Theorem uspgrlimlem3 49182

Description: Lemma 3 for uspgrlim 49184. (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 3933	. . 3 ⊢ (𝑥 = 𝑒 → (𝑥 ⊆ 𝑁 ↔ 𝑒 ⊆ 𝑁))
2		uspgrlim.k	. . 3 ⊢ 𝐾 = {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁}
3	1, 2	elrab2 3648	. 2 ⊢ (𝑒 ∈ 𝐾 ↔ (𝑒 ∈ 𝐼 ∧ 𝑒 ⊆ 𝑁))
4		eqid 2760	. . . . . . . 8 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
5	4	uspgrf1oedg 29760	. . . . . . 7 ⊢ (𝐺 ∈ USPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1-onto→(Edg‘𝐺))
6		f1ocnv 6838	. . . . . . 7 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1-onto→(Edg‘𝐺) → ◡(iEdg‘𝐺):(Edg‘𝐺)–1-1-onto→dom (iEdg‘𝐺))
7		f1of 6825	. . . . . . 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 2852	. . . . . 6 ⊢ (𝑒 ∈ 𝐼 ↔ 𝑒 ∈ (Edg‘𝐺))
12	11	birani 509	. . . . 5 ⊢ ((𝑒 ∈ 𝐼 ∧ 𝑒 ⊆ 𝑁) → 𝑒 ∈ (Edg‘𝐺))
13		fvco3 6986	. . . . 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 608	. . . 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 7294	. . . . . . . . . . . . . 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 608	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑒 ∈ 𝐼 ∧ 𝑒 ⊆ 𝑁)) → (◡(iEdg‘𝐺)‘𝑒) ∈ dom (iEdg‘𝐺))
17		f1ocnvfv2 7286	. . . . . . . . . . . . . . 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 608	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑒 ∈ 𝐼 ∧ 𝑒 ⊆ 𝑁)) → ((iEdg‘𝐺)‘(◡(iEdg‘𝐺)‘𝑒)) = 𝑒)
19		simprr 785	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑒 ∈ 𝐼 ∧ 𝑒 ⊆ 𝑁)) → 𝑒 ⊆ 𝑁)
20	18, 19	eqsstrd 3942	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑒 ∈ 𝐼 ∧ 𝑒 ⊆ 𝑁)) → ((iEdg‘𝐺)‘(◡(iEdg‘𝐺)‘𝑒)) ⊆ 𝑁)
21	16, 20	jca 521	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑒 ∈ 𝐼 ∧ 𝑒 ⊆ 𝑁)) → ((◡(iEdg‘𝐺)‘𝑒) ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘(◡(iEdg‘𝐺)‘𝑒)) ⊆ 𝑁))
22	21	adantlr 728	. . . . . . . . . . 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 6886	. . . . . . . . . . . . 13 ⊢ (𝑥 = (◡(iEdg‘𝐺)‘𝑒) → ((iEdg‘𝐺)‘𝑥) = ((iEdg‘𝐺)‘(◡(iEdg‘𝐺)‘𝑒)))
24	23	sseq1d 3939	. . . . . . . . . . . 12 ⊢ (𝑥 = (◡(iEdg‘𝐺)‘𝑒) → (((iEdg‘𝐺)‘𝑥) ⊆ 𝑁 ↔ ((iEdg‘𝐺)‘(◡(iEdg‘𝐺)‘𝑒)) ⊆ 𝑁))
25	24	elrab 3644	. . . . . . . . . . 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 6886	. . . . . . . . . . . . 13 ⊢ (𝑖 = (◡(iEdg‘𝐺)‘𝑒) → ((iEdg‘𝐺)‘𝑖) = ((iEdg‘𝐺)‘(◡(iEdg‘𝐺)‘𝑒)))
28	27	imaeq2d 6051	. . . . . . . . . . . 12 ⊢ (𝑖 = (◡(iEdg‘𝐺)‘𝑒) → (𝑓 “ ((iEdg‘𝐺)‘𝑖)) = (𝑓 “ ((iEdg‘𝐺)‘(◡(iEdg‘𝐺)‘𝑒))))
29		2fveq3 6891	. . . . . . . . . . . 12 ⊢ (𝑖 = (◡(iEdg‘𝐺)‘𝑒) → ((iEdg‘𝐻)‘(ℎ‘𝑖)) = ((iEdg‘𝐻)‘(ℎ‘(◡(iEdg‘𝐺)‘𝑒))))
30	28, 29	eqeq12d 2776	. . . . . . . . . . 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 3572	. . . . . . . . . 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 2767	. . . . . . . . . 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 6825	. . . . . . . . . . . . . . 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 740	. . . . . . . . . . . . . 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 6987	. . . . . . . . . . . . 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 2766	. . . . . . . . . . . 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 486	. . . . . . . . . . . . . 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 608	. . . . . . . . . . . . 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 6051	. . . . . . . . . . . 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 2776	. . . . . . . . . . 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 418	. . . . . . 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 418	. . . . 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 2796	. . 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 418	. 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)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ∀wral 3076  {crab 3412   ⊆ wss 3895  ^◡ccnv 5649  dom cdm 5650   “ cima 5653   ∘ ccom 5654  ⟶wf 6535  –1-1-onto→wf1o 6538  ‘cfv 6539  (class class class)co 7421  Vtxcvtx 29580  iEdgciedg 29581  Edgcedg 29631  USPGraphcuspgr 29735   ClNeighbVtx cclnbgr 49010

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7752

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-edg 29632  df-uspgr 29737

This theorem is used by:  uspgrlim  49184