Theorem uspgrlimlem3 48573

Description: Lemma 3 for uspgrlim 48575. (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 3959	. . 3 ⊢ (𝑥 = 𝑒 → (𝑥 ⊆ 𝑁 ↔ 𝑒 ⊆ 𝑁))
2		uspgrlim.k	. . 3 ⊢ 𝐾 = {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁}
3	1, 2	elrab2 3652	. 2 ⊢ (𝑒 ∈ 𝐾 ↔ (𝑒 ∈ 𝐼 ∧ 𝑒 ⊆ 𝑁))
4		eqid 2761	. . . . . . . 8 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
5	4	uspgrf1oedg 29331	. . . . . . 7 ⊢ (𝐺 ∈ USPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1-onto→(Edg‘𝐺))
6		f1ocnv 6814	. . . . . . 7 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1-onto→(Edg‘𝐺) → ◡(iEdg‘𝐺):(Edg‘𝐺)–1-1-onto→dom (iEdg‘𝐺))
7		f1of 6801	. . . . . . 7 ⊢ (◡(iEdg‘𝐺):(Edg‘𝐺)–1-1-onto→dom (iEdg‘𝐺) → ◡(iEdg‘𝐺):(Edg‘𝐺)⟶dom (iEdg‘𝐺))
8	5, 6, 7	3syl 18	. . . . . 6 ⊢ (𝐺 ∈ USPGraph → ◡(iEdg‘𝐺):(Edg‘𝐺)⟶dom (iEdg‘𝐺))
9	8	3ad2ant1 1145	. . . . 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 2853	. . . . . 6 ⊢ (𝑒 ∈ 𝐼 ↔ 𝑒 ∈ (Edg‘𝐺))
12	11	birani 507	. . . . 5 ⊢ ((𝑒 ∈ 𝐼 ∧ 𝑒 ⊆ 𝑁) → 𝑒 ∈ (Edg‘𝐺))
13		fvco3 6962	. . . . 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 605	. . . 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 7264	. . . . . . . . . . . . . 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 605	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑒 ∈ 𝐼 ∧ 𝑒 ⊆ 𝑁)) → (◡(iEdg‘𝐺)‘𝑒) ∈ dom (iEdg‘𝐺))
17		f1ocnvfv2 7256	. . . . . . . . . . . . . . 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 605	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑒 ∈ 𝐼 ∧ 𝑒 ⊆ 𝑁)) → ((iEdg‘𝐺)‘(◡(iEdg‘𝐺)‘𝑒)) = 𝑒)
19		simprr 782	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑒 ∈ 𝐼 ∧ 𝑒 ⊆ 𝑁)) → 𝑒 ⊆ 𝑁)
20	18, 19	eqsstrd 3968	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑒 ∈ 𝐼 ∧ 𝑒 ⊆ 𝑁)) → ((iEdg‘𝐺)‘(◡(iEdg‘𝐺)‘𝑒)) ⊆ 𝑁)
21	16, 20	jca 519	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑒 ∈ 𝐼 ∧ 𝑒 ⊆ 𝑁)) → ((◡(iEdg‘𝐺)‘𝑒) ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘(◡(iEdg‘𝐺)‘𝑒)) ⊆ 𝑁))
22	21	adantlr 725	. . . . . . . . . . 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 6862	. . . . . . . . . . . . 13 ⊢ (𝑥 = (◡(iEdg‘𝐺)‘𝑒) → ((iEdg‘𝐺)‘𝑥) = ((iEdg‘𝐺)‘(◡(iEdg‘𝐺)‘𝑒)))
24	23	sseq1d 3965	. . . . . . . . . . . 12 ⊢ (𝑥 = (◡(iEdg‘𝐺)‘𝑒) → (((iEdg‘𝐺)‘𝑥) ⊆ 𝑁 ↔ ((iEdg‘𝐺)‘(◡(iEdg‘𝐺)‘𝑒)) ⊆ 𝑁))
25	24	elrab 3649	. . . . . . . . . . 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 236	. . . . . . . . . 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 6862	. . . . . . . . . . . . 13 ⊢ (𝑖 = (◡(iEdg‘𝐺)‘𝑒) → ((iEdg‘𝐺)‘𝑖) = ((iEdg‘𝐺)‘(◡(iEdg‘𝐺)‘𝑒)))
28	27	imaeq2d 6045	. . . . . . . . . . . 12 ⊢ (𝑖 = (◡(iEdg‘𝐺)‘𝑒) → (𝑓 “ ((iEdg‘𝐺)‘𝑖)) = (𝑓 “ ((iEdg‘𝐺)‘(◡(iEdg‘𝐺)‘𝑒))))
29		2fveq3 6867	. . . . . . . . . . . 12 ⊢ (𝑖 = (◡(iEdg‘𝐺)‘𝑒) → ((iEdg‘𝐻)‘(ℎ‘𝑖)) = ((iEdg‘𝐻)‘(ℎ‘(◡(iEdg‘𝐺)‘𝑒))))
30	28, 29	eqeq12d 2777	. . . . . . . . . . 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 3576	. . . . . . . . . 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 17	. . . . . . . . 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 2768	. . . . . . . . . 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 6801	. . . . . . . . . . . . . . 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 737	. . . . . . . . . . . . . 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 6963	. . . . . . . . . . . . 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 2767	. . . . . . . . . . . 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 484	. . . . . . . . . . . . . 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 605	. . . . . . . . . . . . 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 6045	. . . . . . . . . . . 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 2777	. . . . . . . . . . 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 231	. . . . . . . . . 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 244	. . . . . . . . 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 47	. . . . . . . 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 416	. . . . . . 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 86	. . . . . 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 416	. . . . 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 1360	. . . 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 2797	. . 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 416	. 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 244	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 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141  ∀wral 3075  {crab 3413   ⊆ wss 3902  ^◡ccnv 5642  dom cdm 5643   “ cima 5646   ∘ ccom 5647  ⟶wf 6512  –1-1-onto→wf1o 6515  ‘cfv 6516  (class class class)co 7391  Vtxcvtx 29154  iEdgciedg 29155  Edgcedg 29205  USPGraphcuspgr 29306   ClNeighbVtx cclnbgr 48401

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pr 5387  ax-un 7713

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-sbc 3743  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-edg 29206  df-uspgr 29308

This theorem is referenced by:  uspgrlim  48575