Theorem isubgredg 47860

Description: An edge of an induced subgraph of a hypergraph is an edge of the hypergraph connecting vertices of the subgraph. (Contributed by AV, 24-Sep-2025.)

Hypotheses

Ref	Expression
isubgredg.v	⊢ 𝑉 = (Vtx‘𝐺)
isubgredg.e	⊢ 𝐸 = (Edg‘𝐺)
isubgredg.h	⊢ 𝐻 = (𝐺 ISubGr 𝑆)
isubgredg.i	⊢ 𝐼 = (Edg‘𝐻)

Assertion

Ref	Expression
isubgredg	⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (𝐾 ∈ 𝐼 ↔ (𝐾 ∈ 𝐸 ∧ 𝐾 ⊆ 𝑆)))

Proof of Theorem isubgredg

Dummy variables 𝑥 𝑖 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isubgredg.h	. . . . . . 7 ⊢ 𝐻 = (𝐺 ISubGr 𝑆)
2	1	fveq2i 6825	. . . . . 6 ⊢ (iEdg‘𝐻) = (iEdg‘(𝐺 ISubGr 𝑆))
3		isubgredg.v	. . . . . . 7 ⊢ 𝑉 = (Vtx‘𝐺)
4		eqid 2729	. . . . . . 7 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
5	3, 4	isubgriedg 47857	. . . . . 6 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (iEdg‘(𝐺 ISubGr 𝑆)) = ((iEdg‘𝐺) ↾ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆}))
6	2, 5	eqtrid 2776	. . . . 5 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (iEdg‘𝐻) = ((iEdg‘𝐺) ↾ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆}))
7	6	rneqd 5880	. . . 4 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → ran (iEdg‘𝐻) = ran ((iEdg‘𝐺) ↾ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆}))
8	7	eleq2d 2814	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (𝐾 ∈ ran (iEdg‘𝐻) ↔ 𝐾 ∈ ran ((iEdg‘𝐺) ↾ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆})))
9	3, 4	uhgrf 29007	. . . . . . 7 ⊢ (𝐺 ∈ UHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 𝑉 ∖ {∅}))
10	9	adantr 480	. . . . . 6 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 𝑉 ∖ {∅}))
11	10	ffnd 6653	. . . . 5 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
12		ssrab2 4031	. . . . . 6 ⊢ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆} ⊆ dom (iEdg‘𝐺)
13	12	a1i 11	. . . . 5 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆} ⊆ dom (iEdg‘𝐺))
14	11, 13	fnssresd 6606	. . . 4 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → ((iEdg‘𝐺) ↾ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆}) Fn {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆})
15		fvelrnb 6883	. . . 4 ⊢ (((iEdg‘𝐺) ↾ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆}) Fn {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆} → (𝐾 ∈ ran ((iEdg‘𝐺) ↾ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆}) ↔ ∃𝑥 ∈ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆} (((iEdg‘𝐺) ↾ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆})‘𝑥) = 𝐾))
16	14, 15	syl 17	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (𝐾 ∈ ran ((iEdg‘𝐺) ↾ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆}) ↔ ∃𝑥 ∈ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆} (((iEdg‘𝐺) ↾ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆})‘𝑥) = 𝐾))
17		fvres 6841	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆} → (((iEdg‘𝐺) ↾ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆})‘𝑥) = ((iEdg‘𝐺)‘𝑥))
18	17	adantl 481	. . . . . . 7 ⊢ (((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) ∧ 𝑥 ∈ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆}) → (((iEdg‘𝐺) ↾ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆})‘𝑥) = ((iEdg‘𝐺)‘𝑥))
19	18	eqeq1d 2731	. . . . . 6 ⊢ (((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) ∧ 𝑥 ∈ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆}) → ((((iEdg‘𝐺) ↾ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆})‘𝑥) = 𝐾 ↔ ((iEdg‘𝐺)‘𝑥) = 𝐾))
20		fveq2 6822	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑥 → ((iEdg‘𝐺)‘𝑖) = ((iEdg‘𝐺)‘𝑥))
21	20	sseq1d 3967	. . . . . . . . . 10 ⊢ (𝑖 = 𝑥 → (((iEdg‘𝐺)‘𝑖) ⊆ 𝑆 ↔ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆))
22	21	elrab 3648	. . . . . . . . 9 ⊢ (𝑥 ∈ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆} ↔ (𝑥 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆))
23	4	uhgrfun 29011	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
24	23	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → Fun (iEdg‘𝐺))
25		simpl 482	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆) → 𝑥 ∈ dom (iEdg‘𝐺))
26		fvelrn 7010	. . . . . . . . . . . 12 ⊢ ((Fun (iEdg‘𝐺) ∧ 𝑥 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘𝑥) ∈ ran (iEdg‘𝐺))
27	24, 25, 26	syl2anr 597	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆) ∧ (𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉)) → ((iEdg‘𝐺)‘𝑥) ∈ ran (iEdg‘𝐺))
28		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆) → ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆)
29	28	adantr 480	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆) ∧ (𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉)) → ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆)
30	27, 29	jca 511	. . . . . . . . . 10 ⊢ (((𝑥 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆) ∧ (𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉)) → (((iEdg‘𝐺)‘𝑥) ∈ ran (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆))
31	30	ex 412	. . . . . . . . 9 ⊢ ((𝑥 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆) → ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (((iEdg‘𝐺)‘𝑥) ∈ ran (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆)))
32	22, 31	sylbi 217	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆} → ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (((iEdg‘𝐺)‘𝑥) ∈ ran (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆)))
33	32	impcom 407	. . . . . . 7 ⊢ (((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) ∧ 𝑥 ∈ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆}) → (((iEdg‘𝐺)‘𝑥) ∈ ran (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆))
34		eleq1 2816	. . . . . . . 8 ⊢ (((iEdg‘𝐺)‘𝑥) = 𝐾 → (((iEdg‘𝐺)‘𝑥) ∈ ran (iEdg‘𝐺) ↔ 𝐾 ∈ ran (iEdg‘𝐺)))
35		sseq1 3961	. . . . . . . 8 ⊢ (((iEdg‘𝐺)‘𝑥) = 𝐾 → (((iEdg‘𝐺)‘𝑥) ⊆ 𝑆 ↔ 𝐾 ⊆ 𝑆))
36	34, 35	anbi12d 632	. . . . . . 7 ⊢ (((iEdg‘𝐺)‘𝑥) = 𝐾 → ((((iEdg‘𝐺)‘𝑥) ∈ ran (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆) ↔ (𝐾 ∈ ran (iEdg‘𝐺) ∧ 𝐾 ⊆ 𝑆)))
37	33, 36	syl5ibcom 245	. . . . . 6 ⊢ (((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) ∧ 𝑥 ∈ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆}) → (((iEdg‘𝐺)‘𝑥) = 𝐾 → (𝐾 ∈ ran (iEdg‘𝐺) ∧ 𝐾 ⊆ 𝑆)))
38	19, 37	sylbid 240	. . . . 5 ⊢ (((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) ∧ 𝑥 ∈ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆}) → ((((iEdg‘𝐺) ↾ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆})‘𝑥) = 𝐾 → (𝐾 ∈ ran (iEdg‘𝐺) ∧ 𝐾 ⊆ 𝑆)))
39	38	rexlimdva 3130	. . . 4 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (∃𝑥 ∈ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆} (((iEdg‘𝐺) ↾ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆})‘𝑥) = 𝐾 → (𝐾 ∈ ran (iEdg‘𝐺) ∧ 𝐾 ⊆ 𝑆)))
40		edgval 28994	. . . . . . . . . . 11 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
41	40	eqcomi 2738	. . . . . . . . . 10 ⊢ ran (iEdg‘𝐺) = (Edg‘𝐺)
42	41	eleq2i 2820	. . . . . . . . 9 ⊢ (𝐾 ∈ ran (iEdg‘𝐺) ↔ 𝐾 ∈ (Edg‘𝐺))
43	4	edgiedgb 28999	. . . . . . . . 9 ⊢ (Fun (iEdg‘𝐺) → (𝐾 ∈ (Edg‘𝐺) ↔ ∃𝑥 ∈ dom (iEdg‘𝐺)𝐾 = ((iEdg‘𝐺)‘𝑥)))
44	42, 43	bitrid 283	. . . . . . . 8 ⊢ (Fun (iEdg‘𝐺) → (𝐾 ∈ ran (iEdg‘𝐺) ↔ ∃𝑥 ∈ dom (iEdg‘𝐺)𝐾 = ((iEdg‘𝐺)‘𝑥)))
45	23, 44	syl 17	. . . . . . 7 ⊢ (𝐺 ∈ UHGraph → (𝐾 ∈ ran (iEdg‘𝐺) ↔ ∃𝑥 ∈ dom (iEdg‘𝐺)𝐾 = ((iEdg‘𝐺)‘𝑥)))
46	45	adantr 480	. . . . . 6 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (𝐾 ∈ ran (iEdg‘𝐺) ↔ ∃𝑥 ∈ dom (iEdg‘𝐺)𝐾 = ((iEdg‘𝐺)‘𝑥)))
47		simprl 770	. . . . . . . . . . . . 13 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) ∧ 𝐾 ⊆ 𝑆) ∧ (𝑥 ∈ dom (iEdg‘𝐺) ∧ 𝐾 = ((iEdg‘𝐺)‘𝑥))) → 𝑥 ∈ dom (iEdg‘𝐺))
48		simpr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ dom (iEdg‘𝐺) ∧ 𝐾 = ((iEdg‘𝐺)‘𝑥)) → 𝐾 = ((iEdg‘𝐺)‘𝑥))
49	48	sseq1d 3967	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ dom (iEdg‘𝐺) ∧ 𝐾 = ((iEdg‘𝐺)‘𝑥)) → (𝐾 ⊆ 𝑆 ↔ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆))
50	49	biimpcd 249	. . . . . . . . . . . . . . 15 ⊢ (𝐾 ⊆ 𝑆 → ((𝑥 ∈ dom (iEdg‘𝐺) ∧ 𝐾 = ((iEdg‘𝐺)‘𝑥)) → ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆))
51	50	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) ∧ 𝐾 ⊆ 𝑆) → ((𝑥 ∈ dom (iEdg‘𝐺) ∧ 𝐾 = ((iEdg‘𝐺)‘𝑥)) → ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆))
52	51	imp 406	. . . . . . . . . . . . 13 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) ∧ 𝐾 ⊆ 𝑆) ∧ (𝑥 ∈ dom (iEdg‘𝐺) ∧ 𝐾 = ((iEdg‘𝐺)‘𝑥))) → ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆)
53	47, 52, 22	sylanbrc 583	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) ∧ 𝐾 ⊆ 𝑆) ∧ (𝑥 ∈ dom (iEdg‘𝐺) ∧ 𝐾 = ((iEdg‘𝐺)‘𝑥))) → 𝑥 ∈ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆})
54		simpr 484	. . . . . . . . . . . . 13 ⊢ (((((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) ∧ 𝐾 ⊆ 𝑆) ∧ (𝑥 ∈ dom (iEdg‘𝐺) ∧ 𝐾 = ((iEdg‘𝐺)‘𝑥))) ∧ 𝑥 ∈ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆}) → 𝑥 ∈ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆})
55	48	eqcomd 2735	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ dom (iEdg‘𝐺) ∧ 𝐾 = ((iEdg‘𝐺)‘𝑥)) → ((iEdg‘𝐺)‘𝑥) = 𝐾)
56	55	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) ∧ 𝐾 ⊆ 𝑆) ∧ (𝑥 ∈ dom (iEdg‘𝐺) ∧ 𝐾 = ((iEdg‘𝐺)‘𝑥))) → ((iEdg‘𝐺)‘𝑥) = 𝐾)
57	17, 56	sylan9eqr 2786	. . . . . . . . . . . . 13 ⊢ (((((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) ∧ 𝐾 ⊆ 𝑆) ∧ (𝑥 ∈ dom (iEdg‘𝐺) ∧ 𝐾 = ((iEdg‘𝐺)‘𝑥))) ∧ 𝑥 ∈ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆}) → (((iEdg‘𝐺) ↾ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆})‘𝑥) = 𝐾)
58	54, 57	jca 511	. . . . . . . . . . . 12 ⊢ (((((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) ∧ 𝐾 ⊆ 𝑆) ∧ (𝑥 ∈ dom (iEdg‘𝐺) ∧ 𝐾 = ((iEdg‘𝐺)‘𝑥))) ∧ 𝑥 ∈ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆}) → (𝑥 ∈ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆} ∧ (((iEdg‘𝐺) ↾ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆})‘𝑥) = 𝐾))
59	53, 58	mpdan 687	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) ∧ 𝐾 ⊆ 𝑆) ∧ (𝑥 ∈ dom (iEdg‘𝐺) ∧ 𝐾 = ((iEdg‘𝐺)‘𝑥))) → (𝑥 ∈ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆} ∧ (((iEdg‘𝐺) ↾ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆})‘𝑥) = 𝐾))
60	59	ex 412	. . . . . . . . . 10 ⊢ (((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) ∧ 𝐾 ⊆ 𝑆) → ((𝑥 ∈ dom (iEdg‘𝐺) ∧ 𝐾 = ((iEdg‘𝐺)‘𝑥)) → (𝑥 ∈ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆} ∧ (((iEdg‘𝐺) ↾ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆})‘𝑥) = 𝐾)))
61	60	eximdv 1917	. . . . . . . . 9 ⊢ (((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) ∧ 𝐾 ⊆ 𝑆) → (∃𝑥(𝑥 ∈ dom (iEdg‘𝐺) ∧ 𝐾 = ((iEdg‘𝐺)‘𝑥)) → ∃𝑥(𝑥 ∈ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆} ∧ (((iEdg‘𝐺) ↾ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆})‘𝑥) = 𝐾)))
62		df-rex 3054	. . . . . . . . 9 ⊢ (∃𝑥 ∈ dom (iEdg‘𝐺)𝐾 = ((iEdg‘𝐺)‘𝑥) ↔ ∃𝑥(𝑥 ∈ dom (iEdg‘𝐺) ∧ 𝐾 = ((iEdg‘𝐺)‘𝑥)))
63		df-rex 3054	. . . . . . . . 9 ⊢ (∃𝑥 ∈ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆} (((iEdg‘𝐺) ↾ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆})‘𝑥) = 𝐾 ↔ ∃𝑥(𝑥 ∈ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆} ∧ (((iEdg‘𝐺) ↾ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆})‘𝑥) = 𝐾))
64	61, 62, 63	3imtr4g 296	. . . . . . . 8 ⊢ (((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) ∧ 𝐾 ⊆ 𝑆) → (∃𝑥 ∈ dom (iEdg‘𝐺)𝐾 = ((iEdg‘𝐺)‘𝑥) → ∃𝑥 ∈ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆} (((iEdg‘𝐺) ↾ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆})‘𝑥) = 𝐾))
65	64	ex 412	. . . . . . 7 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (𝐾 ⊆ 𝑆 → (∃𝑥 ∈ dom (iEdg‘𝐺)𝐾 = ((iEdg‘𝐺)‘𝑥) → ∃𝑥 ∈ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆} (((iEdg‘𝐺) ↾ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆})‘𝑥) = 𝐾)))
66	65	com23 86	. . . . . 6 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (∃𝑥 ∈ dom (iEdg‘𝐺)𝐾 = ((iEdg‘𝐺)‘𝑥) → (𝐾 ⊆ 𝑆 → ∃𝑥 ∈ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆} (((iEdg‘𝐺) ↾ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆})‘𝑥) = 𝐾)))
67	46, 66	sylbid 240	. . . . 5 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (𝐾 ∈ ran (iEdg‘𝐺) → (𝐾 ⊆ 𝑆 → ∃𝑥 ∈ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆} (((iEdg‘𝐺) ↾ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆})‘𝑥) = 𝐾)))
68	67	impd 410	. . . 4 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → ((𝐾 ∈ ran (iEdg‘𝐺) ∧ 𝐾 ⊆ 𝑆) → ∃𝑥 ∈ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆} (((iEdg‘𝐺) ↾ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆})‘𝑥) = 𝐾))
69	39, 68	impbid 212	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (∃𝑥 ∈ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆} (((iEdg‘𝐺) ↾ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆})‘𝑥) = 𝐾 ↔ (𝐾 ∈ ran (iEdg‘𝐺) ∧ 𝐾 ⊆ 𝑆)))
70	8, 16, 69	3bitrd 305	. 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (𝐾 ∈ ran (iEdg‘𝐻) ↔ (𝐾 ∈ ran (iEdg‘𝐺) ∧ 𝐾 ⊆ 𝑆)))
71		isubgredg.i	. . . 4 ⊢ 𝐼 = (Edg‘𝐻)
72		edgval 28994	. . . 4 ⊢ (Edg‘𝐻) = ran (iEdg‘𝐻)
73	71, 72	eqtri 2752	. . 3 ⊢ 𝐼 = ran (iEdg‘𝐻)
74	73	eleq2i 2820	. 2 ⊢ (𝐾 ∈ 𝐼 ↔ 𝐾 ∈ ran (iEdg‘𝐻))
75		isubgredg.e	. . . . 5 ⊢ 𝐸 = (Edg‘𝐺)
76	75, 40	eqtri 2752	. . . 4 ⊢ 𝐸 = ran (iEdg‘𝐺)
77	76	eleq2i 2820	. . 3 ⊢ (𝐾 ∈ 𝐸 ↔ 𝐾 ∈ ran (iEdg‘𝐺))
78	77	anbi1i 624	. 2 ⊢ ((𝐾 ∈ 𝐸 ∧ 𝐾 ⊆ 𝑆) ↔ (𝐾 ∈ ran (iEdg‘𝐺) ∧ 𝐾 ⊆ 𝑆))
79	70, 74, 78	3bitr4g 314	1 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (𝐾 ∈ 𝐼 ↔ (𝐾 ∈ 𝐸 ∧ 𝐾 ⊆ 𝑆)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∃wrex 3053  {crab 3394   ∖ cdif 3900   ⊆ wss 3903  ∅c0 4284  𝒫 cpw 4551  {csn 4577  dom cdm 5619  ran crn 5620   ↾ cres 5621  Fun wfun 6476   Fn wfn 6477  ⟶wf 6478  ‘cfv 6482  (class class class)co 7349  Vtxcvtx 28941  iEdgciedg 28942  Edgcedg 28992  UHGraphcuhgr 29001   ISubGr cisubgr 47854

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 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371  ax-un 7671

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-2nd 7925  df-iedg 28944  df-edg 28993  df-uhgr 29003  df-isubgr 47855

This theorem is referenced by:  isubgr3stgrlem6  47965  isubgr3stgrlem7  47966  isubgr3stgrlem8  47967