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 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 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 ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) )
7		upgruhgr 15961	. . . . . . . . . . 11 |- ( G e. UPGraph -> G e. UHGraph )
8		subgruhgrfun 16118	. . . . . . . . . . 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 5357	. . . . . . . 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 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 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 528	. . . . . . . . . . 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 16120	. . . . . . . . . 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 16119	. . . . . . . . . . . . . . 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 490	. . . . . . . . . . . . 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 15927	. . . . . . . . . . . . . . . . . . 19 |- ( G e. UHGraph -> Fun (iEdg ` G ) )
31	7, 30	syl 14	. . . . . . . . . . . . . . . . . 18 |- ( G e. UPGraph -> Fun (iEdg ` G ) )
32	31	funfnd 5357	. . . . . . . . . . . . . . . . 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 15951	. . . . . . . . . . . . . . . 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 1273	. . . . . . . . . . . . . . 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 491	. . . . . . . . . . . . 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 491	. . . . . . . . . . . . . . . 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 1063	. . . . . . . . . . . . . . 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 5665	. . . . . . . . . . . . . . 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 1273	. . . . . . . . . . . . . 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 2237	. . . . . . . . . . . . 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 4098	. . . . . . . . . . . 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 4098	. . . . . . . . . . . 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 800	. . . . . . . . . . 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 2964	. . . . . . . . 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 2605	. . . . . . . 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 5807	. . . . . . . 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 411	. . . . . . 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 5330	. . . . . . 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 417	. . . . . 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 7402	. . . . . . 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 5467	. . . . . . 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 16106	. . . . . . 7 |- ( S SubGraph G -> ( S e. _V /\ G e. _V ) )
62	1, 3	isupgren 15945	. . . . . . . 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 490	. . . . 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 584	1 |- ( ( G e. UPGraph /\ S SubGraph G ) -> S e. UPGraph )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 715   /\ w3a 1004   = wceq 1397   E.wex 1540   e. wcel 2202   A.wral 2510   {crab 2514   _Vcvv 2802   C_ wss 3200   ~Pcpw 3652   class class class wbr 4088   dom cdm 4725   ran crn 4726   Fun wfun 5320   Fn wfn 5321   -->wf 5322   ` cfv 5326   1oc1o 6574   2oc2o 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