Theorem subupgr 16394

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 𝑗 𝑠 𝑒 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2234	. . . 4 ⊢ (Vtx‘𝑆) = (Vtx‘𝑆)
2		eqid 2234	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
3		eqid 2234	. . . 4 ⊢ (iEdg‘𝑆) = (iEdg‘𝑆)
4		eqid 2234	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
5		eqid 2234	. . . 4 ⊢ (Edg‘𝑆) = (Edg‘𝑆)
6	1, 2, 3, 4, 5	subgrprop2 16381	. . 3 ⊢ (𝑆 SubGraph 𝐺 → ((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)))
7		upgruhgr 16232	. . . . . . . . . . 11 ⊢ (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph)
8		subgruhgrfun 16389	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 SubGraph 𝐺) → Fun (iEdg‘𝑆))
9	7, 8	sylan 283	. . . . . . . . . 10 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑆 SubGraph 𝐺) → Fun (iEdg‘𝑆))
10	9	ancoms 268	. . . . . . . . 9 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UPGraph) → Fun (iEdg‘𝑆))
11	10	funfnd 5388	. . . . . . . 8 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UPGraph) → (iEdg‘𝑆) Fn dom (iEdg‘𝑆))
12	11	adantl 277	. . . . . . 7 ⊢ ((((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		breq1 4117	. . . . . . . . . . 11 ⊢ (𝑒 = ((iEdg‘𝑆)‘𝑥) → (𝑒 ≈ 1o ↔ ((iEdg‘𝑆)‘𝑥) ≈ 1o))
14		breq1 4117	. . . . . . . . . . 11 ⊢ (𝑒 = ((iEdg‘𝑆)‘𝑥) → (𝑒 ≈ 2o ↔ ((iEdg‘𝑆)‘𝑥) ≈ 2o))
15	13, 14	orbi12d 801	. . . . . . . . . 10 ⊢ (𝑒 = ((iEdg‘𝑆)‘𝑥) → ((𝑒 ≈ 1o ∨ 𝑒 ≈ 2o) ↔ (((iEdg‘𝑆)‘𝑥) ≈ 1o ∨ ((iEdg‘𝑆)‘𝑥) ≈ 2o)))
16	7	anim2i 342	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UPGraph) → (𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UHGraph))
17	16	adantl 277	. . . . . . . . . . . . . 14 ⊢ ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UPGraph)) → (𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UHGraph))
18	17	ancomd 267	. . . . . . . . . . . . 13 ⊢ ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UPGraph)) → (𝐺 ∈ UHGraph ∧ 𝑆 SubGraph 𝐺))
19	18	anim1i 340	. . . . . . . . . . . 12 ⊢ (((((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)))
20	19	simplld 528	. . . . . . . . . . 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)
21		simpl 109	. . . . . . . . . . . . 13 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UPGraph) → 𝑆 SubGraph 𝐺)
22	21	adantl 277	. . . . . . . . . . . 12 ⊢ ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UPGraph)) → 𝑆 SubGraph 𝐺)
23	22	adantr 276	. . . . . . . . . . 11 ⊢ (((((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)
24		simpr 110	. . . . . . . . . . 11 ⊢ (((((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))
25	1, 3, 20, 23, 24	subgruhgredgdm 16391	. . . . . . . . . 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) ∈ {𝑠 ∈ 𝒫 (Vtx‘𝑆) ∣ ∃𝑗 𝑗 ∈ 𝑠})
26		subgreldmiedg 16390	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝑥 ∈ dom (iEdg‘𝑆)) → 𝑥 ∈ dom (iEdg‘𝐺))
27	26	ex 115	. . . . . . . . . . . . . 14 ⊢ (𝑆 SubGraph 𝐺 → (𝑥 ∈ dom (iEdg‘𝑆) → 𝑥 ∈ dom (iEdg‘𝐺)))
28	27	ad2antrl 490	. . . . . . . . . . . . 13 ⊢ ((((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)))
29		simpr 110	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ dom (iEdg‘𝐺) ∧ 𝐺 ∈ UPGraph) → 𝐺 ∈ UPGraph)
30	4	uhgrfun 16198	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
31	7, 30	syl 14	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺 ∈ UPGraph → Fun (iEdg‘𝐺))
32	31	funfnd 5388	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺 ∈ UPGraph → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
33	32	adantl 277	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ dom (iEdg‘𝐺) ∧ 𝐺 ∈ UPGraph) → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
34		simpl 109	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ dom (iEdg‘𝐺) ∧ 𝐺 ∈ UPGraph) → 𝑥 ∈ dom (iEdg‘𝐺))
35	2, 4	upgr1or2 16222	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 ∈ UPGraph ∧ (iEdg‘𝐺) Fn dom (iEdg‘𝐺) ∧ 𝑥 ∈ dom (iEdg‘𝐺)) → (((iEdg‘𝐺)‘𝑥) ≈ 1o ∨ ((iEdg‘𝐺)‘𝑥) ≈ 2o))
36	29, 33, 34, 35	syl3anc 1274	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ dom (iEdg‘𝐺) ∧ 𝐺 ∈ UPGraph) → (((iEdg‘𝐺)‘𝑥) ≈ 1o ∨ ((iEdg‘𝐺)‘𝑥) ≈ 2o))
37	36	expcom 116	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ UPGraph → (𝑥 ∈ dom (iEdg‘𝐺) → (((iEdg‘𝐺)‘𝑥) ≈ 1o ∨ ((iEdg‘𝐺)‘𝑥) ≈ 2o)))
38	37	ad2antll 491	. . . . . . . . . . . . 13 ⊢ ((((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) ≈ 1o ∨ ((iEdg‘𝐺)‘𝑥) ≈ 2o)))
39	28, 38	syld 45	. . . . . . . . . . . 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) ≈ 1o ∨ ((iEdg‘𝐺)‘𝑥) ≈ 2o)))
40	39	imp 124	. . . . . . . . . . 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) ≈ 1o ∨ ((iEdg‘𝐺)‘𝑥) ≈ 2o))
41	31	ad2antll 491	. . . . . . . . . . . . . . . 16 ⊢ ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UPGraph)) → Fun (iEdg‘𝐺))
42	41	adantr 276	. . . . . . . . . . . . . . 15 ⊢ (((((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))
43		simpll2 1064	. . . . . . . . . . . . . . 15 ⊢ (((((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))
44		funssfv 5701	. . . . . . . . . . . . . . 15 ⊢ ((Fun (iEdg‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ 𝑥 ∈ dom (iEdg‘𝑆)) → ((iEdg‘𝐺)‘𝑥) = ((iEdg‘𝑆)‘𝑥))
45	42, 43, 24, 44	syl3anc 1274	. . . . . . . . . . . . . 14 ⊢ (((((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))
46	45	eqcomd 2240	. . . . . . . . . . . . 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)‘𝑥) = ((iEdg‘𝐺)‘𝑥))
47	46	breq1d 4124	. . . . . . . . . . . 12 ⊢ (((((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) ≈ 1o ↔ ((iEdg‘𝐺)‘𝑥) ≈ 1o))
48	46	breq1d 4124	. . . . . . . . . . . 12 ⊢ (((((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) ≈ 2o ↔ ((iEdg‘𝐺)‘𝑥) ≈ 2o))
49	47, 48	orbi12d 801	. . . . . . . . . . 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) ≈ 1o ∨ ((iEdg‘𝑆)‘𝑥) ≈ 2o) ↔ (((iEdg‘𝐺)‘𝑥) ≈ 1o ∨ ((iEdg‘𝐺)‘𝑥) ≈ 2o)))
50	40, 49	mpbird 167	. . . . . . . . . 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) ≈ 1o ∨ ((iEdg‘𝑆)‘𝑥) ≈ 2o))
51	15, 25, 50	elrabd 2978	. . . . . . . . 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) ∈ {𝑒 ∈ {𝑠 ∈ 𝒫 (Vtx‘𝑆) ∣ ∃𝑗 𝑗 ∈ 𝑠} ∣ (𝑒 ≈ 1o ∨ 𝑒 ≈ 2o)})
52	51	ralrimiva 2617	. . . . . . . 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 ∈ {𝑠 ∈ 𝒫 (Vtx‘𝑆) ∣ ∃𝑗 𝑗 ∈ 𝑠} ∣ (𝑒 ≈ 1o ∨ 𝑒 ≈ 2o)})
53		fnfvrnss 5842	. . . . . . . 8 ⊢ (((iEdg‘𝑆) Fn dom (iEdg‘𝑆) ∧ ∀𝑥 ∈ dom (iEdg‘𝑆)((iEdg‘𝑆)‘𝑥) ∈ {𝑒 ∈ {𝑠 ∈ 𝒫 (Vtx‘𝑆) ∣ ∃𝑗 𝑗 ∈ 𝑠} ∣ (𝑒 ≈ 1o ∨ 𝑒 ≈ 2o)}) → ran (iEdg‘𝑆) ⊆ {𝑒 ∈ {𝑠 ∈ 𝒫 (Vtx‘𝑆) ∣ ∃𝑗 𝑗 ∈ 𝑠} ∣ (𝑒 ≈ 1o ∨ 𝑒 ≈ 2o)})
54	12, 52, 53	syl2anc 411	. . . . . . 7 ⊢ ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UPGraph)) → ran (iEdg‘𝑆) ⊆ {𝑒 ∈ {𝑠 ∈ 𝒫 (Vtx‘𝑆) ∣ ∃𝑗 𝑗 ∈ 𝑠} ∣ (𝑒 ≈ 1o ∨ 𝑒 ≈ 2o)})
55		df-f 5361	. . . . . . 7 ⊢ ((iEdg‘𝑆):dom (iEdg‘𝑆)⟶{𝑒 ∈ {𝑠 ∈ 𝒫 (Vtx‘𝑆) ∣ ∃𝑗 𝑗 ∈ 𝑠} ∣ (𝑒 ≈ 1o ∨ 𝑒 ≈ 2o)} ↔ ((iEdg‘𝑆) Fn dom (iEdg‘𝑆) ∧ ran (iEdg‘𝑆) ⊆ {𝑒 ∈ {𝑠 ∈ 𝒫 (Vtx‘𝑆) ∣ ∃𝑗 𝑗 ∈ 𝑠} ∣ (𝑒 ≈ 1o ∨ 𝑒 ≈ 2o)}))
56	12, 54, 55	sylanbrc 417	. . . . . 6 ⊢ ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UPGraph)) → (iEdg‘𝑆):dom (iEdg‘𝑆)⟶{𝑒 ∈ {𝑠 ∈ 𝒫 (Vtx‘𝑆) ∣ ∃𝑗 𝑗 ∈ 𝑠} ∣ (𝑒 ≈ 1o ∨ 𝑒 ≈ 2o)})
57		sspw1or2 7508	. . . . . . 7 ⊢ {𝑒 ∈ {𝑠 ∈ 𝒫 (Vtx‘𝑆) ∣ ∃𝑗 𝑗 ∈ 𝑠} ∣ (𝑒 ≈ 1o ∨ 𝑒 ≈ 2o)} = {𝑒 ∈ 𝒫 (Vtx‘𝑆) ∣ (𝑒 ≈ 1o ∨ 𝑒 ≈ 2o)}
58		feq3 5498	. . . . . . 7 ⊢ ({𝑒 ∈ {𝑠 ∈ 𝒫 (Vtx‘𝑆) ∣ ∃𝑗 𝑗 ∈ 𝑠} ∣ (𝑒 ≈ 1o ∨ 𝑒 ≈ 2o)} = {𝑒 ∈ 𝒫 (Vtx‘𝑆) ∣ (𝑒 ≈ 1o ∨ 𝑒 ≈ 2o)} → ((iEdg‘𝑆):dom (iEdg‘𝑆)⟶{𝑒 ∈ {𝑠 ∈ 𝒫 (Vtx‘𝑆) ∣ ∃𝑗 𝑗 ∈ 𝑠} ∣ (𝑒 ≈ 1o ∨ 𝑒 ≈ 2o)} ↔ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶{𝑒 ∈ 𝒫 (Vtx‘𝑆) ∣ (𝑒 ≈ 1o ∨ 𝑒 ≈ 2o)}))
59	57, 58	ax-mp 5	. . . . . 6 ⊢ ((iEdg‘𝑆):dom (iEdg‘𝑆)⟶{𝑒 ∈ {𝑠 ∈ 𝒫 (Vtx‘𝑆) ∣ ∃𝑗 𝑗 ∈ 𝑠} ∣ (𝑒 ≈ 1o ∨ 𝑒 ≈ 2o)} ↔ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶{𝑒 ∈ 𝒫 (Vtx‘𝑆) ∣ (𝑒 ≈ 1o ∨ 𝑒 ≈ 2o)})
60	56, 59	sylib 122	. . . . 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 ≈ 1o ∨ 𝑒 ≈ 2o)})
61		subgrv 16377	. . . . . . 7 ⊢ (𝑆 SubGraph 𝐺 → (𝑆 ∈ V ∧ 𝐺 ∈ V))
62	1, 3	isupgren 16216	. . . . . . . 8 ⊢ (𝑆 ∈ V → (𝑆 ∈ UPGraph ↔ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶{𝑒 ∈ 𝒫 (Vtx‘𝑆) ∣ (𝑒 ≈ 1o ∨ 𝑒 ≈ 2o)}))
63	62	adantr 276	. . . . . . 7 ⊢ ((𝑆 ∈ V ∧ 𝐺 ∈ V) → (𝑆 ∈ UPGraph ↔ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶{𝑒 ∈ 𝒫 (Vtx‘𝑆) ∣ (𝑒 ≈ 1o ∨ 𝑒 ≈ 2o)}))
64	61, 63	syl 14	. . . . . 6 ⊢ (𝑆 SubGraph 𝐺 → (𝑆 ∈ UPGraph ↔ (iEdg‘𝑆):dom (iEdg‘𝑆)⟶{𝑒 ∈ 𝒫 (Vtx‘𝑆) ∣ (𝑒 ≈ 1o ∨ 𝑒 ≈ 2o)}))
65	64	ad2antrl 490	. . . . 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 ≈ 1o ∨ 𝑒 ≈ 2o)}))
66	60, 65	mpbird 167	. . . 4 ⊢ ((((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) ∧ (𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UPGraph)) → 𝑆 ∈ UPGraph)
67	66	ex 115	. . 3 ⊢ (((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) → ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UPGraph) → 𝑆 ∈ UPGraph))
68	6, 67	syl 14	. 2 ⊢ (𝑆 SubGraph 𝐺 → ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UPGraph) → 𝑆 ∈ UPGraph))
69	68	anabsi8 584	1 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑆 SubGraph 𝐺) → 𝑆 ∈ UPGraph)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 716   ∧ w3a 1005   = wceq 1398  ∃wex 1541   ∈ wcel 2205  ∀wral 2522  {crab 2526  Vcvv 2815   ⊆ wss 3214  𝒫 cpw 3674   class class class wbr 4114  dom cdm 4754  ran crn 4755  Fun wfun 5351   Fn wfn 5352  ⟶wf 5353  ‘cfv 5357  1_oc1o 6653  2_oc2o 6654   ≈ cen 6986  Vtxcvtx 16133  iEdgciedg 16134  Edgcedg 16178  UHGraphcuhgr 16188  UPGraphcupgr 16212   SubGraph csubgr 16374

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-suc 4497  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-1o 6660  df-2o 6661  df-en 6989  df-sub 8462  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-5 9316  df-6 9317  df-7 9318  df-8 9319  df-9 9320  df-n0 9514  df-dec 9728  df-ndx 13299  df-slot 13300  df-base 13302  df-edgf 16126  df-vtx 16135  df-iedg 16136  df-edg 16179  df-uhgrm 16190  df-upgren 16214  df-subgr 16375

This theorem is referenced by:  upgrspan  16400