Theorem isomuspgrlem2b 42734

Description: Lemma 2 for isomuspgrlem2 42738. (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 26525	. . . . . 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 26486	. . . . 5 ⊢ ((𝐴 ∈ UPGraph ∧ 𝑥 ∈ 𝐸) → ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 𝑥 = {𝑐, 𝑑})
7	3, 6	sylan 575	. . . 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 4501	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 = 𝑐 ∧ 𝑏 = 𝑑) → {𝑎, 𝑏} = {𝑐, 𝑑})
10	9	eleq1d 2843	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 = 𝑐 ∧ 𝑏 = 𝑑) → ({𝑎, 𝑏} ∈ 𝐸 ↔ {𝑐, 𝑑} ∈ 𝐸))
11		fveq2 6446	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = 𝑐 → (𝐹‘𝑎) = (𝐹‘𝑐))
12	11	adantr 474	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎 = 𝑐 ∧ 𝑏 = 𝑑) → (𝐹‘𝑎) = (𝐹‘𝑐))
13		fveq2 6446	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝑑 → (𝐹‘𝑏) = (𝐹‘𝑑))
14	13	adantl 475	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎 = 𝑐 ∧ 𝑏 = 𝑑) → (𝐹‘𝑏) = (𝐹‘𝑑))
15	12, 14	preq12d 4507	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 = 𝑐 ∧ 𝑏 = 𝑑) → {(𝐹‘𝑎), (𝐹‘𝑏)} = {(𝐹‘𝑐), (𝐹‘𝑑)})
16	15	eleq1d 2843	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 = 𝑐 ∧ 𝑏 = 𝑑) → ({(𝐹‘𝑎), (𝐹‘𝑏)} ∈ 𝐾 ↔ {(𝐹‘𝑐), (𝐹‘𝑑)} ∈ 𝐾))
17	10, 16	bibi12d 337	. . . . . . . . . . . . . 14 ⊢ ((𝑎 = 𝑐 ∧ 𝑏 = 𝑑) → (({𝑎, 𝑏} ∈ 𝐸 ↔ {(𝐹‘𝑎), (𝐹‘𝑏)} ∈ 𝐾) ↔ ({𝑐, 𝑑} ∈ 𝐸 ↔ {(𝐹‘𝑐), (𝐹‘𝑑)} ∈ 𝐾)))
18	17	rspc2gv 3522	. . . . . . . . . . . . 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 474	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 = {𝑐, 𝑑}) → ((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) → ({𝑐, 𝑑} ∈ 𝐸 ↔ {(𝐹‘𝑐), (𝐹‘𝑑)} ∈ 𝐾)))
21	20	imp 397	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 = {𝑐, 𝑑}) ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) → ({𝑐, 𝑑} ∈ 𝐸 ↔ {(𝐹‘𝑐), (𝐹‘𝑑)} ∈ 𝐾))
22		bicom 214	. . . . . . . . . . . . 13 ⊢ (({𝑐, 𝑑} ∈ 𝐸 ↔ {(𝐹‘𝑐), (𝐹‘𝑑)} ∈ 𝐾) ↔ ({(𝐹‘𝑐), (𝐹‘𝑑)} ∈ 𝐾 ↔ {𝑐, 𝑑} ∈ 𝐸))
23		bianir 1042	. . . . . . . . . . . . . 14 ⊢ (({𝑐, 𝑑} ∈ 𝐸 ∧ ({(𝐹‘𝑐), (𝐹‘𝑑)} ∈ 𝐾 ↔ {𝑐, 𝑑} ∈ 𝐸)) → {(𝐹‘𝑐), (𝐹‘𝑑)} ∈ 𝐾)
24	23	ex 403	. . . . . . . . . . . . 13 ⊢ ({𝑐, 𝑑} ∈ 𝐸 → (({(𝐹‘𝑐), (𝐹‘𝑑)} ∈ 𝐾 ↔ {𝑐, 𝑑} ∈ 𝐸) → {(𝐹‘𝑐), (𝐹‘𝑑)} ∈ 𝐾))
25	22, 24	syl5bi 234	. . . . . . . . . . . 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 6392	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐹:𝑉–1-1-onto→𝑊 → 𝐹 Fn 𝑉)
28	26, 27	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝐹 Fn 𝑉)
29	28	adantr 474	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 = {𝑐, 𝑑}) → 𝐹 Fn 𝑉)
30	29	adantr 474	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 = {𝑐, 𝑑}) ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) → 𝐹 Fn 𝑉)
31		simprl 761	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 = {𝑐, 𝑑}) ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) → 𝑐 ∈ 𝑉)
32		simprr 763	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 = {𝑐, 𝑑}) ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) → 𝑑 ∈ 𝑉)
33	30, 31, 32	3jca 1119	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 = {𝑐, 𝑑}) ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) → (𝐹 Fn 𝑉 ∧ 𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))
34	33	adantl 475	. . . . . . . . . . . . . . . . . 18 ⊢ (({𝑐, 𝑑} ∈ 𝐸 ∧ ((𝜑 ∧ 𝑥 = {𝑐, 𝑑}) ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))) → (𝐹 Fn 𝑉 ∧ 𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))
35		fnimapr 6522	. . . . . . . . . . . . . . . . . 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 2783	. . . . . . . . . . . . . . . 16 ⊢ (({𝑐, 𝑑} ∈ 𝐸 ∧ ((𝜑 ∧ 𝑥 = {𝑐, 𝑑}) ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))) → {(𝐹‘𝑐), (𝐹‘𝑑)} = (𝐹 “ {𝑐, 𝑑}))
38	37	eleq1d 2843	. . . . . . . . . . . . . . 15 ⊢ (({𝑐, 𝑑} ∈ 𝐸 ∧ ((𝜑 ∧ 𝑥 = {𝑐, 𝑑}) ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))) → ({(𝐹‘𝑐), (𝐹‘𝑑)} ∈ 𝐾 ↔ (𝐹 “ {𝑐, 𝑑}) ∈ 𝐾))
39	38	biimpd 221	. . . . . . . . . . . . . 14 ⊢ (({𝑐, 𝑑} ∈ 𝐸 ∧ ((𝜑 ∧ 𝑥 = {𝑐, 𝑑}) ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))) → ({(𝐹‘𝑐), (𝐹‘𝑑)} ∈ 𝐾 → (𝐹 “ {𝑐, 𝑑}) ∈ 𝐾))
40	39	ex 403	. . . . . . . . . . . . 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 2846	. . . . . . . . . . . 12 ⊢ (𝑥 = {𝑐, 𝑑} → (𝑥 ∈ 𝐸 ↔ {𝑐, 𝑑} ∈ 𝐸))
46		imaeq2 5716	. . . . . . . . . . . . 13 ⊢ (𝑥 = {𝑐, 𝑑} → (𝐹 “ 𝑥) = (𝐹 “ {𝑐, 𝑑}))
47	46	eleq1d 2843	. . . . . . . . . . . 12 ⊢ (𝑥 = {𝑐, 𝑑} → ((𝐹 “ 𝑥) ∈ 𝐾 ↔ (𝐹 “ {𝑐, 𝑑}) ∈ 𝐾))
48	45, 47	imbi12d 336	. . . . . . . . . . 11 ⊢ (𝑥 = {𝑐, 𝑑} → ((𝑥 ∈ 𝐸 → (𝐹 “ 𝑥) ∈ 𝐾) ↔ ({𝑐, 𝑑} ∈ 𝐸 → (𝐹 “ {𝑐, 𝑑}) ∈ 𝐾)))
49	48	adantl 475	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 = {𝑐, 𝑑}) → ((𝑥 ∈ 𝐸 → (𝐹 “ 𝑥) ∈ 𝐾) ↔ ({𝑐, 𝑑} ∈ 𝐸 → (𝐹 “ {𝑐, 𝑑}) ∈ 𝐾)))
50	49	adantr 474	. . . . . . . . 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 249	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 = {𝑐, 𝑑}) ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) → (𝑥 ∈ 𝐸 → (𝐹 “ 𝑥) ∈ 𝐾))
52	51	exp31 412	. . . . . . 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 410	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐸) ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) → (𝑥 = {𝑐, 𝑑} → (𝐹 “ 𝑥) ∈ 𝐾))
55	54	rexlimdvva 3220	. . . 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 3147	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐸 (𝐹 “ 𝑥) ∈ 𝐾)
58		isomuspgrlem2.g	. . 3 ⊢ 𝐺 = (𝑥 ∈ 𝐸 ↦ (𝐹 “ 𝑥))
59	58	fmpt 6644	. 2 ⊢ (∀𝑥 ∈ 𝐸 (𝐹 “ 𝑥) ∈ 𝐾 ↔ 𝐺:𝐸⟶𝐾)
60	57, 59	sylib 210	1 ⊢ (𝜑 → 𝐺:𝐸⟶𝐾)

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   ∧ w3a 1071   = wceq 1601   ∈ wcel 2106  ∀wral 3089  ∃wrex 3090  {cpr 4399   ↦ cmpt 4965   “ cima 5358   Fn wfn 6130  ⟶wf 6131  –1-1-onto→wf1o 6134  ‘cfv 6135  Vtxcvtx 26344  Edgcedg 26395  UPGraphcupgr 26428  USPGraphcuspgr 26497

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-cnex 10328  ax-resscn 10329  ax-1cn 10330  ax-icn 10331  ax-addcl 10332  ax-addrcl 10333  ax-mulcl 10334  ax-mulrcl 10335  ax-mulcom 10336  ax-addass 10337  ax-mulass 10338  ax-distr 10339  ax-i2m1 10340  ax-1ne0 10341  ax-1rid 10342  ax-rnegex 10343  ax-rrecex 10344  ax-cnre 10345  ax-pre-lttri 10346  ax-pre-lttrn 10347  ax-pre-ltadd 10348  ax-pre-mulgt0 10349

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3399  df-sbc 3652  df-csb 3751  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-pss 3807  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4672  df-int 4711  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-1st 7445  df-2nd 7446  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-1o 7843  df-2o 7844  df-oadd 7847  df-er 8026  df-en 8242  df-dom 8243  df-sdom 8244  df-fin 8245  df-card 9098  df-cda 9325  df-pnf 10413  df-mnf 10414  df-xr 10415  df-ltxr 10416  df-le 10417  df-sub 10608  df-neg 10609  df-nn 11375  df-2 11438  df-n0 11643  df-xnn0 11715  df-z 11729  df-uz 11993  df-fz 12644  df-hash 13436  df-edg 26396  df-upgr 26430  df-uspgr 26499

This theorem is referenced by:  isomuspgrlem2c  42735  isomuspgrlem2d  42736