Theorem subupgr 29214

Description: A subgraph of a pseudograph is a pseudograph. (Contributed by AV, 16-Nov-2020.) (Proof shortened by AV, 21-Nov-2020.)

Assertion

Ref	Expression
subupgr	⊢ ((𝐺 ∈ UPGraph ∧ 𝑆 SubGraph 𝐺) → 𝑆 ∈ UPGraph)

Proof of Theorem subupgr

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

Step	Hyp	Ref	Expression
1		eqid 2729	. . . 4 ⊢ (Vtx‘𝑆) = (Vtx‘𝑆)
2		eqid 2729	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
3		eqid 2729	. . . 4 ⊢ (iEdg‘𝑆) = (iEdg‘𝑆)
4		eqid 2729	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
5		eqid 2729	. . . 4 ⊢ (Edg‘𝑆) = (Edg‘𝑆)
6	1, 2, 3, 4, 5	subgrprop2 29201	. . 3 ⊢ (𝑆 SubGraph 𝐺 → ((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)))
7		upgruhgr 29029	. . . . . . . . . 10 ⊢ (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph)
8		subgruhgrfun 29209	. . . . . . . . . 10 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 SubGraph 𝐺) → Fun (iEdg‘𝑆))
9	7, 8	sylan 580	. . . . . . . . 9 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑆 SubGraph 𝐺) → Fun (iEdg‘𝑆))
10	9	ancoms 458	. . . . . . . 8 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UPGraph) → Fun (iEdg‘𝑆))
11	10	funfnd 6547	. . . . . . 7 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UPGraph) → (iEdg‘𝑆) Fn dom (iEdg‘𝑆))
12	11	adantl 481	. . . . . 6 ⊢ ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UPGraph)) → (iEdg‘𝑆) Fn dom (iEdg‘𝑆))
13		fveq2 6858	. . . . . . . . . 10 ⊢ (𝑒 = ((iEdg‘𝑆)‘𝑥) → (♯‘𝑒) = (♯‘((iEdg‘𝑆)‘𝑥)))
14	13	breq1d 5117	. . . . . . . . 9 ⊢ (𝑒 = ((iEdg‘𝑆)‘𝑥) → ((♯‘𝑒) ≤ 2 ↔ (♯‘((iEdg‘𝑆)‘𝑥)) ≤ 2))
15	7	anim2i 617	. . . . . . . . . . . . . 14 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UPGraph) → (𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UHGraph))
16	15	adantl 481	. . . . . . . . . . . . 13 ⊢ ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UPGraph)) → (𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UHGraph))
17	16	ancomd 461	. . . . . . . . . . . 12 ⊢ ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UPGraph)) → (𝐺 ∈ UHGraph ∧ 𝑆 SubGraph 𝐺))
18	17	anim1i 615	. . . . . . . . . . 11 ⊢ (((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UPGraph)) ∧ 𝑥 ∈ dom (iEdg‘𝑆)) → ((𝐺 ∈ UHGraph ∧ 𝑆 SubGraph 𝐺) ∧ 𝑥 ∈ dom (iEdg‘𝑆)))
19	18	simplld 767	. . . . . . . . . 10 ⊢ (((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UPGraph)) ∧ 𝑥 ∈ dom (iEdg‘𝑆)) → 𝐺 ∈ UHGraph)
20		simpl 482	. . . . . . . . . . . 12 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UPGraph) → 𝑆 SubGraph 𝐺)
21	20	adantl 481	. . . . . . . . . . 11 ⊢ ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UPGraph)) → 𝑆 SubGraph 𝐺)
22	21	adantr 480	. . . . . . . . . 10 ⊢ (((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UPGraph)) ∧ 𝑥 ∈ dom (iEdg‘𝑆)) → 𝑆 SubGraph 𝐺)
23		simpr 484	. . . . . . . . . 10 ⊢ (((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UPGraph)) ∧ 𝑥 ∈ dom (iEdg‘𝑆)) → 𝑥 ∈ dom (iEdg‘𝑆))
24	1, 3, 19, 22, 23	subgruhgredgd 29211	. . . . . . . . 9 ⊢ (((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UPGraph)) ∧ 𝑥 ∈ dom (iEdg‘𝑆)) → ((iEdg‘𝑆)‘𝑥) ∈ (𝒫 (Vtx‘𝑆) ∖ {∅}))
25	4	uhgrfun 28993	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
26	7, 25	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ UPGraph → Fun (iEdg‘𝐺))
27	26	ad2antll 729	. . . . . . . . . . . . . 14 ⊢ ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UPGraph)) → Fun (iEdg‘𝐺))
28	27	adantr 480	. . . . . . . . . . . . 13 ⊢ (((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UPGraph)) ∧ 𝑥 ∈ dom (iEdg‘𝑆)) → Fun (iEdg‘𝐺))
29		simpll2 1214	. . . . . . . . . . . . 13 ⊢ (((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UPGraph)) ∧ 𝑥 ∈ dom (iEdg‘𝑆)) → (iEdg‘𝑆) ⊆ (iEdg‘𝐺))
30		funssfv 6879	. . . . . . . . . . . . 13 ⊢ ((Fun (iEdg‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ 𝑥 ∈ dom (iEdg‘𝑆)) → ((iEdg‘𝐺)‘𝑥) = ((iEdg‘𝑆)‘𝑥))
31	28, 29, 23, 30	syl3anc 1373	. . . . . . . . . . . 12 ⊢ (((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UPGraph)) ∧ 𝑥 ∈ dom (iEdg‘𝑆)) → ((iEdg‘𝐺)‘𝑥) = ((iEdg‘𝑆)‘𝑥))
32	31	eqcomd 2735	. . . . . . . . . . 11 ⊢ (((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UPGraph)) ∧ 𝑥 ∈ dom (iEdg‘𝑆)) → ((iEdg‘𝑆)‘𝑥) = ((iEdg‘𝐺)‘𝑥))
33	32	fveq2d 6862	. . . . . . . . . 10 ⊢ (((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UPGraph)) ∧ 𝑥 ∈ dom (iEdg‘𝑆)) → (♯‘((iEdg‘𝑆)‘𝑥)) = (♯‘((iEdg‘𝐺)‘𝑥)))
34		subgreldmiedg 29210	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝑥 ∈ dom (iEdg‘𝑆)) → 𝑥 ∈ dom (iEdg‘𝐺))
35	34	ex 412	. . . . . . . . . . . . . 14 ⊢ (𝑆 SubGraph 𝐺 → (𝑥 ∈ dom (iEdg‘𝑆) → 𝑥 ∈ dom (iEdg‘𝐺)))
36	35	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UPGraph) → (𝑥 ∈ dom (iEdg‘𝑆) → 𝑥 ∈ dom (iEdg‘𝐺)))
37	36	adantl 481	. . . . . . . . . . . 12 ⊢ ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UPGraph)) → (𝑥 ∈ dom (iEdg‘𝑆) → 𝑥 ∈ dom (iEdg‘𝐺)))
38		simpr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ dom (iEdg‘𝐺) ∧ 𝐺 ∈ UPGraph) → 𝐺 ∈ UPGraph)
39	26	funfnd 6547	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ UPGraph → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
40	39	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ dom (iEdg‘𝐺) ∧ 𝐺 ∈ UPGraph) → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
41		simpl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ dom (iEdg‘𝐺) ∧ 𝐺 ∈ UPGraph) → 𝑥 ∈ dom (iEdg‘𝐺))
42	2, 4	upgrle 29017	. . . . . . . . . . . . . . 15 ⊢ ((𝐺 ∈ UPGraph ∧ (iEdg‘𝐺) Fn dom (iEdg‘𝐺) ∧ 𝑥 ∈ dom (iEdg‘𝐺)) → (♯‘((iEdg‘𝐺)‘𝑥)) ≤ 2)
43	38, 40, 41, 42	syl3anc 1373	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ dom (iEdg‘𝐺) ∧ 𝐺 ∈ UPGraph) → (♯‘((iEdg‘𝐺)‘𝑥)) ≤ 2)
44	43	expcom 413	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ UPGraph → (𝑥 ∈ dom (iEdg‘𝐺) → (♯‘((iEdg‘𝐺)‘𝑥)) ≤ 2))
45	44	ad2antll 729	. . . . . . . . . . . 12 ⊢ ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UPGraph)) → (𝑥 ∈ dom (iEdg‘𝐺) → (♯‘((iEdg‘𝐺)‘𝑥)) ≤ 2))
46	37, 45	syld 47	. . . . . . . . . . 11 ⊢ ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UPGraph)) → (𝑥 ∈ dom (iEdg‘𝑆) → (♯‘((iEdg‘𝐺)‘𝑥)) ≤ 2))
47	46	imp 406	. . . . . . . . . 10 ⊢ (((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UPGraph)) ∧ 𝑥 ∈ dom (iEdg‘𝑆)) → (♯‘((iEdg‘𝐺)‘𝑥)) ≤ 2)
48	33, 47	eqbrtrd 5129	. . . . . . . . 9 ⊢ (((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UPGraph)) ∧ 𝑥 ∈ dom (iEdg‘𝑆)) → (♯‘((iEdg‘𝑆)‘𝑥)) ≤ 2)
49	14, 24, 48	elrabd 3661	. . . . . . . 8 ⊢ (((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UPGraph)) ∧ 𝑥 ∈ dom (iEdg‘𝑆)) → ((iEdg‘𝑆)‘𝑥) ∈ {𝑒 ∈ (𝒫 (Vtx‘𝑆) ∖ {∅}) ∣ (♯‘𝑒) ≤ 2})
50	49	ralrimiva 3125	. . . . . . 7 ⊢ ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UPGraph)) → ∀𝑥 ∈ dom (iEdg‘𝑆)((iEdg‘𝑆)‘𝑥) ∈ {𝑒 ∈ (𝒫 (Vtx‘𝑆) ∖ {∅}) ∣ (♯‘𝑒) ≤ 2})
51		fnfvrnss 7093	. . . . . . 7 ⊢ (((iEdg‘𝑆) Fn dom (iEdg‘𝑆) ∧ ∀𝑥 ∈ dom (iEdg‘𝑆)((iEdg‘𝑆)‘𝑥) ∈ {𝑒 ∈ (𝒫 (Vtx‘𝑆) ∖ {∅}) ∣ (♯‘𝑒) ≤ 2}) → ran (iEdg‘𝑆) ⊆ {𝑒 ∈ (𝒫 (Vtx‘𝑆) ∖ {∅}) ∣ (♯‘𝑒) ≤ 2})
52	12, 50, 51	syl2anc 584	. . . . . 6 ⊢ ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UPGraph)) → ran (iEdg‘𝑆) ⊆ {𝑒 ∈ (𝒫 (Vtx‘𝑆) ∖ {∅}) ∣ (♯‘𝑒) ≤ 2})
53		df-f 6515	. . . . . 6 ⊢ ((iEdg‘𝑆):dom (iEdg‘𝑆)⟶{𝑒 ∈ (𝒫 (Vtx‘𝑆) ∖ {∅}) ∣ (♯‘𝑒) ≤ 2} ↔ ((iEdg‘𝑆) Fn dom (iEdg‘𝑆) ∧ ran (iEdg‘𝑆) ⊆ {𝑒 ∈ (𝒫 (Vtx‘𝑆) ∖ {∅}) ∣ (♯‘𝑒) ≤ 2}))
54	12, 52, 53	sylanbrc 583	. . . . 5 ⊢ ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UPGraph)) → (iEdg‘𝑆):dom (iEdg‘𝑆)⟶{𝑒 ∈ (𝒫 (Vtx‘𝑆) ∖ {∅}) ∣ (♯‘𝑒) ≤ 2})
55		subgrv 29197	. . . . . . . 8 ⊢ (𝑆 SubGraph 𝐺 → (𝑆 ∈ V ∧ 𝐺 ∈ V))
56	1, 3	isupgr 29011	. . . . . . . . 9 ⊢ (𝑆 ∈ V → (𝑆 ∈ UPGraph ↔ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶{𝑒 ∈ (𝒫 (Vtx‘𝑆) ∖ {∅}) ∣ (♯‘𝑒) ≤ 2}))
57	56	adantr 480	. . . . . . . 8 ⊢ ((𝑆 ∈ V ∧ 𝐺 ∈ V) → (𝑆 ∈ UPGraph ↔ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶{𝑒 ∈ (𝒫 (Vtx‘𝑆) ∖ {∅}) ∣ (♯‘𝑒) ≤ 2}))
58	55, 57	syl 17	. . . . . . 7 ⊢ (𝑆 SubGraph 𝐺 → (𝑆 ∈ UPGraph ↔ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶{𝑒 ∈ (𝒫 (Vtx‘𝑆) ∖ {∅}) ∣ (♯‘𝑒) ≤ 2}))
59	58	adantr 480	. . . . . 6 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UPGraph) → (𝑆 ∈ UPGraph ↔ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶{𝑒 ∈ (𝒫 (Vtx‘𝑆) ∖ {∅}) ∣ (♯‘𝑒) ≤ 2}))
60	59	adantl 481	. . . . 5 ⊢ ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UPGraph)) → (𝑆 ∈ UPGraph ↔ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶{𝑒 ∈ (𝒫 (Vtx‘𝑆) ∖ {∅}) ∣ (♯‘𝑒) ≤ 2}))
61	54, 60	mpbird 257	. . . 4 ⊢ ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UPGraph)) → 𝑆 ∈ UPGraph)
62	61	ex 412	. . 3 ⊢ (((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) → ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UPGraph) → 𝑆 ∈ UPGraph))
63	6, 62	syl 17	. 2 ⊢ (𝑆 SubGraph 𝐺 → ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UPGraph) → 𝑆 ∈ UPGraph))
64	63	anabsi8 672	1 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑆 SubGraph 𝐺) → 𝑆 ∈ UPGraph)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044  {crab 3405  Vcvv 3447   ∖ cdif 3911   ⊆ wss 3914  ∅c0 4296  𝒫 cpw 4563  {csn 4589   class class class wbr 5107  dom cdm 5638  ran crn 5639  Fun wfun 6505   Fn wfn 6506  ⟶wf 6507  ‘cfv 6511   ≤ cle 11209  2c2 12241  ♯chash 14295  Vtxcvtx 28923  iEdgciedg 28924  Edgcedg 28974  UHGraphcuhgr 28983  UPGraphcupgr 29007   SubGraph csubgr 29194

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 5251  ax-nul 5261  ax-pr 5387  ax-un 7711

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 3406  df-v 3449  df-sbc 3754  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-fv 6519  df-edg 28975  df-uhgr 28985  df-upgr 29009  df-subgr 29195

This theorem is referenced by:  upgrspan  29220  isubgrupgr  47870