Theorem subupgr 16428

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 e. UPGraph /\ S SubGraph G ) -> S e. 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 16415	. . 3 |- ( S SubGraph G -> ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) )
7		upgruhgr 16266	. . . . . . . . . . 11 |- ( G e. UPGraph -> G e. UHGraph )
8		subgruhgrfun 16423	. . . . . . . . . . 11 |- ( ( G e. UHGraph /\ S SubGraph G ) -> Fun (iEdg ` S ) )
9	7, 8	sylan 283	. . . . . . . . . 10 |- ( ( G e. UPGraph /\ S SubGraph G ) -> Fun (iEdg ` S ) )
10	9	ancoms 268	. . . . . . . . 9 |- ( ( S SubGraph G /\ G e. UPGraph ) -> Fun (iEdg ` S ) )
11	10	funfnd 5403	. . . . . . . 8 |- ( ( S SubGraph G /\ G e. UPGraph ) -> (iEdg ` S ) Fn dom (iEdg ` S ) )
12	11	adantl 277	. . . . . . 7 |- ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. UPGraph ) ) -> (iEdg ` S ) Fn dom (iEdg ` S ) )
13		breq1 4128	. . . . . . . . . . 11 |- ( e = ( (iEdg ` S ) ` x ) -> ( e ~~ 1o <-> ( (iEdg ` S ) ` x ) ~~ 1o ) )
14		breq1 4128	. . . . . . . . . . 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 e. UPGraph ) -> ( S SubGraph G /\ G e. UHGraph ) )
17	16	adantl 277	. . . . . . . . . . . . . 14 |- ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. UPGraph ) ) -> ( S SubGraph G /\ G e. UHGraph ) )
18	17	ancomd 267	. . . . . . . . . . . . 13 |- ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. UPGraph ) ) -> ( G e. UHGraph /\ S SubGraph G ) )
19	18	anim1i 340	. . . . . . . . . . . 12 |- ( ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. UPGraph ) ) /\ x e. dom (iEdg ` S ) ) -> ( ( G e. UHGraph /\ S SubGraph G ) /\ x e. dom (iEdg ` S ) ) )
20	19	simplld 532	. . . . . . . . . . 11 |- ( ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. UPGraph ) ) /\ x e. dom (iEdg ` S ) ) -> G e. UHGraph )
21		simpl 109	. . . . . . . . . . . . 13 |- ( ( S SubGraph G /\ G e. UPGraph ) -> S SubGraph G )
22	21	adantl 277	. . . . . . . . . . . 12 |- ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. UPGraph ) ) -> S SubGraph G )
23	22	adantr 276	. . . . . . . . . . 11 |- ( ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. UPGraph ) ) /\ x e. dom (iEdg ` S ) ) -> S SubGraph G )
24		simpr 110	. . . . . . . . . . 11 |- ( ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. UPGraph ) ) /\ x e. dom (iEdg ` S ) ) -> x e. dom (iEdg ` S ) )
25	1, 3, 20, 23, 24	subgruhgredgdm 16425	. . . . . . . . . 10 |- ( ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. UPGraph ) ) /\ x e. dom (iEdg ` S ) ) -> ( (iEdg ` S ) ` x ) e. { s e. ~P (Vtx ` S ) | E. j j e. s } )
26		subgreldmiedg 16424	. . . . . . . . . . . . . . 15 |- ( ( S SubGraph G /\ x e. dom (iEdg ` S ) ) -> x e. dom (iEdg ` G ) )
27	26	ex 115	. . . . . . . . . . . . . 14 |- ( S SubGraph G -> ( x e. dom (iEdg ` S ) -> x e. dom (iEdg ` G ) ) )
28	27	ad2antrl 494	. . . . . . . . . . . . 13 |- ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. UPGraph ) ) -> ( x e. dom (iEdg ` S ) -> x e. dom (iEdg ` G ) ) )
29		simpr 110	. . . . . . . . . . . . . . . 16 |- ( ( x e. dom (iEdg ` G ) /\ G e. UPGraph ) -> G e. UPGraph )
30	4	uhgrfun 16232	. . . . . . . . . . . . . . . . . . 19 |- ( G e. UHGraph -> Fun (iEdg ` G ) )
31	7, 30	syl 14	. . . . . . . . . . . . . . . . . 18 |- ( G e. UPGraph -> Fun (iEdg ` G ) )
32	31	funfnd 5403	. . . . . . . . . . . . . . . . 17 |- ( G e. UPGraph -> (iEdg ` G ) Fn dom (iEdg ` G ) )
33	32	adantl 277	. . . . . . . . . . . . . . . 16 |- ( ( x e. dom (iEdg ` G ) /\ G e. UPGraph ) -> (iEdg ` G ) Fn dom (iEdg ` G ) )
34		simpl 109	. . . . . . . . . . . . . . . 16 |- ( ( x e. dom (iEdg ` G ) /\ G e. UPGraph ) -> x e. dom (iEdg ` G ) )
35	2, 4	upgr1or2 16256	. . . . . . . . . . . . . . . 16 |- ( ( G e. UPGraph /\ (iEdg ` G ) Fn dom (iEdg ` G ) /\ x e. dom (iEdg ` G ) ) -> ( ( (iEdg ` G ) ` x ) ~~ 1o \/ ( (iEdg ` G ) ` x ) ~~ 2o ) )
36	29, 33, 34, 35	syl3anc 1278	. . . . . . . . . . . . . . 15 |- ( ( x e. dom (iEdg ` G ) /\ G e. UPGraph ) -> ( ( (iEdg ` G ) ` x ) ~~ 1o \/ ( (iEdg ` G ) ` x ) ~~ 2o ) )
37	36	expcom 116	. . . . . . . . . . . . . 14 |- ( G e. UPGraph -> ( x e. dom (iEdg ` G ) -> ( ( (iEdg ` G ) ` x ) ~~ 1o \/ ( (iEdg ` G ) ` x ) ~~ 2o ) ) )
38	37	ad2antll 495	. . . . . . . . . . . . 13 |- ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. UPGraph ) ) -> ( x e. dom (iEdg ` G ) -> ( ( (iEdg ` G ) ` x ) ~~ 1o \/ ( (iEdg ` G ) ` x ) ~~ 2o ) ) )
39	28, 38	syld 45	. . . . . . . . . . . 12 |- ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. UPGraph ) ) -> ( x e. dom (iEdg ` S ) -> ( ( (iEdg ` G ) ` x ) ~~ 1o \/ ( (iEdg ` G ) ` x ) ~~ 2o ) ) )
40	39	imp 124	. . . . . . . . . . 11 |- ( ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. UPGraph ) ) /\ x e. dom (iEdg ` S ) ) -> ( ( (iEdg ` G ) ` x ) ~~ 1o \/ ( (iEdg ` G ) ` x ) ~~ 2o ) )
41	31	ad2antll 495	. . . . . . . . . . . . . . . 16 |- ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. UPGraph ) ) -> Fun (iEdg ` G ) )
42	41	adantr 276	. . . . . . . . . . . . . . 15 |- ( ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. UPGraph ) ) /\ x e. dom (iEdg ` S ) ) -> Fun (iEdg ` G ) )
43		simpll2 1068	. . . . . . . . . . . . . . 15 |- ( ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. UPGraph ) ) /\ x e. dom (iEdg ` S ) ) -> (iEdg ` S ) C_ (iEdg ` G ) )
44		funssfv 5716	. . . . . . . . . . . . . . 15 |- ( ( Fun (iEdg ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ x e. dom (iEdg ` S ) ) -> ( (iEdg ` G ) ` x ) = ( (iEdg ` S ) ` x ) )
45	42, 43, 24, 44	syl3anc 1278	. . . . . . . . . . . . . 14 |- ( ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. UPGraph ) ) /\ x e. dom (iEdg ` S ) ) -> ( (iEdg ` G ) ` x ) = ( (iEdg ` S ) ` x ) )
46	45	eqcomd 2244	. . . . . . . . . . . . 13 |- ( ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. UPGraph ) ) /\ x e. dom (iEdg ` S ) ) -> ( (iEdg ` S ) ` x ) = ( (iEdg ` G ) ` x ) )
47	46	breq1d 4135	. . . . . . . . . . . 12 |- ( ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. UPGraph ) ) /\ x e. dom (iEdg ` S ) ) -> ( ( (iEdg ` S ) ` x ) ~~ 1o <-> ( (iEdg ` G ) ` x ) ~~ 1o ) )
48	46	breq1d 4135	. . . . . . . . . . . 12 |- ( ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. UPGraph ) ) /\ x e. dom (iEdg ` S ) ) -> ( ( (iEdg ` S ) ` x ) ~~ 2o <-> ( (iEdg ` G ) ` x ) ~~ 2o ) )
49	47, 48	orbi12d 805	. . . . . . . . . . 11 |- ( ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. UPGraph ) ) /\ x e. 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 ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. UPGraph ) ) /\ x e. dom (iEdg ` S ) ) -> ( ( (iEdg ` S ) ` x ) ~~ 1o \/ ( (iEdg ` S ) ` x ) ~~ 2o ) )
51	15, 25, 50	elrabd 2984	. . . . . . . . 9 |- ( ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. UPGraph ) ) /\ x e. dom (iEdg ` S ) ) -> ( (iEdg ` S ) ` x ) e. { e e. { s e. ~P (Vtx ` S ) | E. j j e. s } | ( e ~~ 1o \/ e ~~ 2o ) } )
52	51	ralrimiva 2623	. . . . . . . 8 |- ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. UPGraph ) ) -> A. x e. dom (iEdg ` S ) ( (iEdg ` S ) ` x ) e. { e e. { s e. ~P (Vtx ` S ) | E. j j e. s } | ( e ~~ 1o \/ e ~~ 2o ) } )
53		fnfvrnss 5859	. . . . . . . 8 |- ( ( (iEdg ` S ) Fn dom (iEdg ` S ) /\ A. x e. dom (iEdg ` S ) ( (iEdg ` S ) ` x ) e. { e e. { s e. ~P (Vtx ` S ) | E. j j e. s } | ( e ~~ 1o \/ e ~~ 2o ) } ) -> ran (iEdg ` S ) C_ { e e. { s e. ~P (Vtx ` S ) | E. j j e. s } | ( e ~~ 1o \/ e ~~ 2o ) } )
54	12, 52, 53	syl2anc 415	. . . . . . 7 |- ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. UPGraph ) ) -> ran (iEdg ` S ) C_ { e e. { s e. ~P (Vtx ` S ) | E. j j e. s } | ( e ~~ 1o \/ e ~~ 2o ) } )
55		df-f 5376	. . . . . . 7 |- ( (iEdg ` S ) : dom (iEdg ` S ) --> { e e. { s e. ~P (Vtx ` S ) | E. j j e. s } | ( e ~~ 1o \/ e ~~ 2o ) } <-> ( (iEdg ` S ) Fn dom (iEdg ` S ) /\ ran (iEdg ` S ) C_ { e e. { s e. ~P (Vtx ` S ) | E. j j e. s } | ( e ~~ 1o \/ e ~~ 2o ) } ) )
56	12, 54, 55	sylanbrc 421	. . . . . 6 |- ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. UPGraph ) ) -> (iEdg ` S ) : dom (iEdg ` S ) --> { e e. { s e. ~P (Vtx ` S ) | E. j j e. s } | ( e ~~ 1o \/ e ~~ 2o ) } )
57		sspw1or2 7534	. . . . . . 7 |- { e e. { s e. ~P (Vtx ` S ) | E. j j e. s } | ( e ~~ 1o \/ e ~~ 2o ) } = { e e. ~P (Vtx ` S ) | ( e ~~ 1o \/ e ~~ 2o ) }
58		feq3 5513	. . . . . . 7 |- ( { e e. { s e. ~P (Vtx ` S ) | E. j j e. s } | ( e ~~ 1o \/ e ~~ 2o ) } = { e e. ~P (Vtx ` S ) | ( e ~~ 1o \/ e ~~ 2o ) } -> ( (iEdg ` S ) : dom (iEdg ` S ) --> { e e. { s e. ~P (Vtx ` S ) | E. j j e. s } | ( e ~~ 1o \/ e ~~ 2o ) } <-> (iEdg ` S ) : dom (iEdg ` S ) --> { e e. ~P (Vtx ` S ) | ( e ~~ 1o \/ e ~~ 2o ) } ) )
59	57, 58	ax-mp 5	. . . . . 6 |- ( (iEdg ` S ) : dom (iEdg ` S ) --> { e e. { s e. ~P (Vtx ` S ) | E. j j e. s } | ( e ~~ 1o \/ e ~~ 2o ) } <-> (iEdg ` S ) : dom (iEdg ` S ) --> { e e. ~P (Vtx ` S ) | ( e ~~ 1o \/ e ~~ 2o ) } )
60	56, 59	sylib 122	. . . . 5 |- ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. UPGraph ) ) -> (iEdg ` S ) : dom (iEdg ` S ) --> { e e. ~P (Vtx ` S ) | ( e ~~ 1o \/ e ~~ 2o ) } )
61		subgrv 16411	. . . . . . 7 |- ( S SubGraph G -> ( S e. _V /\ G e. _V ) )
62	1, 3	isupgren 16250	. . . . . . . 8 |- ( S e. _V -> ( S e. UPGraph <-> (iEdg ` S ) : dom (iEdg ` S ) --> { e e. ~P (Vtx ` S ) | ( e ~~ 1o \/ e ~~ 2o ) } ) )
63	62	adantr 276	. . . . . . 7 |- ( ( S e. _V /\ G e. _V ) -> ( S e. UPGraph <-> (iEdg ` S ) : dom (iEdg ` S ) --> { e e. ~P (Vtx ` S ) | ( e ~~ 1o \/ e ~~ 2o ) } ) )
64	61, 63	syl 14	. . . . . 6 |- ( S SubGraph G -> ( S e. UPGraph <-> (iEdg ` S ) : dom (iEdg ` S ) --> { e e. ~P (Vtx ` S ) | ( e ~~ 1o \/ e ~~ 2o ) } ) )
65	64	ad2antrl 494	. . . . 5 |- ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. UPGraph ) ) -> ( S e. UPGraph <-> (iEdg ` S ) : dom (iEdg ` S ) --> { e e. ~P (Vtx ` S ) | ( e ~~ 1o \/ e ~~ 2o ) } ) )
66	60, 65	mpbird 167	. . . 4 |- ( ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) /\ ( S SubGraph G /\ G e. UPGraph ) ) -> S e. UPGraph )
67	66	ex 115	. . 3 |- ( ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) -> ( ( S SubGraph G /\ G e. UPGraph ) -> S e. UPGraph ) )
68	6, 67	syl 14	. 2 |- ( S SubGraph G -> ( ( S SubGraph G /\ G e. UPGraph ) -> S e. UPGraph ) )
69	68	anabsi8 588	1 |- ( ( G e. UPGraph /\ S SubGraph G ) -> S e. UPGraph )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720   /\ w3a 1009   = wceq 1402   E.wex 1545   e. wcel 2209   A.wral 2528   {crab 2532   _Vcvv 2821   C_ wss 3220   ~Pcpw 3685   class class class wbr 4125   dom cdm 4769   ran crn 4770   Fun wfun 5366   Fn wfn 5367   -->wf 5368   ` cfv 5372   1oc1o 6670   2oc2o 6671   ~~ cen 7010  Vtxcvtx 16167  iEdgciedg 16168  Edgcedg 16212  UHGraphcuhgr 16222  UPGraphcupgr 16246   SubGraph csubgr 16408

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 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 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280

This theorem 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 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-suc 4511  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-1o 6677  df-2o 6678  df-en 7013  df-sub 8489  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-5 9345  df-6 9346  df-7 9347  df-8 9348  df-9 9349  df-n0 9543  df-dec 9757  df-ndx 13333  df-slot 13334  df-base 13336  df-edgf 16160  df-vtx 16169  df-iedg 16170  df-edg 16213  df-uhgrm 16224  df-upgren 16248  df-subgr 16409

This theorem is referenced by:  upgrspan  16434