Theorem uspgrlimlem4 48789

Description: Lemma 4 for uspgrlim 48790. (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
uspgrlimlem4	⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑒 ∈ 𝐾 (𝑓 “ 𝑒) = (𝑔‘𝑒))) → ((𝑖 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑁) → (𝑓 “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(((◡(iEdg‘𝐻) ∘ 𝑔) ∘ (iEdg‘𝐺))‘𝑖))))

Distinct variable groups:   𝑖,𝐺,𝑥   𝑖,𝐻,𝑥   𝑥,𝐼   𝑥,𝐽   𝑥,𝑀   𝑖,𝑁,𝑥   𝑒,𝑖,𝑥   𝑓,𝑖   𝑒,𝐺   𝑒,𝐾,𝑥   𝑥,𝐿   𝑒,𝑓   𝑒,𝑔

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

Proof of Theorem uspgrlimlem4

Step	Hyp	Ref	Expression
1		eqid 2765	. . . . . . 7 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
2	1	uspgrf1oedg 29557	. . . . . 6 ⊢ (𝐺 ∈ USPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1-onto→(Edg‘𝐺))
3		f1of 6824	. . . . . 6 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1-onto→(Edg‘𝐺) → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(Edg‘𝐺))
4	2, 3	syl 18	. . . . 5 ⊢ (𝐺 ∈ USPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(Edg‘𝐺))
5	4	ad2antrr 739	. . . 4 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑒 ∈ 𝐾 (𝑓 “ 𝑒) = (𝑔‘𝑒))) → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(Edg‘𝐺))
6		simpl 488	. . . 4 ⊢ ((𝑖 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑁) → 𝑖 ∈ dom (iEdg‘𝐺))
7		fvco3 6985	. . . . 5 ⊢ (((iEdg‘𝐺):dom (iEdg‘𝐺)⟶(Edg‘𝐺) ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → (((◡(iEdg‘𝐻) ∘ 𝑔) ∘ (iEdg‘𝐺))‘𝑖) = ((◡(iEdg‘𝐻) ∘ 𝑔)‘((iEdg‘𝐺)‘𝑖)))
8	7	fveq2d 6889	. . . 4 ⊢ (((iEdg‘𝐺):dom (iEdg‘𝐺)⟶(Edg‘𝐺) ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐻)‘(((◡(iEdg‘𝐻) ∘ 𝑔) ∘ (iEdg‘𝐺))‘𝑖)) = ((iEdg‘𝐻)‘((◡(iEdg‘𝐻) ∘ 𝑔)‘((iEdg‘𝐺)‘𝑖))))
9	5, 6, 8	syl2an 608	. . 3 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑒 ∈ 𝐾 (𝑓 “ 𝑒) = (𝑔‘𝑒))) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑁)) → ((iEdg‘𝐻)‘(((◡(iEdg‘𝐻) ∘ 𝑔) ∘ (iEdg‘𝐺))‘𝑖)) = ((iEdg‘𝐻)‘((◡(iEdg‘𝐻) ∘ 𝑔)‘((iEdg‘𝐺)‘𝑖))))
10		eqid 2765	. . . . . . 7 ⊢ (iEdg‘𝐻) = (iEdg‘𝐻)
11	10	uspgrf1oedg 29557	. . . . . 6 ⊢ (𝐻 ∈ USPGraph → (iEdg‘𝐻):dom (iEdg‘𝐻)–1-1-onto→(Edg‘𝐻))
12	11	ad3antlr 744	. . . . 5 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑒 ∈ 𝐾 (𝑓 “ 𝑒) = (𝑔‘𝑒))) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑁)) → (iEdg‘𝐻):dom (iEdg‘𝐻)–1-1-onto→(Edg‘𝐻))
13		ssrab2 4035	. . . . . . 7 ⊢ {𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ 𝑀} ⊆ 𝐽
14		uspgrlim.l	. . . . . . 7 ⊢ 𝐿 = {𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ 𝑀}
15		uspgrlim.j	. . . . . . . 8 ⊢ 𝐽 = (Edg‘𝐻)
16	15	eqcomi 2774	. . . . . . 7 ⊢ (Edg‘𝐻) = 𝐽
17	13, 14, 16	3sstr4i 3989	. . . . . 6 ⊢ 𝐿 ⊆ (Edg‘𝐻)
18		f1of 6824	. . . . . . . . . 10 ⊢ (𝑔:𝐾–1-1-onto→𝐿 → 𝑔:𝐾⟶𝐿)
19	18	adantr 486	. . . . . . . . 9 ⊢ ((𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑒 ∈ 𝐾 (𝑓 “ 𝑒) = (𝑔‘𝑒)) → 𝑔:𝐾⟶𝐿)
20	19	adantl 487	. . . . . . . 8 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑒 ∈ 𝐾 (𝑓 “ 𝑒) = (𝑔‘𝑒))) → 𝑔:𝐾⟶𝐿)
21	20	adantr 486	. . . . . . 7 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑒 ∈ 𝐾 (𝑓 “ 𝑒) = (𝑔‘𝑒))) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑁)) → 𝑔:𝐾⟶𝐿)
22	5	ffund 6714	. . . . . . . . . 10 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑒 ∈ 𝐾 (𝑓 “ 𝑒) = (𝑔‘𝑒))) → Fun (iEdg‘𝐺))
23	1	iedgedg 29431	. . . . . . . . . 10 ⊢ ((Fun (iEdg‘𝐺) ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘𝑖) ∈ (Edg‘𝐺))
24	22, 6, 23	syl2an 608	. . . . . . . . 9 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑒 ∈ 𝐾 (𝑓 “ 𝑒) = (𝑔‘𝑒))) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑁)) → ((iEdg‘𝐺)‘𝑖) ∈ (Edg‘𝐺))
25		uspgrlim.i	. . . . . . . . 9 ⊢ 𝐼 = (Edg‘𝐺)
26	24, 25	eleqtrrdi 2876	. . . . . . . 8 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑒 ∈ 𝐾 (𝑓 “ 𝑒) = (𝑔‘𝑒))) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑁)) → ((iEdg‘𝐺)‘𝑖) ∈ 𝐼)
27		simprr 785	. . . . . . . 8 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑒 ∈ 𝐾 (𝑓 “ 𝑒) = (𝑔‘𝑒))) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑁)) → ((iEdg‘𝐺)‘𝑖) ⊆ 𝑁)
28		sseq1 3963	. . . . . . . . 9 ⊢ (𝑥 = ((iEdg‘𝐺)‘𝑖) → (𝑥 ⊆ 𝑁 ↔ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑁))
29		uspgrlim.k	. . . . . . . . 9 ⊢ 𝐾 = {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁}
30	28, 29	elrab2 3656	. . . . . . . 8 ⊢ (((iEdg‘𝐺)‘𝑖) ∈ 𝐾 ↔ (((iEdg‘𝐺)‘𝑖) ∈ 𝐼 ∧ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑁))
31	26, 27, 30	sylanbrc 595	. . . . . . 7 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑒 ∈ 𝐾 (𝑓 “ 𝑒) = (𝑔‘𝑒))) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑁)) → ((iEdg‘𝐺)‘𝑖) ∈ 𝐾)
32	21, 31	ffvelcdmd 7084	. . . . . 6 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑒 ∈ 𝐾 (𝑓 “ 𝑒) = (𝑔‘𝑒))) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑁)) → (𝑔‘((iEdg‘𝐺)‘𝑖)) ∈ 𝐿)
33	17, 32	sselid 3936	. . . . 5 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑒 ∈ 𝐾 (𝑓 “ 𝑒) = (𝑔‘𝑒))) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑁)) → (𝑔‘((iEdg‘𝐺)‘𝑖)) ∈ (Edg‘𝐻))
34		f1ocnvfv2 7281	. . . . 5 ⊢ (((iEdg‘𝐻):dom (iEdg‘𝐻)–1-1-onto→(Edg‘𝐻) ∧ (𝑔‘((iEdg‘𝐺)‘𝑖)) ∈ (Edg‘𝐻)) → ((iEdg‘𝐻)‘(◡(iEdg‘𝐻)‘(𝑔‘((iEdg‘𝐺)‘𝑖)))) = (𝑔‘((iEdg‘𝐺)‘𝑖)))
35	12, 33, 34	syl2anc 596	. . . 4 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑒 ∈ 𝐾 (𝑓 “ 𝑒) = (𝑔‘𝑒))) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑁)) → ((iEdg‘𝐻)‘(◡(iEdg‘𝐻)‘(𝑔‘((iEdg‘𝐺)‘𝑖)))) = (𝑔‘((iEdg‘𝐺)‘𝑖)))
36		fvco3 6985	. . . . . 6 ⊢ ((𝑔:𝐾⟶𝐿 ∧ ((iEdg‘𝐺)‘𝑖) ∈ 𝐾) → ((◡(iEdg‘𝐻) ∘ 𝑔)‘((iEdg‘𝐺)‘𝑖)) = (◡(iEdg‘𝐻)‘(𝑔‘((iEdg‘𝐺)‘𝑖))))
37	36	fveq2d 6889	. . . . 5 ⊢ ((𝑔:𝐾⟶𝐿 ∧ ((iEdg‘𝐺)‘𝑖) ∈ 𝐾) → ((iEdg‘𝐻)‘((◡(iEdg‘𝐻) ∘ 𝑔)‘((iEdg‘𝐺)‘𝑖))) = ((iEdg‘𝐻)‘(◡(iEdg‘𝐻)‘(𝑔‘((iEdg‘𝐺)‘𝑖)))))
38	21, 31, 37	syl2anc 596	. . . 4 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑒 ∈ 𝐾 (𝑓 “ 𝑒) = (𝑔‘𝑒))) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑁)) → ((iEdg‘𝐻)‘((◡(iEdg‘𝐻) ∘ 𝑔)‘((iEdg‘𝐺)‘𝑖))) = ((iEdg‘𝐻)‘(◡(iEdg‘𝐻)‘(𝑔‘((iEdg‘𝐺)‘𝑖)))))
39	25	eqcomi 2774	. . . . . . . . . . . . . . 15 ⊢ (Edg‘𝐺) = 𝐼
40		feq3 6689	. . . . . . . . . . . . . . 15 ⊢ ((Edg‘𝐺) = 𝐼 → ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶(Edg‘𝐺) ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶𝐼))
41	39, 40	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶(Edg‘𝐺) ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶𝐼)
42	41	biimpi 219	. . . . . . . . . . . . 13 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶(Edg‘𝐺) → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶𝐼)
43	2, 3, 42	3syl 19	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ USPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶𝐼)
44	43	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑁)) → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶𝐼)
45	6	adantl 487	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑁)) → 𝑖 ∈ dom (iEdg‘𝐺))
46	44, 45	ffvelcdmd 7084	. . . . . . . . . 10 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑁)) → ((iEdg‘𝐺)‘𝑖) ∈ 𝐼)
47		simprr 785	. . . . . . . . . 10 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑁)) → ((iEdg‘𝐺)‘𝑖) ⊆ 𝑁)
48	46, 47, 30	sylanbrc 595	. . . . . . . . 9 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑁)) → ((iEdg‘𝐺)‘𝑖) ∈ 𝐾)
49		imaeq2 6060	. . . . . . . . . . 11 ⊢ (𝑒 = ((iEdg‘𝐺)‘𝑖) → (𝑓 “ 𝑒) = (𝑓 “ ((iEdg‘𝐺)‘𝑖)))
50		fveq2 6885	. . . . . . . . . . 11 ⊢ (𝑒 = ((iEdg‘𝐺)‘𝑖) → (𝑔‘𝑒) = (𝑔‘((iEdg‘𝐺)‘𝑖)))
51	49, 50	eqeq12d 2781	. . . . . . . . . 10 ⊢ (𝑒 = ((iEdg‘𝐺)‘𝑖) → ((𝑓 “ 𝑒) = (𝑔‘𝑒) ↔ (𝑓 “ ((iEdg‘𝐺)‘𝑖)) = (𝑔‘((iEdg‘𝐺)‘𝑖))))
52	51	rspcv 3579	. . . . . . . . 9 ⊢ (((iEdg‘𝐺)‘𝑖) ∈ 𝐾 → (∀𝑒 ∈ 𝐾 (𝑓 “ 𝑒) = (𝑔‘𝑒) → (𝑓 “ ((iEdg‘𝐺)‘𝑖)) = (𝑔‘((iEdg‘𝐺)‘𝑖))))
53	48, 52	syl 18	. . . . . . . 8 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑁)) → (∀𝑒 ∈ 𝐾 (𝑓 “ 𝑒) = (𝑔‘𝑒) → (𝑓 “ ((iEdg‘𝐺)‘𝑖)) = (𝑔‘((iEdg‘𝐺)‘𝑖))))
54	53	ex 418	. . . . . . 7 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) → ((𝑖 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑁) → (∀𝑒 ∈ 𝐾 (𝑓 “ 𝑒) = (𝑔‘𝑒) → (𝑓 “ ((iEdg‘𝐺)‘𝑖)) = (𝑔‘((iEdg‘𝐺)‘𝑖)))))
55	54	com23 87	. . . . . 6 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) → (∀𝑒 ∈ 𝐾 (𝑓 “ 𝑒) = (𝑔‘𝑒) → ((𝑖 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑁) → (𝑓 “ ((iEdg‘𝐺)‘𝑖)) = (𝑔‘((iEdg‘𝐺)‘𝑖)))))
56	55	adantld 496	. . . . 5 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) → ((𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑒 ∈ 𝐾 (𝑓 “ 𝑒) = (𝑔‘𝑒)) → ((𝑖 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑁) → (𝑓 “ ((iEdg‘𝐺)‘𝑖)) = (𝑔‘((iEdg‘𝐺)‘𝑖)))))
57	56	imp31 423	. . . 4 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑒 ∈ 𝐾 (𝑓 “ 𝑒) = (𝑔‘𝑒))) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑁)) → (𝑓 “ ((iEdg‘𝐺)‘𝑖)) = (𝑔‘((iEdg‘𝐺)‘𝑖)))
58	35, 38, 57	3eqtr4d 2810	. . 3 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑒 ∈ 𝐾 (𝑓 “ 𝑒) = (𝑔‘𝑒))) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑁)) → ((iEdg‘𝐻)‘((◡(iEdg‘𝐻) ∘ 𝑔)‘((iEdg‘𝐺)‘𝑖))) = (𝑓 “ ((iEdg‘𝐺)‘𝑖)))
59	9, 58	eqtr2d 2801	. 2 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑒 ∈ 𝐾 (𝑓 “ 𝑒) = (𝑔‘𝑒))) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑁)) → (𝑓 “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(((◡(iEdg‘𝐻) ∘ 𝑔) ∘ (iEdg‘𝐺))‘𝑖)))
60	59	ex 418	1 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑒 ∈ 𝐾 (𝑓 “ 𝑒) = (𝑔‘𝑒))) → ((𝑖 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑁) → (𝑓 “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(((◡(iEdg‘𝐻) ∘ 𝑔) ∘ (iEdg‘𝐺))‘𝑖))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∀wral 3081  {crab 3418   ⊆ wss 3906  ^◡ccnv 5662  dom cdm 5663   “ cima 5666   ∘ ccom 5667  Fun wfun 6534  ⟶wf 6536  –1-1-onto→wf1o 6539  ‘cfv 6540  (class class class)co 7416  Vtxcvtx 29377  iEdgciedg 29378  Edgcedg 29428  USPGraphcuspgr 29532   ClNeighbVtx cclnbgr 48616

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406  ax-un 7738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-edg 29429  df-uspgr 29534

This theorem is used by:  uspgrlim  48790