Theorem isubgredg 48964

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 6888	. . . . . 6 ⊢ (iEdg‘𝐻) = (iEdg‘(𝐺 ISubGr 𝑆))
3		isubgredg.v	. . . . . . 7 ⊢ 𝑉 = (Vtx‘𝐺)
4		eqid 2760	. . . . . . 7 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
5	3, 4	isubgriedg 48961	. . . . . 6 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (iEdg‘(𝐺 ISubGr 𝑆)) = ((iEdg‘𝐺) ↾ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆}))
6	2, 5	eqtrid 2807	. . . . 5 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (iEdg‘𝐻) = ((iEdg‘𝐺) ↾ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆}))
7	6	rneqd 5919	. . . 4 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → ran (iEdg‘𝐻) = ran ((iEdg‘𝐺) ↾ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆}))
8	7	eleq2d 2846	. . 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 29640	. . . . . . 7 ⊢ (𝐺 ∈ UHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 𝑉 ∖ {∅}))
10	9	adantr 486	. . . . . 6 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 𝑉 ∖ {∅}))
11	10	ffnd 6710	. . . . 5 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
12		ssrab2 4005	. . . . . 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 6663	. . . 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 6945	. . . 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 18	. . 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 6904	. . . . . . . 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 487	. . . . . . 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 2762	. . . . . 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 6885	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑥 → ((iEdg‘𝐺)‘𝑖) = ((iEdg‘𝐺)‘𝑥))
21	20	sseq1d 3939	. . . . . . . . . 10 ⊢ (𝑖 = 𝑥 → (((iEdg‘𝐺)‘𝑖) ⊆ 𝑆 ↔ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆))
22	21	elrab 3644	. . . . . . . . 9 ⊢ (𝑥 ∈ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆} ↔ (𝑥 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆))
23	4	uhgrfun 29644	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
24	23	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → Fun (iEdg‘𝐺))
25		simpl 488	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆) → 𝑥 ∈ dom (iEdg‘𝐺))
26		fvelrn 7076	. . . . . . . . . . . 12 ⊢ ((Fun (iEdg‘𝐺) ∧ 𝑥 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘𝑥) ∈ ran (iEdg‘𝐺))
27	24, 25, 26	syl2anr 609	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆) ∧ (𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉)) → ((iEdg‘𝐺)‘𝑥) ∈ ran (iEdg‘𝐺))
28		simpr 490	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆) → ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆)
29	28	adantr 486	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆) ∧ (𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉)) → ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆)
30	27, 29	jca 521	. . . . . . . . . 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 418	. . . . . . . . 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 220	. . . . . . . 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 413	. . . . . . 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 2848	. . . . . . . 8 ⊢ (((iEdg‘𝐺)‘𝑥) = 𝐾 → (((iEdg‘𝐺)‘𝑥) ∈ ran (iEdg‘𝐺) ↔ 𝐾 ∈ ran (iEdg‘𝐺)))
35		sseq1 3933	. . . . . . . 8 ⊢ (((iEdg‘𝐺)‘𝑥) = 𝐾 → (((iEdg‘𝐺)‘𝑥) ⊆ 𝑆 ↔ 𝐾 ⊆ 𝑆))
36	34, 35	anbi12d 644	. . . . . . 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 248	. . . . . 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 243	. . . . 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 3163	. . . 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 29627	. . . . . . . . . . 11 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
41	40	eqcomi 2769	. . . . . . . . . 10 ⊢ ran (iEdg‘𝐺) = (Edg‘𝐺)
42	41	eleq2i 2852	. . . . . . . . 9 ⊢ (𝐾 ∈ ran (iEdg‘𝐺) ↔ 𝐾 ∈ (Edg‘𝐺))
43	4	edgiedgb 29632	. . . . . . . . 9 ⊢ (Fun (iEdg‘𝐺) → (𝐾 ∈ (Edg‘𝐺) ↔ ∃𝑥 ∈ dom (iEdg‘𝐺)𝐾 = ((iEdg‘𝐺)‘𝑥)))
44	42, 43	bitrid 286	. . . . . . . 8 ⊢ (Fun (iEdg‘𝐺) → (𝐾 ∈ ran (iEdg‘𝐺) ↔ ∃𝑥 ∈ dom (iEdg‘𝐺)𝐾 = ((iEdg‘𝐺)‘𝑥)))
45	23, 44	syl 18	. . . . . . 7 ⊢ (𝐺 ∈ UHGraph → (𝐾 ∈ ran (iEdg‘𝐺) ↔ ∃𝑥 ∈ dom (iEdg‘𝐺)𝐾 = ((iEdg‘𝐺)‘𝑥)))
46	45	adantr 486	. . . . . 6 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (𝐾 ∈ ran (iEdg‘𝐺) ↔ ∃𝑥 ∈ dom (iEdg‘𝐺)𝐾 = ((iEdg‘𝐺)‘𝑥)))
47		simprl 783	. . . . . . . . . . . . 13 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) ∧ 𝐾 ⊆ 𝑆) ∧ (𝑥 ∈ dom (iEdg‘𝐺) ∧ 𝐾 = ((iEdg‘𝐺)‘𝑥))) → 𝑥 ∈ dom (iEdg‘𝐺))
48		simpr 490	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ dom (iEdg‘𝐺) ∧ 𝐾 = ((iEdg‘𝐺)‘𝑥)) → 𝐾 = ((iEdg‘𝐺)‘𝑥))
49	48	sseq1d 3939	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ dom (iEdg‘𝐺) ∧ 𝐾 = ((iEdg‘𝐺)‘𝑥)) → (𝐾 ⊆ 𝑆 ↔ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆))
50	49	biimpcd 252	. . . . . . . . . . . . . . 15 ⊢ (𝐾 ⊆ 𝑆 → ((𝑥 ∈ dom (iEdg‘𝐺) ∧ 𝐾 = ((iEdg‘𝐺)‘𝑥)) → ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆))
51	50	adantl 487	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) ∧ 𝐾 ⊆ 𝑆) → ((𝑥 ∈ dom (iEdg‘𝐺) ∧ 𝐾 = ((iEdg‘𝐺)‘𝑥)) → ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆))
52	51	imp 412	. . . . . . . . . . . . 13 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) ∧ 𝐾 ⊆ 𝑆) ∧ (𝑥 ∈ dom (iEdg‘𝐺) ∧ 𝐾 = ((iEdg‘𝐺)‘𝑥))) → ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆)
53	47, 52, 22	sylanbrc 595	. . . . . . . . . . . 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 490	. . . . . . . . . . . . 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 2766	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ dom (iEdg‘𝐺) ∧ 𝐾 = ((iEdg‘𝐺)‘𝑥)) → ((iEdg‘𝐺)‘𝑥) = 𝐾)
56	55	adantl 487	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) ∧ 𝐾 ⊆ 𝑆) ∧ (𝑥 ∈ dom (iEdg‘𝐺) ∧ 𝐾 = ((iEdg‘𝐺)‘𝑥))) → ((iEdg‘𝐺)‘𝑥) = 𝐾)
57	17, 56	sylan9eqr 2817	. . . . . . . . . . . . 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 521	. . . . . . . . . . . 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 700	. . . . . . . . . . 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 418	. . . . . . . . . 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 1949	. . . . . . . . 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 3087	. . . . . . . . 9 ⊢ (∃𝑥 ∈ dom (iEdg‘𝐺)𝐾 = ((iEdg‘𝐺)‘𝑥) ↔ ∃𝑥(𝑥 ∈ dom (iEdg‘𝐺) ∧ 𝐾 = ((iEdg‘𝐺)‘𝑥)))
63		df-rex 3087	. . . . . . . . 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 299	. . . . . . . 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 418	. . . . . . 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 87	. . . . . 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 243	. . . . 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 416	. . . 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 215	. . 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 308	. 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (𝐾 ∈ ran (iEdg‘𝐻) ↔ (𝐾 ∈ ran (iEdg‘𝐺) ∧ 𝐾 ⊆ 𝑆)))
71		isubgredg.i	. . . 4 ⊢ 𝐼 = (Edg‘𝐻)
72		edgval 29627	. . . 4 ⊢ (Edg‘𝐻) = ran (iEdg‘𝐻)
73	71, 72	eqtri 2783	. . 3 ⊢ 𝐼 = ran (iEdg‘𝐻)
74	73	eleq2i 2852	. 2 ⊢ (𝐾 ∈ 𝐼 ↔ 𝐾 ∈ ran (iEdg‘𝐻))
75		isubgredg.e	. . . . 5 ⊢ 𝐸 = (Edg‘𝐺)
76	75, 40	eqtri 2783	. . . 4 ⊢ 𝐸 = ran (iEdg‘𝐺)
77	76	eleq2i 2852	. . 3 ⊢ (𝐾 ∈ 𝐸 ↔ 𝐾 ∈ ran (iEdg‘𝐺))
78	77	anbi1i 636	. 2 ⊢ ((𝐾 ∈ 𝐸 ∧ 𝐾 ⊆ 𝑆) ↔ (𝐾 ∈ ran (iEdg‘𝐺) ∧ 𝐾 ⊆ 𝑆))
79	70, 74, 78	3bitr4g 317	1 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (𝐾 ∈ 𝐼 ↔ (𝐾 ∈ 𝐸 ∧ 𝐾 ⊆ 𝑆)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569  ∃wex 1811   ∈ wcel 2144  ∃wrex 3086  {crab 3412   ⊆ wss 3895   ∖ cdif 4045  ∅c0 4278  𝒫 cpw 4556  {csn 4583  dom cdm 5650  ran crn 5651   ↾ cres 5652  Fun wfun 6532   Fn wfn 6533  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420  Vtxcvtx 29574  iEdgciedg 29575  Edgcedg 29625  UHGraphcuhgr 29634   ISubGr cisubgr 48958

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 7751

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-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  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-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-2nd 8002  df-iedg 29577  df-edg 29626  df-uhgr 29636  df-isubgr 48959

This theorem is used by:  isubgr3stgrlem6  49069  isubgr3stgrlem7  49070  isubgr3stgrlem8  49071