Theorem isomuspgrlem2b 44347

Description: Lemma 2 for isomuspgrlem2 44351. (Contributed by AV, 29-Nov-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))

Assertion

Ref	Expression
isomuspgrlem2b	⊢ (𝜑 → 𝐺:𝐸⟶𝐾)

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

Proof of Theorem isomuspgrlem2b

Dummy variables 𝑐 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isomuspgrlem2.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ USPGraph)
2		uspgrupgr 26969	. . . . . 6 ⊢ (𝐴 ∈ USPGraph → 𝐴 ∈ UPGraph)
3	1, 2	syl 17	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ UPGraph)
4		isomushgr.v	. . . . . 6 ⊢ 𝑉 = (Vtx‘𝐴)
5		isomushgr.e	. . . . . 6 ⊢ 𝐸 = (Edg‘𝐴)
6	4, 5	upgredg 26930	. . . . 5 ⊢ ((𝐴 ∈ UPGraph ∧ 𝑥 ∈ 𝐸) → ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 𝑥 = {𝑐, 𝑑})
7	3, 6	sylan 583	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐸) → ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 𝑥 = {𝑐, 𝑑})
8		isomuspgrlem2.i	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ({𝑎, 𝑏} ∈ 𝐸 ↔ {(𝐹‘𝑎), (𝐹‘𝑏)} ∈ 𝐾))
9		preq12 4631	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 = 𝑐 ∧ 𝑏 = 𝑑) → {𝑎, 𝑏} = {𝑐, 𝑑})
10	9	eleq1d 2874	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 = 𝑐 ∧ 𝑏 = 𝑑) → ({𝑎, 𝑏} ∈ 𝐸 ↔ {𝑐, 𝑑} ∈ 𝐸))
11		fveq2 6645	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = 𝑐 → (𝐹‘𝑎) = (𝐹‘𝑐))
12	11	adantr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎 = 𝑐 ∧ 𝑏 = 𝑑) → (𝐹‘𝑎) = (𝐹‘𝑐))
13		fveq2 6645	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝑑 → (𝐹‘𝑏) = (𝐹‘𝑑))
14	13	adantl 485	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎 = 𝑐 ∧ 𝑏 = 𝑑) → (𝐹‘𝑏) = (𝐹‘𝑑))
15	12, 14	preq12d 4637	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 = 𝑐 ∧ 𝑏 = 𝑑) → {(𝐹‘𝑎), (𝐹‘𝑏)} = {(𝐹‘𝑐), (𝐹‘𝑑)})
16	15	eleq1d 2874	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 = 𝑐 ∧ 𝑏 = 𝑑) → ({(𝐹‘𝑎), (𝐹‘𝑏)} ∈ 𝐾 ↔ {(𝐹‘𝑐), (𝐹‘𝑑)} ∈ 𝐾))
17	10, 16	bibi12d 349	. . . . . . . . . . . . . 14 ⊢ ((𝑎 = 𝑐 ∧ 𝑏 = 𝑑) → (({𝑎, 𝑏} ∈ 𝐸 ↔ {(𝐹‘𝑎), (𝐹‘𝑏)} ∈ 𝐾) ↔ ({𝑐, 𝑑} ∈ 𝐸 ↔ {(𝐹‘𝑐), (𝐹‘𝑑)} ∈ 𝐾)))
18	17	rspc2gv 3580	. . . . . . . . . . . . 13 ⊢ ((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) → (∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ({𝑎, 𝑏} ∈ 𝐸 ↔ {(𝐹‘𝑎), (𝐹‘𝑏)} ∈ 𝐾) → ({𝑐, 𝑑} ∈ 𝐸 ↔ {(𝐹‘𝑐), (𝐹‘𝑑)} ∈ 𝐾)))
19	8, 18	syl5com 31	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) → ({𝑐, 𝑑} ∈ 𝐸 ↔ {(𝐹‘𝑐), (𝐹‘𝑑)} ∈ 𝐾)))
20	19	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 = {𝑐, 𝑑}) → ((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) → ({𝑐, 𝑑} ∈ 𝐸 ↔ {(𝐹‘𝑐), (𝐹‘𝑑)} ∈ 𝐾)))
21	20	imp 410	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 = {𝑐, 𝑑}) ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) → ({𝑐, 𝑑} ∈ 𝐸 ↔ {(𝐹‘𝑐), (𝐹‘𝑑)} ∈ 𝐾))
22		bicom 225	. . . . . . . . . . . . 13 ⊢ (({𝑐, 𝑑} ∈ 𝐸 ↔ {(𝐹‘𝑐), (𝐹‘𝑑)} ∈ 𝐾) ↔ ({(𝐹‘𝑐), (𝐹‘𝑑)} ∈ 𝐾 ↔ {𝑐, 𝑑} ∈ 𝐸))
23		bianir 1054	. . . . . . . . . . . . . 14 ⊢ (({𝑐, 𝑑} ∈ 𝐸 ∧ ({(𝐹‘𝑐), (𝐹‘𝑑)} ∈ 𝐾 ↔ {𝑐, 𝑑} ∈ 𝐸)) → {(𝐹‘𝑐), (𝐹‘𝑑)} ∈ 𝐾)
24	23	ex 416	. . . . . . . . . . . . 13 ⊢ ({𝑐, 𝑑} ∈ 𝐸 → (({(𝐹‘𝑐), (𝐹‘𝑑)} ∈ 𝐾 ↔ {𝑐, 𝑑} ∈ 𝐸) → {(𝐹‘𝑐), (𝐹‘𝑑)} ∈ 𝐾))
25	22, 24	syl5bi 245	. . . . . . . . . . . 12 ⊢ ({𝑐, 𝑑} ∈ 𝐸 → (({𝑐, 𝑑} ∈ 𝐸 ↔ {(𝐹‘𝑐), (𝐹‘𝑑)} ∈ 𝐾) → {(𝐹‘𝑐), (𝐹‘𝑑)} ∈ 𝐾))
26		isomuspgrlem2.f	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝑊)
27		f1ofn 6591	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐹:𝑉–1-1-onto→𝑊 → 𝐹 Fn 𝑉)
28	26, 27	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝐹 Fn 𝑉)
29	28	adantr 484	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 = {𝑐, 𝑑}) → 𝐹 Fn 𝑉)
30	29	adantr 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 = {𝑐, 𝑑}) ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) → 𝐹 Fn 𝑉)
31		simprl 770	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 = {𝑐, 𝑑}) ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) → 𝑐 ∈ 𝑉)
32		simprr 772	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 = {𝑐, 𝑑}) ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) → 𝑑 ∈ 𝑉)
33	30, 31, 32	3jca 1125	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 = {𝑐, 𝑑}) ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) → (𝐹 Fn 𝑉 ∧ 𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))
34	33	adantl 485	. . . . . . . . . . . . . . . . . 18 ⊢ (({𝑐, 𝑑} ∈ 𝐸 ∧ ((𝜑 ∧ 𝑥 = {𝑐, 𝑑}) ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))) → (𝐹 Fn 𝑉 ∧ 𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))
35		fnimapr 6722	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹 Fn 𝑉 ∧ 𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) → (𝐹 “ {𝑐, 𝑑}) = {(𝐹‘𝑐), (𝐹‘𝑑)})
36	34, 35	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (({𝑐, 𝑑} ∈ 𝐸 ∧ ((𝜑 ∧ 𝑥 = {𝑐, 𝑑}) ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))) → (𝐹 “ {𝑐, 𝑑}) = {(𝐹‘𝑐), (𝐹‘𝑑)})
37	36	eqcomd 2804	. . . . . . . . . . . . . . . 16 ⊢ (({𝑐, 𝑑} ∈ 𝐸 ∧ ((𝜑 ∧ 𝑥 = {𝑐, 𝑑}) ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))) → {(𝐹‘𝑐), (𝐹‘𝑑)} = (𝐹 “ {𝑐, 𝑑}))
38	37	eleq1d 2874	. . . . . . . . . . . . . . 15 ⊢ (({𝑐, 𝑑} ∈ 𝐸 ∧ ((𝜑 ∧ 𝑥 = {𝑐, 𝑑}) ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))) → ({(𝐹‘𝑐), (𝐹‘𝑑)} ∈ 𝐾 ↔ (𝐹 “ {𝑐, 𝑑}) ∈ 𝐾))
39	38	biimpd 232	. . . . . . . . . . . . . 14 ⊢ (({𝑐, 𝑑} ∈ 𝐸 ∧ ((𝜑 ∧ 𝑥 = {𝑐, 𝑑}) ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))) → ({(𝐹‘𝑐), (𝐹‘𝑑)} ∈ 𝐾 → (𝐹 “ {𝑐, 𝑑}) ∈ 𝐾))
40	39	ex 416	. . . . . . . . . . . . 13 ⊢ ({𝑐, 𝑑} ∈ 𝐸 → (((𝜑 ∧ 𝑥 = {𝑐, 𝑑}) ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) → ({(𝐹‘𝑐), (𝐹‘𝑑)} ∈ 𝐾 → (𝐹 “ {𝑐, 𝑑}) ∈ 𝐾)))
41	40	com23 86	. . . . . . . . . . . 12 ⊢ ({𝑐, 𝑑} ∈ 𝐸 → ({(𝐹‘𝑐), (𝐹‘𝑑)} ∈ 𝐾 → (((𝜑 ∧ 𝑥 = {𝑐, 𝑑}) ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) → (𝐹 “ {𝑐, 𝑑}) ∈ 𝐾)))
42	25, 41	syld 47	. . . . . . . . . . 11 ⊢ ({𝑐, 𝑑} ∈ 𝐸 → (({𝑐, 𝑑} ∈ 𝐸 ↔ {(𝐹‘𝑐), (𝐹‘𝑑)} ∈ 𝐾) → (((𝜑 ∧ 𝑥 = {𝑐, 𝑑}) ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) → (𝐹 “ {𝑐, 𝑑}) ∈ 𝐾)))
43	42	com13 88	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 = {𝑐, 𝑑}) ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) → (({𝑐, 𝑑} ∈ 𝐸 ↔ {(𝐹‘𝑐), (𝐹‘𝑑)} ∈ 𝐾) → ({𝑐, 𝑑} ∈ 𝐸 → (𝐹 “ {𝑐, 𝑑}) ∈ 𝐾)))
44	21, 43	mpd 15	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 = {𝑐, 𝑑}) ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) → ({𝑐, 𝑑} ∈ 𝐸 → (𝐹 “ {𝑐, 𝑑}) ∈ 𝐾))
45		eleq1 2877	. . . . . . . . . . . 12 ⊢ (𝑥 = {𝑐, 𝑑} → (𝑥 ∈ 𝐸 ↔ {𝑐, 𝑑} ∈ 𝐸))
46		imaeq2 5892	. . . . . . . . . . . . 13 ⊢ (𝑥 = {𝑐, 𝑑} → (𝐹 “ 𝑥) = (𝐹 “ {𝑐, 𝑑}))
47	46	eleq1d 2874	. . . . . . . . . . . 12 ⊢ (𝑥 = {𝑐, 𝑑} → ((𝐹 “ 𝑥) ∈ 𝐾 ↔ (𝐹 “ {𝑐, 𝑑}) ∈ 𝐾))
48	45, 47	imbi12d 348	. . . . . . . . . . 11 ⊢ (𝑥 = {𝑐, 𝑑} → ((𝑥 ∈ 𝐸 → (𝐹 “ 𝑥) ∈ 𝐾) ↔ ({𝑐, 𝑑} ∈ 𝐸 → (𝐹 “ {𝑐, 𝑑}) ∈ 𝐾)))
49	48	adantl 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 = {𝑐, 𝑑}) → ((𝑥 ∈ 𝐸 → (𝐹 “ 𝑥) ∈ 𝐾) ↔ ({𝑐, 𝑑} ∈ 𝐸 → (𝐹 “ {𝑐, 𝑑}) ∈ 𝐾)))
50	49	adantr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 = {𝑐, 𝑑}) ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) → ((𝑥 ∈ 𝐸 → (𝐹 “ 𝑥) ∈ 𝐾) ↔ ({𝑐, 𝑑} ∈ 𝐸 → (𝐹 “ {𝑐, 𝑑}) ∈ 𝐾)))
51	44, 50	mpbird 260	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 = {𝑐, 𝑑}) ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) → (𝑥 ∈ 𝐸 → (𝐹 “ 𝑥) ∈ 𝐾))
52	51	exp31 423	. . . . . . 7 ⊢ (𝜑 → (𝑥 = {𝑐, 𝑑} → ((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) → (𝑥 ∈ 𝐸 → (𝐹 “ 𝑥) ∈ 𝐾))))
53	52	com24 95	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐸 → ((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) → (𝑥 = {𝑐, 𝑑} → (𝐹 “ 𝑥) ∈ 𝐾))))
54	53	imp31 421	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐸) ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) → (𝑥 = {𝑐, 𝑑} → (𝐹 “ 𝑥) ∈ 𝐾))
55	54	rexlimdvva 3253	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐸) → (∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 𝑥 = {𝑐, 𝑑} → (𝐹 “ 𝑥) ∈ 𝐾))
56	7, 55	mpd 15	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐸) → (𝐹 “ 𝑥) ∈ 𝐾)
57	56	ralrimiva 3149	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐸 (𝐹 “ 𝑥) ∈ 𝐾)
58		isomuspgrlem2.g	. . 3 ⊢ 𝐺 = (𝑥 ∈ 𝐸 ↦ (𝐹 “ 𝑥))
59	58	fmpt 6851	. 2 ⊢ (∀𝑥 ∈ 𝐸 (𝐹 “ 𝑥) ∈ 𝐾 ↔ 𝐺:𝐸⟶𝐾)
60	57, 59	sylib 221	1 ⊢ (𝜑 → 𝐺:𝐸⟶𝐾)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107  {cpr 4527   ↦ cmpt 5110   “ cima 5522   Fn wfn 6319  ⟶wf 6320  –1-1-onto→wf1o 6323  ‘cfv 6324  Vtxcvtx 26789  Edgcedg 26840  UPGraphcupgr 26873  USPGraphcuspgr 26941

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-2o 8086  df-oadd 8089  df-er 8272  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-dju 9314  df-card 9352  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-2 11688  df-n0 11886  df-xnn0 11956  df-z 11970  df-uz 12232  df-fz 12886  df-hash 13687  df-edg 26841  df-upgr 26875  df-uspgr 26943

This theorem is referenced by:  isomuspgrlem2c  44348  isomuspgrlem2d  44349