Theorem uspgrlimlem3 47930

Description: Lemma 3 for uspgrlim 47932. (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 4008	. . 3 ⊢ (𝑥 = 𝑒 → (𝑥 ⊆ 𝑁 ↔ 𝑒 ⊆ 𝑁))
2		uspgrlim.k	. . 3 ⊢ 𝐾 = {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁}
3	1, 2	elrab2 3694	. 2 ⊢ (𝑒 ∈ 𝐾 ↔ (𝑒 ∈ 𝐼 ∧ 𝑒 ⊆ 𝑁))
4		eqid 2736	. . . . . . . 8 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
5	4	uspgrf1oedg 29180	. . . . . . 7 ⊢ (𝐺 ∈ USPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1-onto→(Edg‘𝐺))
6		f1ocnv 6858	. . . . . . 7 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1-onto→(Edg‘𝐺) → ◡(iEdg‘𝐺):(Edg‘𝐺)–1-1-onto→dom (iEdg‘𝐺))
7		f1of 6846	. . . . . . 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 1134	. . . . 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	. . . . . . . 8 ⊢ 𝐼 = (Edg‘𝐺)
11	10	eleq2i 2832	. . . . . . 7 ⊢ (𝑒 ∈ 𝐼 ↔ 𝑒 ∈ (Edg‘𝐺))
12	11	biimpi 216	. . . . . 6 ⊢ (𝑒 ∈ 𝐼 → 𝑒 ∈ (Edg‘𝐺))
13	12	adantr 480	. . . . 5 ⊢ ((𝑒 ∈ 𝐼 ∧ 𝑒 ⊆ 𝑁) → 𝑒 ∈ (Edg‘𝐺))
14		fvco3 7006	. . . . 5 ⊢ ((◡(iEdg‘𝐺):(Edg‘𝐺)⟶dom (iEdg‘𝐺) ∧ 𝑒 ∈ (Edg‘𝐺)) → ((((iEdg‘𝐻) ∘ ℎ) ∘ ◡(iEdg‘𝐺))‘𝑒) = (((iEdg‘𝐻) ∘ ℎ)‘(◡(iEdg‘𝐺)‘𝑒)))
15	9, 13, 14	syl2an 596	. . . 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)))
16		f1ocnvdm 7303	. . . . . . . . . . . . . 14 ⊢ (((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1-onto→(Edg‘𝐺) ∧ 𝑒 ∈ (Edg‘𝐺)) → (◡(iEdg‘𝐺)‘𝑒) ∈ dom (iEdg‘𝐺))
17	5, 13, 16	syl2an 596	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑒 ∈ 𝐼 ∧ 𝑒 ⊆ 𝑁)) → (◡(iEdg‘𝐺)‘𝑒) ∈ dom (iEdg‘𝐺))
18		f1ocnvfv2 7295	. . . . . . . . . . . . . . 15 ⊢ (((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1-onto→(Edg‘𝐺) ∧ 𝑒 ∈ (Edg‘𝐺)) → ((iEdg‘𝐺)‘(◡(iEdg‘𝐺)‘𝑒)) = 𝑒)
19	5, 13, 18	syl2an 596	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑒 ∈ 𝐼 ∧ 𝑒 ⊆ 𝑁)) → ((iEdg‘𝐺)‘(◡(iEdg‘𝐺)‘𝑒)) = 𝑒)
20		simprr 773	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑒 ∈ 𝐼 ∧ 𝑒 ⊆ 𝑁)) → 𝑒 ⊆ 𝑁)
21	19, 20	eqsstrd 4017	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑒 ∈ 𝐼 ∧ 𝑒 ⊆ 𝑁)) → ((iEdg‘𝐺)‘(◡(iEdg‘𝐺)‘𝑒)) ⊆ 𝑁)
22	17, 21	jca 511	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑒 ∈ 𝐼 ∧ 𝑒 ⊆ 𝑁)) → ((◡(iEdg‘𝐺)‘𝑒) ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘(◡(iEdg‘𝐺)‘𝑒)) ⊆ 𝑁))
23	22	adantlr 715	. . . . . . . . . . 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))
24		fveq2 6904	. . . . . . . . . . . . 13 ⊢ (𝑥 = (◡(iEdg‘𝐺)‘𝑒) → ((iEdg‘𝐺)‘𝑥) = ((iEdg‘𝐺)‘(◡(iEdg‘𝐺)‘𝑒)))
25	24	sseq1d 4014	. . . . . . . . . . . 12 ⊢ (𝑥 = (◡(iEdg‘𝐺)‘𝑒) → (((iEdg‘𝐺)‘𝑥) ⊆ 𝑁 ↔ ((iEdg‘𝐺)‘(◡(iEdg‘𝐺)‘𝑒)) ⊆ 𝑁))
26	25	elrab 3691	. . . . . . . . . . 11 ⊢ ((◡(iEdg‘𝐺)‘𝑒) ∈ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑁} ↔ ((◡(iEdg‘𝐺)‘𝑒) ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘(◡(iEdg‘𝐺)‘𝑒)) ⊆ 𝑁))
27	23, 26	sylibr 234	. . . . . . . . . 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})
28		fveq2 6904	. . . . . . . . . . . . 13 ⊢ (𝑖 = (◡(iEdg‘𝐺)‘𝑒) → ((iEdg‘𝐺)‘𝑖) = ((iEdg‘𝐺)‘(◡(iEdg‘𝐺)‘𝑒)))
29	28	imaeq2d 6076	. . . . . . . . . . . 12 ⊢ (𝑖 = (◡(iEdg‘𝐺)‘𝑒) → (𝑓 “ ((iEdg‘𝐺)‘𝑖)) = (𝑓 “ ((iEdg‘𝐺)‘(◡(iEdg‘𝐺)‘𝑒))))
30		2fveq3 6909	. . . . . . . . . . . 12 ⊢ (𝑖 = (◡(iEdg‘𝐺)‘𝑒) → ((iEdg‘𝐻)‘(ℎ‘𝑖)) = ((iEdg‘𝐻)‘(ℎ‘(◡(iEdg‘𝐺)‘𝑒))))
31	29, 30	eqeq12d 2752	. . . . . . . . . . 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)))))
32	31	rspcv 3617	. . . . . . . . . 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)))))
33	27, 32	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)))))
34		eqcom 2743	. . . . . . . . . 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))))
35		f1of 6846	. . . . . . . . . . . . . . 15 ⊢ (ℎ:{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑁}–1-1-onto→𝑅 → ℎ:{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑁}⟶𝑅)
36	35	ad2antlr 727	. . . . . . . . . . . . . 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)
37	36, 27	fvco3d 7007	. . . . . . . . . . . . 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))))
38	37	eqcomd 2742	. . . . . . . . . . . 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)))
39	5	adantr 480	. . . . . . . . . . . . . 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))
40	39, 13, 18	syl2an 596	. . . . . . . . . . . . 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)
41	40	imaeq2d 6076	. . . . . . . . . . . 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))
42	38, 41	eqeq12d 2752	. . . . . . . . . . 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)))
43	42	biimpd 229	. . . . . . . . . 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)))
44	34, 43	biimtrid 242	. . . . . . . . 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)))
45	33, 44	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)))
46	45	ex 412	. . . . . . 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))))
47	46	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))))
48	47	ex 412	. . . . 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)))))
49	48	3imp1 1348	. . . 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))
50	15, 49	eqtr2d 2777	. . 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))
51	50	ex 412	. 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)))
52	3, 51	biimtrid 242	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 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  ∀wral 3060  {crab 3435   ⊆ wss 3950  ^◡ccnv 5682  dom cdm 5683   “ cima 5686   ∘ ccom 5687  ⟶wf 6555  –1-1-onto→wf1o 6558  ‘cfv 6559  (class class class)co 7429  Vtxcvtx 29003  iEdgciedg 29004  Edgcedg 29054  USPGraphcuspgr 29155   ClNeighbVtx cclnbgr 47778

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5294  ax-nul 5304  ax-pr 5430  ax-un 7751

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-br 5142  df-opab 5204  df-mpt 5224  df-id 5576  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-iota 6512  df-fun 6561  df-fn 6562  df-f 6563  df-f1 6564  df-fo 6565  df-f1o 6566  df-fv 6567  df-edg 29055  df-uspgr 29157

This theorem is referenced by:  uspgrlim  47932