Theorem subupgr 16514

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 2238	. . . 4 ⊢ (Vtx‘𝑆) = (Vtx‘𝑆)
2		eqid 2238	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
3		eqid 2238	. . . 4 ⊢ (iEdg‘𝑆) = (iEdg‘𝑆)
4		eqid 2238	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
5		eqid 2238	. . . 4 ⊢ (Edg‘𝑆) = (Edg‘𝑆)
6	1, 2, 3, 4, 5	subgrprop2 16501	. . 3 ⊢ (𝑆 SubGraph 𝐺 → ((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)))
7		upgruhgr 16352	. . . . . . . . . . 11 ⊢ (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph)
8		subgruhgrfun 16509	. . . . . . . . . . 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 5408	. . . . . . . 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 4133	. . . . . . . . . . 11 ⊢ (𝑒 = ((iEdg‘𝑆)‘𝑥) → (𝑒 ≈ 1o ↔ ((iEdg‘𝑆)‘𝑥) ≈ 1o))
14		breq1 4133	. . . . . . . . . . 11 ⊢ (𝑒 = ((iEdg‘𝑆)‘𝑥) → (𝑒 ≈ 2o ↔ ((iEdg‘𝑆)‘𝑥) ≈ 2o))
15	13, 14	orbi12d 805	. . . . . . . . . 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 532	. . . . . . . . . . 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 16511	. . . . . . . . . 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 16510	. . . . . . . . . . . . . . 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 494	. . . . . . . . . . . . 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 16318	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
31	7, 30	syl 14	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺 ∈ UPGraph → Fun (iEdg‘𝐺))
32	31	funfnd 5408	. . . . . . . . . . . . . . . . 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 16342	. . . . . . . . . . . . . . . 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 1278	. . . . . . . . . . . . . . 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 495	. . . . . . . . . . . . 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 495	. . . . . . . . . . . . . . . 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 1068	. . . . . . . . . . . . . . 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 5721	. . . . . . . . . . . . . . 15 ⊢ ((Fun (iEdg‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ 𝑥 ∈ dom (iEdg‘𝑆)) → ((iEdg‘𝐺)‘𝑥) = ((iEdg‘𝑆)‘𝑥))
45	42, 43, 24, 44	syl3anc 1278	. . . . . . . . . . . . . 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 2244	. . . . . . . . . . . . 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 4140	. . . . . . . . . . . 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 4140	. . . . . . . . . . . 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 805	. . . . . . . . . . 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 2984	. . . . . . . . 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 2623	. . . . . . . 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 5868	. . . . . . . 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 415	. . . . . . 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 5381	. . . . . . 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 421	. . . . . 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 7544	. . . . . . 7 ⊢ {𝑒 ∈ {𝑠 ∈ 𝒫 (Vtx‘𝑆) ∣ ∃𝑗 𝑗 ∈ 𝑠} ∣ (𝑒 ≈ 1o ∨ 𝑒 ≈ 2o)} = {𝑒 ∈ 𝒫 (Vtx‘𝑆) ∣ (𝑒 ≈ 1o ∨ 𝑒 ≈ 2o)}
58		feq3 5518	. . . . . . 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 16497	. . . . . . 7 ⊢ (𝑆 SubGraph 𝐺 → (𝑆 ∈ V ∧ 𝐺 ∈ V))
62	1, 3	isupgren 16336	. . . . . . . 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 494	. . . . 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 588	1 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑆 SubGraph 𝐺) → 𝑆 ∈ UPGraph)

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 720   ∧ w3a 1009   = wceq 1402  ∃wex 1545   ∈ wcel 2209  ∀wral 2528  {crab 2532  Vcvv 2821   ⊆ wss 3220  𝒫 cpw 3688   class class class wbr 4130  dom cdm 4774  ran crn 4775  Fun wfun 5371   Fn wfn 5372  ⟶wf 5373  ‘cfv 5377  1_oc1o 6680  2_oc2o 6681   ≈ cen 7020  Vtxcvtx 16253  iEdgciedg 16254  Edgcedg 16298  UHGraphcuhgr 16308  UPGraphcupgr 16332   SubGraph csubgr 16494

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-suc 4516  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-1o 6687  df-2o 6688  df-en 7023  df-sub 8499  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-5 9366  df-6 9367  df-7 9368  df-8 9369  df-9 9370  df-n0 9564  df-dec 9778  df-ndx 13355  df-slot 13356  df-base 13358  df-edgf 16246  df-vtx 16255  df-iedg 16256  df-edg 16299  df-uhgrm 16310  df-upgren 16334  df-subgr 16495

This theorem is used by:  upgrspan  16520