Theorem subupgr 16123

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 2231	. . . 4 ⊢ (Vtx‘𝑆) = (Vtx‘𝑆)
2		eqid 2231	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
3		eqid 2231	. . . 4 ⊢ (iEdg‘𝑆) = (iEdg‘𝑆)
4		eqid 2231	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
5		eqid 2231	. . . 4 ⊢ (Edg‘𝑆) = (Edg‘𝑆)
6	1, 2, 3, 4, 5	subgrprop2 16110	. . 3 ⊢ (𝑆 SubGraph 𝐺 → ((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)))
7		upgruhgr 15961	. . . . . . . . . . 11 ⊢ (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph)
8		subgruhgrfun 16118	. . . . . . . . . . 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 5357	. . . . . . . 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 4091	. . . . . . . . . . 11 ⊢ (𝑒 = ((iEdg‘𝑆)‘𝑥) → (𝑒 ≈ 1o ↔ ((iEdg‘𝑆)‘𝑥) ≈ 1o))
14		breq1 4091	. . . . . . . . . . 11 ⊢ (𝑒 = ((iEdg‘𝑆)‘𝑥) → (𝑒 ≈ 2o ↔ ((iEdg‘𝑆)‘𝑥) ≈ 2o))
15	13, 14	orbi12d 800	. . . . . . . . . 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 16120	. . . . . . . . . 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 16119	. . . . . . . . . . . . . . 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 15927	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
31	7, 30	syl 14	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺 ∈ UPGraph → Fun (iEdg‘𝐺))
32	31	funfnd 5357	. . . . . . . . . . . . . . . . 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 15951	. . . . . . . . . . . . . . . 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 1273	. . . . . . . . . . . . . . 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 1063	. . . . . . . . . . . . . . 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 5665	. . . . . . . . . . . . . . 15 ⊢ ((Fun (iEdg‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ 𝑥 ∈ dom (iEdg‘𝑆)) → ((iEdg‘𝐺)‘𝑥) = ((iEdg‘𝑆)‘𝑥))
45	42, 43, 24, 44	syl3anc 1273	. . . . . . . . . . . . . 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 2237	. . . . . . . . . . . . 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 4098	. . . . . . . . . . . 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 4098	. . . . . . . . . . . 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 800	. . . . . . . . . . 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 2964	. . . . . . . . 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 2605	. . . . . . . 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 5807	. . . . . . . 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 5330	. . . . . . 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 7402	. . . . . . 7 ⊢ {𝑒 ∈ {𝑠 ∈ 𝒫 (Vtx‘𝑆) ∣ ∃𝑗 𝑗 ∈ 𝑠} ∣ (𝑒 ≈ 1o ∨ 𝑒 ≈ 2o)} = {𝑒 ∈ 𝒫 (Vtx‘𝑆) ∣ (𝑒 ≈ 1o ∨ 𝑒 ≈ 2o)}
58		feq3 5467	. . . . . . 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 16106	. . . . . . 7 ⊢ (𝑆 SubGraph 𝐺 → (𝑆 ∈ V ∧ 𝐺 ∈ V))
62	1, 3	isupgren 15945	. . . . . . . 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 715   ∧ w3a 1004   = wceq 1397  ∃wex 1540   ∈ wcel 2202  ∀wral 2510  {crab 2514  Vcvv 2802   ⊆ wss 3200  𝒫 cpw 3652   class class class wbr 4088  dom cdm 4725  ran crn 4726  Fun wfun 5320   Fn wfn 5321  ⟶wf 5322  ‘cfv 5326  1_oc1o 6574  2_oc2o 6575   ≈ cen 6906  Vtxcvtx 15862  iEdgciedg 15863  Edgcedg 15907  UHGraphcuhgr 15917  UPGraphcupgr 15941   SubGraph csubgr 16103

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-cnre 8142

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-1o 6581  df-2o 6582  df-en 6909  df-sub 8351  df-inn 9143  df-2 9201  df-3 9202  df-4 9203  df-5 9204  df-6 9205  df-7 9206  df-8 9207  df-9 9208  df-n0 9402  df-dec 9611  df-ndx 13084  df-slot 13085  df-base 13087  df-edgf 15855  df-vtx 15864  df-iedg 15865  df-edg 15908  df-uhgrm 15919  df-upgren 15943  df-subgr 16104

This theorem is referenced by:  upgrspan  16129