uspgrlimlem4 - Mathbox for Alexander van der Vekens

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Theorem uspgrlimlem4 47980

Description: Lemma 4 for uspgrlim 47981. (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 2730	. . . . . . 7 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
2	1	uspgrf1oedg 29106	. . . . . 6 ⊢ (𝐺 ∈ USPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1-onto→(Edg‘𝐺))
3		f1of 6802	. . . . . 6 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1-onto→(Edg‘𝐺) → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(Edg‘𝐺))
4	2, 3	syl 17	. . . . 5 ⊢ (𝐺 ∈ USPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(Edg‘𝐺))
5	4	ad2antrr 726	. . . 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 482	. . . 4 ⊢ ((𝑖 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑁) → 𝑖 ∈ dom (iEdg‘𝐺))
7		fvco3 6962	. . . . 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 6864	. . . 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 596	. . 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 2730	. . . . . . 7 ⊢ (iEdg‘𝐻) = (iEdg‘𝐻)
11	10	uspgrf1oedg 29106	. . . . . 6 ⊢ (𝐻 ∈ USPGraph → (iEdg‘𝐻):dom (iEdg‘𝐻)–1-1-onto→(Edg‘𝐻))
12	11	ad3antlr 731	. . . . 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 4045	. . . . . . 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 2739	. . . . . . 7 ⊢ (Edg‘𝐻) = 𝐽
17	13, 14, 16	3sstr4i 4000	. . . . . 6 ⊢ 𝐿 ⊆ (Edg‘𝐻)
18		f1of 6802	. . . . . . . . . 10 ⊢ (𝑔:𝐾–1-1-onto→𝐿 → 𝑔:𝐾⟶𝐿)
19	18	adantr 480	. . . . . . . . 9 ⊢ ((𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑒 ∈ 𝐾 (𝑓 “ 𝑒) = (𝑔‘𝑒)) → 𝑔:𝐾⟶𝐿)
20	19	adantl 481	. . . . . . . 8 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑒 ∈ 𝐾 (𝑓 “ 𝑒) = (𝑔‘𝑒))) → 𝑔:𝐾⟶𝐿)
21	20	adantr 480	. . . . . . 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 6694	. . . . . . . . . 10 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑒 ∈ 𝐾 (𝑓 “ 𝑒) = (𝑔‘𝑒))) → Fun (iEdg‘𝐺))
23	1	iedgedg 28983	. . . . . . . . . 10 ⊢ ((Fun (iEdg‘𝐺) ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘𝑖) ∈ (Edg‘𝐺))
24	22, 6, 23	syl2an 596	. . . . . . . . 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 2840	. . . . . . . 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 772	. . . . . . . 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 3974	. . . . . . . . 9 ⊢ (𝑥 = ((iEdg‘𝐺)‘𝑖) → (𝑥 ⊆ 𝑁 ↔ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑁))
29		uspgrlim.k	. . . . . . . . 9 ⊢ 𝐾 = {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁}
30	28, 29	elrab2 3664	. . . . . . . 8 ⊢ (((iEdg‘𝐺)‘𝑖) ∈ 𝐾 ↔ (((iEdg‘𝐺)‘𝑖) ∈ 𝐼 ∧ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑁))
31	26, 27, 30	sylanbrc 583	. . . . . . 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 7059	. . . . . 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 3946	. . . . 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 7254	. . . . 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 584	. . . 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 6962	. . . . . 6 ⊢ ((𝑔:𝐾⟶𝐿 ∧ ((iEdg‘𝐺)‘𝑖) ∈ 𝐾) → ((^◡(iEdg‘𝐻) ∘ 𝑔)‘((iEdg‘𝐺)‘𝑖)) = (^◡(iEdg‘𝐻)‘(𝑔‘((iEdg‘𝐺)‘𝑖))))
37	36	fveq2d 6864	. . . . 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 584	. . . 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 2739	. . . . . . . . . . . . . . 15 ⊢ (Edg‘𝐺) = 𝐼
40		feq3 6670	. . . . . . . . . . . . . . 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 216	. . . . . . . . . . . . 13 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶(Edg‘𝐺) → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶𝐼)
43	2, 3, 42	3syl 18	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ USPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶𝐼)
44	43	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑁)) → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶𝐼)
45	6	adantl 481	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑁)) → 𝑖 ∈ dom (iEdg‘𝐺))
46	44, 45	ffvelcdmd 7059	. . . . . . . . . 10 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑁)) → ((iEdg‘𝐺)‘𝑖) ∈ 𝐼)
47		simprr 772	. . . . . . . . . 10 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑁)) → ((iEdg‘𝐺)‘𝑖) ⊆ 𝑁)
48	46, 47, 30	sylanbrc 583	. . . . . . . . 9 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑖 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑁)) → ((iEdg‘𝐺)‘𝑖) ∈ 𝐾)
49		imaeq2 6029	. . . . . . . . . . 11 ⊢ (𝑒 = ((iEdg‘𝐺)‘𝑖) → (𝑓 “ 𝑒) = (𝑓 “ ((iEdg‘𝐺)‘𝑖)))
50		fveq2 6860	. . . . . . . . . . 11 ⊢ (𝑒 = ((iEdg‘𝐺)‘𝑖) → (𝑔‘𝑒) = (𝑔‘((iEdg‘𝐺)‘𝑖)))
51	49, 50	eqeq12d 2746	. . . . . . . . . 10 ⊢ (𝑒 = ((iEdg‘𝐺)‘𝑖) → ((𝑓 “ 𝑒) = (𝑔‘𝑒) ↔ (𝑓 “ ((iEdg‘𝐺)‘𝑖)) = (𝑔‘((iEdg‘𝐺)‘𝑖))))
52	51	rspcv 3587	. . . . . . . . 9 ⊢ (((iEdg‘𝐺)‘𝑖) ∈ 𝐾 → (∀𝑒 ∈ 𝐾 (𝑓 “ 𝑒) = (𝑔‘𝑒) → (𝑓 “ ((iEdg‘𝐺)‘𝑖)) = (𝑔‘((iEdg‘𝐺)‘𝑖))))
53	48, 52	syl 17	. . . . . . . 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 412	. . . . . . 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 86	. . . . . 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 490	. . . . 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 417	. . . 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 2775	. . 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 2766	. 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 412	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))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ∀wral 3045 {crab 3408 ⊆ wss 3916 ^◡ccnv 5639 dom cdm 5640 “ cima 5643 ∘ ccom 5644 Fun wfun 6507 ⟶wf 6509 –1-1-onto→wf1o 6512 ‘cfv 6513 (class class class)co 7389 Vtxcvtx 28929 iEdgciedg 28930 Edgcedg 28980 USPGraphcuspgr 29081 ClNeighbVtx cclnbgr 47809

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-sep 5253 ax-nul 5263 ax-pr 5389 ax-un 7713

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-ral 3046 df-rex 3055 df-rab 3409 df-v 3452 df-sbc 3756 df-dif 3919 df-un 3921 df-in 3923 df-ss 3933 df-nul 4299 df-if 4491 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4874 df-br 5110 df-opab 5172 df-mpt 5191 df-id 5535 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-rn 5651 df-res 5652 df-ima 5653 df-iota 6466 df-fun 6515 df-fn 6516 df-f 6517 df-f1 6518 df-fo 6519 df-f1o 6520 df-fv 6521 df-edg 28981 df-uspgr 29083

This theorem is referenced by: uspgrlim 47981

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