Theorem isomuspgrlem2b 44043

Description: Lemma 2 for isomuspgrlem2 44047. (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 26961	. . . . . 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 26922	. . . . 5 ⊢ ((𝐴 ∈ UPGraph ∧ 𝑥 ∈ 𝐸) → ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 𝑥 = {𝑐, 𝑑})
7	3, 6	sylan 582	. . . 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 4671	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 = 𝑐 ∧ 𝑏 = 𝑑) → {𝑎, 𝑏} = {𝑐, 𝑑})
10	9	eleq1d 2897	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 = 𝑐 ∧ 𝑏 = 𝑑) → ({𝑎, 𝑏} ∈ 𝐸 ↔ {𝑐, 𝑑} ∈ 𝐸))
11		fveq2 6670	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = 𝑐 → (𝐹‘𝑎) = (𝐹‘𝑐))
12	11	adantr 483	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎 = 𝑐 ∧ 𝑏 = 𝑑) → (𝐹‘𝑎) = (𝐹‘𝑐))
13		fveq2 6670	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝑑 → (𝐹‘𝑏) = (𝐹‘𝑑))
14	13	adantl 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎 = 𝑐 ∧ 𝑏 = 𝑑) → (𝐹‘𝑏) = (𝐹‘𝑑))
15	12, 14	preq12d 4677	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 = 𝑐 ∧ 𝑏 = 𝑑) → {(𝐹‘𝑎), (𝐹‘𝑏)} = {(𝐹‘𝑐), (𝐹‘𝑑)})
16	15	eleq1d 2897	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 = 𝑐 ∧ 𝑏 = 𝑑) → ({(𝐹‘𝑎), (𝐹‘𝑏)} ∈ 𝐾 ↔ {(𝐹‘𝑐), (𝐹‘𝑑)} ∈ 𝐾))
17	10, 16	bibi12d 348	. . . . . . . . . . . . . 14 ⊢ ((𝑎 = 𝑐 ∧ 𝑏 = 𝑑) → (({𝑎, 𝑏} ∈ 𝐸 ↔ {(𝐹‘𝑎), (𝐹‘𝑏)} ∈ 𝐾) ↔ ({𝑐, 𝑑} ∈ 𝐸 ↔ {(𝐹‘𝑐), (𝐹‘𝑑)} ∈ 𝐾)))
18	17	rspc2gv 3632	. . . . . . . . . . . . 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 483	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 = {𝑐, 𝑑}) → ((𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) → ({𝑐, 𝑑} ∈ 𝐸 ↔ {(𝐹‘𝑐), (𝐹‘𝑑)} ∈ 𝐾)))
21	20	imp 409	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 = {𝑐, 𝑑}) ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) → ({𝑐, 𝑑} ∈ 𝐸 ↔ {(𝐹‘𝑐), (𝐹‘𝑑)} ∈ 𝐾))
22		bicom 224	. . . . . . . . . . . . 13 ⊢ (({𝑐, 𝑑} ∈ 𝐸 ↔ {(𝐹‘𝑐), (𝐹‘𝑑)} ∈ 𝐾) ↔ ({(𝐹‘𝑐), (𝐹‘𝑑)} ∈ 𝐾 ↔ {𝑐, 𝑑} ∈ 𝐸))
23		bianir 1053	. . . . . . . . . . . . . 14 ⊢ (({𝑐, 𝑑} ∈ 𝐸 ∧ ({(𝐹‘𝑐), (𝐹‘𝑑)} ∈ 𝐾 ↔ {𝑐, 𝑑} ∈ 𝐸)) → {(𝐹‘𝑐), (𝐹‘𝑑)} ∈ 𝐾)
24	23	ex 415	. . . . . . . . . . . . 13 ⊢ ({𝑐, 𝑑} ∈ 𝐸 → (({(𝐹‘𝑐), (𝐹‘𝑑)} ∈ 𝐾 ↔ {𝑐, 𝑑} ∈ 𝐸) → {(𝐹‘𝑐), (𝐹‘𝑑)} ∈ 𝐾))
25	22, 24	syl5bi 244	. . . . . . . . . . . 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 6616	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐹:𝑉–1-1-onto→𝑊 → 𝐹 Fn 𝑉)
28	26, 27	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝐹 Fn 𝑉)
29	28	adantr 483	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 = {𝑐, 𝑑}) → 𝐹 Fn 𝑉)
30	29	adantr 483	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 = {𝑐, 𝑑}) ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) → 𝐹 Fn 𝑉)
31		simprl 769	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 = {𝑐, 𝑑}) ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) → 𝑐 ∈ 𝑉)
32		simprr 771	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 = {𝑐, 𝑑}) ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) → 𝑑 ∈ 𝑉)
33	30, 31, 32	3jca 1124	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 = {𝑐, 𝑑}) ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) → (𝐹 Fn 𝑉 ∧ 𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))
34	33	adantl 484	. . . . . . . . . . . . . . . . . 18 ⊢ (({𝑐, 𝑑} ∈ 𝐸 ∧ ((𝜑 ∧ 𝑥 = {𝑐, 𝑑}) ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))) → (𝐹 Fn 𝑉 ∧ 𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))
35		fnimapr 6747	. . . . . . . . . . . . . . . . . 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 2827	. . . . . . . . . . . . . . . 16 ⊢ (({𝑐, 𝑑} ∈ 𝐸 ∧ ((𝜑 ∧ 𝑥 = {𝑐, 𝑑}) ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))) → {(𝐹‘𝑐), (𝐹‘𝑑)} = (𝐹 “ {𝑐, 𝑑}))
38	37	eleq1d 2897	. . . . . . . . . . . . . . 15 ⊢ (({𝑐, 𝑑} ∈ 𝐸 ∧ ((𝜑 ∧ 𝑥 = {𝑐, 𝑑}) ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))) → ({(𝐹‘𝑐), (𝐹‘𝑑)} ∈ 𝐾 ↔ (𝐹 “ {𝑐, 𝑑}) ∈ 𝐾))
39	38	biimpd 231	. . . . . . . . . . . . . 14 ⊢ (({𝑐, 𝑑} ∈ 𝐸 ∧ ((𝜑 ∧ 𝑥 = {𝑐, 𝑑}) ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉))) → ({(𝐹‘𝑐), (𝐹‘𝑑)} ∈ 𝐾 → (𝐹 “ {𝑐, 𝑑}) ∈ 𝐾))
40	39	ex 415	. . . . . . . . . . . . 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 2900	. . . . . . . . . . . 12 ⊢ (𝑥 = {𝑐, 𝑑} → (𝑥 ∈ 𝐸 ↔ {𝑐, 𝑑} ∈ 𝐸))
46		imaeq2 5925	. . . . . . . . . . . . 13 ⊢ (𝑥 = {𝑐, 𝑑} → (𝐹 “ 𝑥) = (𝐹 “ {𝑐, 𝑑}))
47	46	eleq1d 2897	. . . . . . . . . . . 12 ⊢ (𝑥 = {𝑐, 𝑑} → ((𝐹 “ 𝑥) ∈ 𝐾 ↔ (𝐹 “ {𝑐, 𝑑}) ∈ 𝐾))
48	45, 47	imbi12d 347	. . . . . . . . . . 11 ⊢ (𝑥 = {𝑐, 𝑑} → ((𝑥 ∈ 𝐸 → (𝐹 “ 𝑥) ∈ 𝐾) ↔ ({𝑐, 𝑑} ∈ 𝐸 → (𝐹 “ {𝑐, 𝑑}) ∈ 𝐾)))
49	48	adantl 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 = {𝑐, 𝑑}) → ((𝑥 ∈ 𝐸 → (𝐹 “ 𝑥) ∈ 𝐾) ↔ ({𝑐, 𝑑} ∈ 𝐸 → (𝐹 “ {𝑐, 𝑑}) ∈ 𝐾)))
50	49	adantr 483	. . . . . . . . 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 259	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 = {𝑐, 𝑑}) ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) → (𝑥 ∈ 𝐸 → (𝐹 “ 𝑥) ∈ 𝐾))
52	51	exp31 422	. . . . . . 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 420	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐸) ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) → (𝑥 = {𝑐, 𝑑} → (𝐹 “ 𝑥) ∈ 𝐾))
55	54	rexlimdvva 3294	. . . 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 3182	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐸 (𝐹 “ 𝑥) ∈ 𝐾)
58		isomuspgrlem2.g	. . 3 ⊢ 𝐺 = (𝑥 ∈ 𝐸 ↦ (𝐹 “ 𝑥))
59	58	fmpt 6874	. 2 ⊢ (∀𝑥 ∈ 𝐸 (𝐹 “ 𝑥) ∈ 𝐾 ↔ 𝐺:𝐸⟶𝐾)
60	57, 59	sylib 220	1 ⊢ (𝜑 → 𝐺:𝐸⟶𝐾)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  ∀wral 3138  ∃wrex 3139  {cpr 4569   ↦ cmpt 5146   “ cima 5558   Fn wfn 6350  ⟶wf 6351  –1-1-onto→wf1o 6354  ‘cfv 6355  Vtxcvtx 26781  Edgcedg 26832  UPGraphcupgr 26865  USPGraphcuspgr 26933

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-1st 7689  df-2nd 7690  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-1o 8102  df-2o 8103  df-oadd 8106  df-er 8289  df-en 8510  df-dom 8511  df-sdom 8512  df-fin 8513  df-dju 9330  df-card 9368  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-nn 11639  df-2 11701  df-n0 11899  df-xnn0 11969  df-z 11983  df-uz 12245  df-fz 12894  df-hash 13692  df-edg 26833  df-upgr 26867  df-uspgr 26935

This theorem is referenced by:  isomuspgrlem2c  44044  isomuspgrlem2d  44045