . . . . . . 11 ⊢ (𝑒 ∈ 𝐸 → (∀𝑦 ∈ 𝐸 (𝐹 “ 𝑦) = (𝐺‘𝑦) → (𝐺‘𝑒) = (𝐹 “ 𝑒)))
94	93	adantl 481	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑦 ∈ 𝐾) ∧ (𝑐 ∈ 𝑊 ∧ 𝑑 ∈ 𝑊)) ∧ 𝑦 = {𝑐, 𝑑}) ∧ 𝑒 ∈ 𝐸) → (∀𝑦 ∈ 𝐸 (𝐹 “ 𝑦) = (𝐺‘𝑦) → (𝐺‘𝑒) = (𝐹 “ 𝑒)))
95	87, 94	mpd 15	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑦 ∈ 𝐾) ∧ (𝑐 ∈ 𝑊 ∧ 𝑑 ∈ 𝑊)) ∧ 𝑦 = {𝑐, 𝑑}) ∧ 𝑒 ∈ 𝐸) → (𝐺‘𝑒) = (𝐹 “ 𝑒))
96	83, 95	eqeq12d 2754	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑦 ∈ 𝐾) ∧ (𝑐 ∈ 𝑊 ∧ 𝑑 ∈ 𝑊)) ∧ 𝑦 = {𝑐, 𝑑}) ∧ 𝑒 ∈ 𝐸) → (𝑦 = (𝐺‘𝑒) ↔ {𝑐, 𝑑} = (𝐹 “ 𝑒)))
97	96	rexbidva 3224	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐾) ∧ (𝑐 ∈ 𝑊 ∧ 𝑑 ∈ 𝑊)) ∧ 𝑦 = {𝑐, 𝑑}) → (∃𝑒 ∈ 𝐸 𝑦 = (𝐺‘𝑒) ↔ ∃𝑒 ∈ 𝐸 {𝑐, 𝑑} = (𝐹 “ 𝑒)))
98	81, 97	mpbird 256	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐾) ∧ (𝑐 ∈ 𝑊 ∧ 𝑑 ∈ 𝑊)) ∧ 𝑦 = {𝑐, 𝑑}) → ∃𝑒 ∈ 𝐸 𝑦 = (𝐺‘𝑒))
99	98	ex 412	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐾) ∧ (𝑐 ∈ 𝑊 ∧ 𝑑 ∈ 𝑊)) → (𝑦 = {𝑐, 𝑑} → ∃𝑒 ∈ 𝐸 𝑦 = (𝐺‘𝑒)))
100	99	rexlimdvva 3222	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐾) → (∃𝑐 ∈ 𝑊 ∃𝑑 ∈ 𝑊 𝑦 = {𝑐, 𝑑} → ∃𝑒 ∈ 𝐸 𝑦 = (𝐺‘𝑒)))
101	14, 100	mpd 15	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐾) → ∃𝑒 ∈ 𝐸 𝑦 = (𝐺‘𝑒))
102	101	ralrimiva 3107	. 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝐾 ∃𝑒 ∈ 𝐸 𝑦 = (𝐺‘𝑒))
103		dffo3 6960	. 2 ⊢ (𝐺:𝐸–onto→𝐾 ↔ (𝐺:𝐸⟶𝐾 ∧ ∀𝑦 ∈ 𝐾 ∃𝑒 ∈ 𝐸 𝑦 = (𝐺‘𝑒)))
104	9, 102, 103	sylanbrc 582	1 ⊢ (𝜑 → 𝐺:𝐸–onto→𝐾)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  ∀wral 3063  ∃wrex 3064  {cpr 4560   ↦ cmpt 5153   “ cima 5583   Fn wfn 6413  ⟶wf 6414  –onto→wfo 6416  –1-1-onto→wf1o 6417  ‘cfv 6418  Vtxcvtx 27269  Edgcedg 27320  UPGraphcupgr 27353  USPGraphcuspgr 27421

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-2o 8268  df-oadd 8271  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-dju 9590  df-card 9628  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-2 11966  df-n0 12164  df-xnn0 12236  df-z 12250  df-uz 12512  df-fz 13169  df-hash 13973  df-edg 27321  df-upgr 27355  df-uspgr 27423

This theorem is referenced by:  isomuspgrlem2e  45172