Theorem uspgrlimlem3 48755

Description: Lemma 3 for uspgrlim 48757. (Contributed by AV, 16-Aug-2025.)

Hypotheses

Ref	Expression
uspgrlim.v	⊢ 𝑉 = (Vtx‘𝐺)
uspgrlim.w	⊢ 𝑊 = (Vtx‘𝐻)
uspgrlim.n	⊢ 𝑁 = (𝐺 ClNeighbVtx 𝑣)
uspgrlim.m	⊢ 𝑀 = (𝐻 ClNeighbVtx (𝐹‘𝑣))
uspgrlim.i	⊢ 𝐼 = (Edg‘𝐺)
uspgrlim.j	⊢ 𝐽 = (Edg‘𝐻)
uspgrlim.k	⊢ 𝐾 = {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁}
uspgrlim.l	⊢ 𝐿 = {𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ 𝑀}

Assertion

Ref	Expression
uspgrlimlem3	⊢ ((𝐺 ∈ USPGraph ∧ ℎ:{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑁}–1-1-onto→𝑅 ∧ ∀𝑖 ∈ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑁} (𝑓 “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(ℎ‘𝑖))) → (𝑒 ∈ 𝐾 → (𝑓 “ 𝑒) = ((((iEdg‘𝐻) ∘ ℎ) ∘ ◡(iEdg‘𝐺))‘𝑒)))

Distinct variable groups:   𝑖,𝐺,𝑥   𝑖,𝐻,𝑥   𝑥,𝐼   𝑥,𝐽   𝑥,𝑀   𝑖,𝑁,𝑥   𝑒,𝑖,𝑥   𝑓,𝑖   ℎ,𝑖

Allowed substitution hints:   𝑅(𝑥,𝑣,𝑒,𝑓,ℎ,𝑖)   𝐹(𝑥,𝑣,𝑒,𝑓,ℎ,𝑖)   𝐺(𝑣,𝑒,𝑓,ℎ)   𝐻(𝑣,𝑒,𝑓,ℎ)   𝐼(𝑣,𝑒,𝑓,ℎ,𝑖)   𝐽(𝑣,𝑒,𝑓,ℎ,𝑖)   𝐾(𝑥,𝑣,𝑒,𝑓,ℎ,𝑖)   𝐿(𝑥,𝑣,𝑒,𝑓,ℎ,𝑖)   𝑀(𝑣,𝑒,𝑓,ℎ,𝑖)   𝑁(𝑣,𝑒,𝑓,ℎ)   𝑉(𝑥,𝑣,𝑒,𝑓,ℎ,𝑖)   𝑊(𝑥,𝑣,𝑒,𝑓,ℎ,𝑖)

Proof of Theorem uspgrlimlem3

Step	Hyp	Ref	Expression
1		sseq1 3962	. . 3 ⊢ (𝑥 = 𝑒 → (𝑥 ⊆ 𝑁 ↔ 𝑒 ⊆ 𝑁))
2		uspgrlim.k	. . 3 ⊢ 𝐾 = {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁}
3	1, 2	elrab2 3654	. 2 ⊢ (𝑒 ∈ 𝐾 ↔ (𝑒 ∈ 𝐼 ∧ 𝑒 ⊆ 𝑁))
4		eqid 2763	. . . . . . . 8 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
5	4	uspgrf1oedg 29523	. . . . . . 7 ⊢ (𝐺 ∈ USPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1-onto→(Edg‘𝐺))
6		f1ocnv 6833	. . . . . . 7 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1-onto→(Edg‘𝐺) → ◡(iEdg‘𝐺):(Edg‘𝐺)–1-1-onto→dom (iEdg‘𝐺))
7		f1of 6820	. . . . . . 7 ⊢ (◡(iEdg‘𝐺):(Edg‘𝐺)–1-1-onto→dom (iEdg‘𝐺) → ◡(iEdg‘𝐺):(Edg‘𝐺)⟶dom (iEdg‘𝐺))
8	5, 6, 7	3syl 19	. . . . . 6 ⊢ (𝐺 ∈ USPGraph → ◡(iEdg‘𝐺):(Edg‘𝐺)⟶dom (iEdg‘𝐺))
9	8	3ad2ant1 1151	. . . . 5 ⊢ ((𝐺 ∈ USPGraph ∧ ℎ:{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑁}–1-1-onto→𝑅 ∧ ∀𝑖 ∈ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑁} (𝑓 “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(ℎ‘𝑖))) → ◡(iEdg‘𝐺):(Edg‘𝐺)⟶dom (iEdg‘𝐺))
10		uspgrlim.i	. . . . . . 7 ⊢ 𝐼 = (Edg‘𝐺)
11	10	eleq2i 2855	. . . . . 6 ⊢ (𝑒 ∈ 𝐼 ↔ 𝑒 ∈ (Edg‘𝐺))
12	11	birani 508	. . . . 5 ⊢ ((𝑒 ∈ 𝐼 ∧ 𝑒 ⊆ 𝑁) → 𝑒 ∈ (Edg‘𝐺))
13		fvco3 6981	. . . . 5 ⊢ ((◡(iEdg‘𝐺):(Edg‘𝐺)⟶dom (iEdg‘𝐺) ∧ 𝑒 ∈ (Edg‘𝐺)) → ((((iEdg‘𝐻) ∘ ℎ) ∘ ◡(iEdg‘𝐺))‘𝑒) = (((iEdg‘𝐻) ∘ ℎ)‘(◡(iEdg‘𝐺)‘𝑒)))
14	9, 12, 13	syl2an 607	. . . 4 ⊢ (((𝐺 ∈ USPGraph ∧ ℎ:{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑁}–1-1-onto→𝑅 ∧ ∀𝑖 ∈ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑁} (𝑓 “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(ℎ‘𝑖))) ∧ (𝑒 ∈ 𝐼 ∧ 𝑒 ⊆ 𝑁)) → ((((iEdg‘𝐻) ∘ ℎ) ∘ ◡(iEdg‘𝐺))‘𝑒) = (((iEdg‘𝐻) ∘ ℎ)‘(◡(iEdg‘𝐺)‘𝑒)))
15		f1ocnvdm 7283	. . . . . . . . . . . . . 14 ⊢ (((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1-onto→(Edg‘𝐺) ∧ 𝑒 ∈ (Edg‘𝐺)) → (◡(iEdg‘𝐺)‘𝑒) ∈ dom (iEdg‘𝐺))
16	5, 12, 15	syl2an 607	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑒 ∈ 𝐼 ∧ 𝑒 ⊆ 𝑁)) → (◡(iEdg‘𝐺)‘𝑒) ∈ dom (iEdg‘𝐺))
17		f1ocnvfv2 7275	. . . . . . . . . . . . . . 15 ⊢ (((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1-onto→(Edg‘𝐺) ∧ 𝑒 ∈ (Edg‘𝐺)) → ((iEdg‘𝐺)‘(◡(iEdg‘𝐺)‘𝑒)) = 𝑒)
18	5, 12, 17	syl2an 607	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑒 ∈ 𝐼 ∧ 𝑒 ⊆ 𝑁)) → ((iEdg‘𝐺)‘(◡(iEdg‘𝐺)‘𝑒)) = 𝑒)
19		simprr 784	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑒 ∈ 𝐼 ∧ 𝑒 ⊆ 𝑁)) → 𝑒 ⊆ 𝑁)
20	18, 19	eqsstrd 3971	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑒 ∈ 𝐼 ∧ 𝑒 ⊆ 𝑁)) → ((iEdg‘𝐺)‘(◡(iEdg‘𝐺)‘𝑒)) ⊆ 𝑁)
21	16, 20	jca 520	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑒 ∈ 𝐼 ∧ 𝑒 ⊆ 𝑁)) → ((◡(iEdg‘𝐺)‘𝑒) ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘(◡(iEdg‘𝐺)‘𝑒)) ⊆ 𝑁))
22	21	adantlr 727	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ USPGraph ∧ ℎ:{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑁}–1-1-onto→𝑅) ∧ (𝑒 ∈ 𝐼 ∧ 𝑒 ⊆ 𝑁)) → ((◡(iEdg‘𝐺)‘𝑒) ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘(◡(iEdg‘𝐺)‘𝑒)) ⊆ 𝑁))
23		fveq2 6881	. . . . . . . . . . . . 13 ⊢ (𝑥 = (◡(iEdg‘𝐺)‘𝑒) → ((iEdg‘𝐺)‘𝑥) = ((iEdg‘𝐺)‘(◡(iEdg‘𝐺)‘𝑒)))
24	23	sseq1d 3968	. . . . . . . . . . . 12 ⊢ (𝑥 = (◡(iEdg‘𝐺)‘𝑒) → (((iEdg‘𝐺)‘𝑥) ⊆ 𝑁 ↔ ((iEdg‘𝐺)‘(◡(iEdg‘𝐺)‘𝑒)) ⊆ 𝑁))
25	24	elrab 3650	. . . . . . . . . . 11 ⊢ ((◡(iEdg‘𝐺)‘𝑒) ∈ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑁} ↔ ((◡(iEdg‘𝐺)‘𝑒) ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘(◡(iEdg‘𝐺)‘𝑒)) ⊆ 𝑁))
26	22, 25	sylibr 237	. . . . . . . . . 10 ⊢ (((𝐺 ∈ USPGraph ∧ ℎ:{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑁}–1-1-onto→𝑅) ∧ (𝑒 ∈ 𝐼 ∧ 𝑒 ⊆ 𝑁)) → (◡(iEdg‘𝐺)‘𝑒) ∈ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑁})
27		fveq2 6881	. . . . . . . . . . . . 13 ⊢ (𝑖 = (◡(iEdg‘𝐺)‘𝑒) → ((iEdg‘𝐺)‘𝑖) = ((iEdg‘𝐺)‘(◡(iEdg‘𝐺)‘𝑒)))
28	27	imaeq2d 6062	. . . . . . . . . . . 12 ⊢ (𝑖 = (◡(iEdg‘𝐺)‘𝑒) → (𝑓 “ ((iEdg‘𝐺)‘𝑖)) = (𝑓 “ ((iEdg‘𝐺)‘(◡(iEdg‘𝐺)‘𝑒))))
29		2fveq3 6886	. . . . . . . . . . . 12 ⊢ (𝑖 = (◡(iEdg‘𝐺)‘𝑒) → ((iEdg‘𝐻)‘(ℎ‘𝑖)) = ((iEdg‘𝐻)‘(ℎ‘(◡(iEdg‘𝐺)‘𝑒))))
30	28, 29	eqeq12d 2779	. . . . . . . . . . 11 ⊢ (𝑖 = (◡(iEdg‘𝐺)‘𝑒) → ((𝑓 “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(ℎ‘𝑖)) ↔ (𝑓 “ ((iEdg‘𝐺)‘(◡(iEdg‘𝐺)‘𝑒))) = ((iEdg‘𝐻)‘(ℎ‘(◡(iEdg‘𝐺)‘𝑒)))))
31	30	rspcv 3577	. . . . . . . . . 10 ⊢ ((◡(iEdg‘𝐺)‘𝑒) ∈ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑁} → (∀𝑖 ∈ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑁} (𝑓 “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(ℎ‘𝑖)) → (𝑓 “ ((iEdg‘𝐺)‘(◡(iEdg‘𝐺)‘𝑒))) = ((iEdg‘𝐻)‘(ℎ‘(◡(iEdg‘𝐺)‘𝑒)))))
32	26, 31	syl 18	. . . . . . . . 9 ⊢ (((𝐺 ∈ USPGraph ∧ ℎ:{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑁}–1-1-onto→𝑅) ∧ (𝑒 ∈ 𝐼 ∧ 𝑒 ⊆ 𝑁)) → (∀𝑖 ∈ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑁} (𝑓 “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(ℎ‘𝑖)) → (𝑓 “ ((iEdg‘𝐺)‘(◡(iEdg‘𝐺)‘𝑒))) = ((iEdg‘𝐻)‘(ℎ‘(◡(iEdg‘𝐺)‘𝑒)))))
33		eqcom 2770	. . . . . . . . . 10 ⊢ ((𝑓 “ ((iEdg‘𝐺)‘(◡(iEdg‘𝐺)‘𝑒))) = ((iEdg‘𝐻)‘(ℎ‘(◡(iEdg‘𝐺)‘𝑒))) ↔ ((iEdg‘𝐻)‘(ℎ‘(◡(iEdg‘𝐺)‘𝑒))) = (𝑓 “ ((iEdg‘𝐺)‘(◡(iEdg‘𝐺)‘𝑒))))
34		f1of 6820	. . . . . . . . . . . . . . 15 ⊢ (ℎ:{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑁}–1-1-onto→𝑅 → ℎ:{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑁}⟶𝑅)
35	34	ad2antlr 739	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ USPGraph ∧ ℎ:{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑁}–1-1-onto→𝑅) ∧ (𝑒 ∈ 𝐼 ∧ 𝑒 ⊆ 𝑁)) → ℎ:{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑁}⟶𝑅)
36	35, 26	fvco3d 6982	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ USPGraph ∧ ℎ:{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑁}–1-1-onto→𝑅) ∧ (𝑒 ∈ 𝐼 ∧ 𝑒 ⊆ 𝑁)) → (((iEdg‘𝐻) ∘ ℎ)‘(◡(iEdg‘𝐺)‘𝑒)) = ((iEdg‘𝐻)‘(ℎ‘(◡(iEdg‘𝐺)‘𝑒))))
37	36	eqcomd 2769	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ USPGraph ∧ ℎ:{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑁}–1-1-onto→𝑅) ∧ (𝑒 ∈ 𝐼 ∧ 𝑒 ⊆ 𝑁)) → ((iEdg‘𝐻)‘(ℎ‘(◡(iEdg‘𝐺)‘𝑒))) = (((iEdg‘𝐻) ∘ ℎ)‘(◡(iEdg‘𝐺)‘𝑒)))
38	5	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ USPGraph ∧ ℎ:{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑁}–1-1-onto→𝑅) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1-onto→(Edg‘𝐺))
39	38, 12, 17	syl2an 607	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ USPGraph ∧ ℎ:{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑁}–1-1-onto→𝑅) ∧ (𝑒 ∈ 𝐼 ∧ 𝑒 ⊆ 𝑁)) → ((iEdg‘𝐺)‘(◡(iEdg‘𝐺)‘𝑒)) = 𝑒)
40	39	imaeq2d 6062	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ USPGraph ∧ ℎ:{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑁}–1-1-onto→𝑅) ∧ (𝑒 ∈ 𝐼 ∧ 𝑒 ⊆ 𝑁)) → (𝑓 “ ((iEdg‘𝐺)‘(◡(iEdg‘𝐺)‘𝑒))) = (𝑓 “ 𝑒))
41	37, 40	eqeq12d 2779	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ USPGraph ∧ ℎ:{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑁}–1-1-onto→𝑅) ∧ (𝑒 ∈ 𝐼 ∧ 𝑒 ⊆ 𝑁)) → (((iEdg‘𝐻)‘(ℎ‘(◡(iEdg‘𝐺)‘𝑒))) = (𝑓 “ ((iEdg‘𝐺)‘(◡(iEdg‘𝐺)‘𝑒))) ↔ (((iEdg‘𝐻) ∘ ℎ)‘(◡(iEdg‘𝐺)‘𝑒)) = (𝑓 “ 𝑒)))
42	41	biimpd 232	. . . . . . . . . 10 ⊢ (((𝐺 ∈ USPGraph ∧ ℎ:{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑁}–1-1-onto→𝑅) ∧ (𝑒 ∈ 𝐼 ∧ 𝑒 ⊆ 𝑁)) → (((iEdg‘𝐻)‘(ℎ‘(◡(iEdg‘𝐺)‘𝑒))) = (𝑓 “ ((iEdg‘𝐺)‘(◡(iEdg‘𝐺)‘𝑒))) → (((iEdg‘𝐻) ∘ ℎ)‘(◡(iEdg‘𝐺)‘𝑒)) = (𝑓 “ 𝑒)))
43	33, 42	biimtrid 245	. . . . . . . . 9 ⊢ (((𝐺 ∈ USPGraph ∧ ℎ:{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑁}–1-1-onto→𝑅) ∧ (𝑒 ∈ 𝐼 ∧ 𝑒 ⊆ 𝑁)) → ((𝑓 “ ((iEdg‘𝐺)‘(◡(iEdg‘𝐺)‘𝑒))) = ((iEdg‘𝐻)‘(ℎ‘(◡(iEdg‘𝐺)‘𝑒))) → (((iEdg‘𝐻) ∘ ℎ)‘(◡(iEdg‘𝐺)‘𝑒)) = (𝑓 “ 𝑒)))
44	32, 43	syld 48	. . . . . . . 8 ⊢ (((𝐺 ∈ USPGraph ∧ ℎ:{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑁}–1-1-onto→𝑅) ∧ (𝑒 ∈ 𝐼 ∧ 𝑒 ⊆ 𝑁)) → (∀𝑖 ∈ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑁} (𝑓 “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(ℎ‘𝑖)) → (((iEdg‘𝐻) ∘ ℎ)‘(◡(iEdg‘𝐺)‘𝑒)) = (𝑓 “ 𝑒)))
45	44	ex 417	. . . . . . 7 ⊢ ((𝐺 ∈ USPGraph ∧ ℎ:{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑁}–1-1-onto→𝑅) → ((𝑒 ∈ 𝐼 ∧ 𝑒 ⊆ 𝑁) → (∀𝑖 ∈ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑁} (𝑓 “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(ℎ‘𝑖)) → (((iEdg‘𝐻) ∘ ℎ)‘(◡(iEdg‘𝐺)‘𝑒)) = (𝑓 “ 𝑒))))
46	45	com23 87	. . . . . 6 ⊢ ((𝐺 ∈ USPGraph ∧ ℎ:{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑁}–1-1-onto→𝑅) → (∀𝑖 ∈ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑁} (𝑓 “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(ℎ‘𝑖)) → ((𝑒 ∈ 𝐼 ∧ 𝑒 ⊆ 𝑁) → (((iEdg‘𝐻) ∘ ℎ)‘(◡(iEdg‘𝐺)‘𝑒)) = (𝑓 “ 𝑒))))
47	46	ex 417	. . . . 5 ⊢ (𝐺 ∈ USPGraph → (ℎ:{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑁}–1-1-onto→𝑅 → (∀𝑖 ∈ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑁} (𝑓 “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(ℎ‘𝑖)) → ((𝑒 ∈ 𝐼 ∧ 𝑒 ⊆ 𝑁) → (((iEdg‘𝐻) ∘ ℎ)‘(◡(iEdg‘𝐺)‘𝑒)) = (𝑓 “ 𝑒)))))
48	47	3imp1 1366	. . . 4 ⊢ (((𝐺 ∈ USPGraph ∧ ℎ:{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑁}–1-1-onto→𝑅 ∧ ∀𝑖 ∈ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑁} (𝑓 “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(ℎ‘𝑖))) ∧ (𝑒 ∈ 𝐼 ∧ 𝑒 ⊆ 𝑁)) → (((iEdg‘𝐻) ∘ ℎ)‘(◡(iEdg‘𝐺)‘𝑒)) = (𝑓 “ 𝑒))
49	14, 48	eqtr2d 2799	. . 3 ⊢ (((𝐺 ∈ USPGraph ∧ ℎ:{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑁}–1-1-onto→𝑅 ∧ ∀𝑖 ∈ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑁} (𝑓 “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(ℎ‘𝑖))) ∧ (𝑒 ∈ 𝐼 ∧ 𝑒 ⊆ 𝑁)) → (𝑓 “ 𝑒) = ((((iEdg‘𝐻) ∘ ℎ) ∘ ◡(iEdg‘𝐺))‘𝑒))
50	49	ex 417	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ ℎ:{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑁}–1-1-onto→𝑅 ∧ ∀𝑖 ∈ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑁} (𝑓 “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(ℎ‘𝑖))) → ((𝑒 ∈ 𝐼 ∧ 𝑒 ⊆ 𝑁) → (𝑓 “ 𝑒) = ((((iEdg‘𝐻) ∘ ℎ) ∘ ◡(iEdg‘𝐺))‘𝑒)))
51	3, 50	biimtrid 245	1 ⊢ ((𝐺 ∈ USPGraph ∧ ℎ:{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑁}–1-1-onto→𝑅 ∧ ∀𝑖 ∈ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑁} (𝑓 “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(ℎ‘𝑖))) → (𝑒 ∈ 𝐾 → (𝑓 “ 𝑒) = ((((iEdg‘𝐻) ∘ ℎ) ∘ ◡(iEdg‘𝐺))‘𝑒)))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ∀wral 3079  {crab 3416   ⊆ wss 3905  ^◡ccnv 5660  dom cdm 5661   “ cima 5664   ∘ ccom 5665  ⟶wf 6532  –1-1-onto→wf1o 6535  ‘cfv 6536  (class class class)co 7410  Vtxcvtx 29346  iEdgciedg 29347  Edgcedg 29397  USPGraphcuspgr 29498   ClNeighbVtx cclnbgr 48583

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3745  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-edg 29398  df-uspgr 29500

This theorem is referenced by:  uspgrlim  48757