Theorem subupgr 29674

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 2766	. . . 4 ⊢ (Vtx‘𝑆) = (Vtx‘𝑆)
2		eqid 2766	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
3		eqid 2766	. . . 4 ⊢ (iEdg‘𝑆) = (iEdg‘𝑆)
4		eqid 2766	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
5		eqid 2766	. . . 4 ⊢ (Edg‘𝑆) = (Edg‘𝑆)
6	1, 2, 3, 4, 5	subgrprop2 29661	. . 3 ⊢ (𝑆 SubGraph 𝐺 → ((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)))
7		upgruhgr 29489	. . . . . . . . . 10 ⊢ (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph)
8		subgruhgrfun 29669	. . . . . . . . . 10 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 SubGraph 𝐺) → Fun (iEdg‘𝑆))
9	7, 8	sylan 592	. . . . . . . . 9 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑆 SubGraph 𝐺) → Fun (iEdg‘𝑆))
10	9	ancoms 464	. . . . . . . 8 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UPGraph) → Fun (iEdg‘𝑆))
11	10	funfnd 6574	. . . . . . 7 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UPGraph) → (iEdg‘𝑆) Fn dom (iEdg‘𝑆))
12	11	adantl 487	. . . . . 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 6888	. . . . . . . . . 10 ⊢ (𝑒 = ((iEdg‘𝑆)‘𝑥) → (♯‘𝑒) = (♯‘((iEdg‘𝑆)‘𝑥)))
14	13	breq1d 5124	. . . . . . . . 9 ⊢ (𝑒 = ((iEdg‘𝑆)‘𝑥) → ((♯‘𝑒) ≤ 2 ↔ (♯‘((iEdg‘𝑆)‘𝑥)) ≤ 2))
15	7	anim2i 629	. . . . . . . . . . . . . 14 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UPGraph) → (𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UHGraph))
16	15	adantl 487	. . . . . . . . . . . . 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 467	. . . . . . . . . . . 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 627	. . . . . . . . . . 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 780	. . . . . . . . . 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 488	. . . . . . . . . . . 12 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UPGraph) → 𝑆 SubGraph 𝐺)
21	20	adantl 487	. . . . . . . . . . 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 486	. . . . . . . . . 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 490	. . . . . . . . . 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 29671	. . . . . . . . 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 29453	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
26	7, 25	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ UPGraph → Fun (iEdg‘𝐺))
27	26	ad2antll 742	. . . . . . . . . . . . . 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 486	. . . . . . . . . . . . 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 1232	. . . . . . . . . . . . 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 6909	. . . . . . . . . . . . 13 ⊢ ((Fun (iEdg‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ 𝑥 ∈ dom (iEdg‘𝑆)) → ((iEdg‘𝐺)‘𝑥) = ((iEdg‘𝑆)‘𝑥))
31	28, 29, 23, 30	syl3anc 1398	. . . . . . . . . . . 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 2772	. . . . . . . . . . 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 6892	. . . . . . . . . 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 29670	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝑥 ∈ dom (iEdg‘𝑆)) → 𝑥 ∈ dom (iEdg‘𝐺))
35	34	ex 418	. . . . . . . . . . . . . 14 ⊢ (𝑆 SubGraph 𝐺 → (𝑥 ∈ dom (iEdg‘𝑆) → 𝑥 ∈ dom (iEdg‘𝐺)))
36	35	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UPGraph) → (𝑥 ∈ dom (iEdg‘𝑆) → 𝑥 ∈ dom (iEdg‘𝐺)))
37	36	adantl 487	. . . . . . . . . . . 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 490	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ dom (iEdg‘𝐺) ∧ 𝐺 ∈ UPGraph) → 𝐺 ∈ UPGraph)
39	26	funfnd 6574	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ UPGraph → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
40	39	adantl 487	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ dom (iEdg‘𝐺) ∧ 𝐺 ∈ UPGraph) → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
41		simpl 488	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ dom (iEdg‘𝐺) ∧ 𝐺 ∈ UPGraph) → 𝑥 ∈ dom (iEdg‘𝐺))
42	2, 4	upgrle 29477	. . . . . . . . . . . . . . 15 ⊢ ((𝐺 ∈ UPGraph ∧ (iEdg‘𝐺) Fn dom (iEdg‘𝐺) ∧ 𝑥 ∈ dom (iEdg‘𝐺)) → (♯‘((iEdg‘𝐺)‘𝑥)) ≤ 2)
43	38, 40, 41, 42	syl3anc 1398	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ dom (iEdg‘𝐺) ∧ 𝐺 ∈ UPGraph) → (♯‘((iEdg‘𝐺)‘𝑥)) ≤ 2)
44	43	expcom 419	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ UPGraph → (𝑥 ∈ dom (iEdg‘𝐺) → (♯‘((iEdg‘𝐺)‘𝑥)) ≤ 2))
45	44	ad2antll 742	. . . . . . . . . . . 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 48	. . . . . . . . . . 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 412	. . . . . . . . . 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 5138	. . . . . . . . 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 3655	. . . . . . . 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 3160	. . . . . . 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 7123	. . . . . . 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 596	. . . . . 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 6547	. . . . . 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 595	. . . . 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 29657	. . . . . . . 8 ⊢ (𝑆 SubGraph 𝐺 → (𝑆 ∈ V ∧ 𝐺 ∈ V))
56	1, 3	isupgr 29471	. . . . . . . . 9 ⊢ (𝑆 ∈ V → (𝑆 ∈ UPGraph ↔ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶{𝑒 ∈ (𝒫 (Vtx‘𝑆) ∖ {∅}) ∣ (♯‘𝑒) ≤ 2}))
57	56	adantr 486	. . . . . . . 8 ⊢ ((𝑆 ∈ V ∧ 𝐺 ∈ V) → (𝑆 ∈ UPGraph ↔ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶{𝑒 ∈ (𝒫 (Vtx‘𝑆) ∖ {∅}) ∣ (♯‘𝑒) ≤ 2}))
58	55, 57	syl 18	. . . . . . 7 ⊢ (𝑆 SubGraph 𝐺 → (𝑆 ∈ UPGraph ↔ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶{𝑒 ∈ (𝒫 (Vtx‘𝑆) ∖ {∅}) ∣ (♯‘𝑒) ≤ 2}))
59	58	adantr 486	. . . . . 6 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UPGraph) → (𝑆 ∈ UPGraph ↔ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶{𝑒 ∈ (𝒫 (Vtx‘𝑆) ∖ {∅}) ∣ (♯‘𝑒) ≤ 2}))
60	59	adantl 487	. . . . 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 260	. . . 4 ⊢ ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UPGraph)) → 𝑆 ∈ UPGraph)
62	61	ex 418	. . 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 18	. 2 ⊢ (𝑆 SubGraph 𝐺 → ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UPGraph) → 𝑆 ∈ UPGraph))
64	63	anabsi8 685	1 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑆 SubGraph 𝐺) → 𝑆 ∈ UPGraph)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ∀wral 3082  {crab 3419  Vcvv 3458   ∖ cdif 3905   ⊆ wss 3908  ∅c0 4289  𝒫 cpw 4567  {csn 4594   class class class wbr 5114  dom cdm 5666  ran crn 5667  Fun wfun 6537   Fn wfn 6538  ⟶wf 6539  ‘cfv 6543   ≤ cle 11262  2c2 12313  ♯chash 14386  Vtxcvtx 29383  iEdgciedg 29384  Edgcedg 29434  UHGraphcuhgr 29443  UPGraphcupgr 29467   SubGraph csubgr 29654

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 2738  ax-sep 5262  ax-nul 5274  ax-pr 5409  ax-un 7745

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 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-edg 29435  df-uhgr 29445  df-upgr 29469  df-subgr 29655

This theorem is used by:  upgrspan  29680  isubgrupgr  48676