Theorem subupgr 16197

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 16184	. . 3 |- ( S SubGraph G -> ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) )
7		upgruhgr 16035	. . . . . . . . . . 11 |- ( G e. UPGraph -> G e. UHGraph )
8		subgruhgrfun 16192	. . . . . . . . . . 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 5364	. . . . . . . 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 4096	. . . . . . . . . . 11 |- ( e = ( (iEdg ` S ) ` x ) -> ( e ~~ 1o <-> ( (iEdg ` S ) ` x ) ~~ 1o ) )
14		breq1 4096	. . . . . . . . . . 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 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 16194	. . . . . . . . . 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 16193	. . . . . . . . . . . . . . 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 16001	. . . . . . . . . . . . . . . . . . 19 |- ( G e. UHGraph -> Fun (iEdg ` G ) )
31	7, 30	syl 14	. . . . . . . . . . . . . . . . . 18 |- ( G e. UPGraph -> Fun (iEdg ` G ) )
32	31	funfnd 5364	. . . . . . . . . . . . . . . . 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 16025	. . . . . . . . . . . . . . . 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 1274	. . . . . . . . . . . . . . 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 1064	. . . . . . . . . . . . . . 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 5674	. . . . . . . . . . . . . . 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 1274	. . . . . . . . . . . . . 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 4103	. . . . . . . . . . . 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 4103	. . . . . . . . . . . 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 801	. . . . . . . . . . 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 2965	. . . . . . . . 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 2606	. . . . . . . 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 5815	. . . . . . . 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 5337	. . . . . . 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 7446	. . . . . . 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 5474	. . . . . . 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 16180	. . . . . . 7 |- ( S SubGraph G -> ( S e. _V /\ G e. _V ) )
62	1, 3	isupgren 16019	. . . . . . . 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 716   /\ w3a 1005   = wceq 1398   E.wex 1541   e. wcel 2202   A.wral 2511   {crab 2515   _Vcvv 2803   C_ wss 3201   ~Pcpw 3656   class class class wbr 4093   dom cdm 4731   ran crn 4732   Fun wfun 5327   Fn wfn 5328   -->wf 5329   ` cfv 5333   1oc1o 6618   2oc2o 6619   ~~ cen 6950  Vtxcvtx 15936  iEdgciedg 15937  Edgcedg 15981  UHGraphcuhgr 15991  UPGraphcupgr 16015   SubGraph csubgr 16177

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 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-cnre 8186

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-suc 4474  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-1o 6625  df-2o 6626  df-en 6953  df-sub 8394  df-inn 9186  df-2 9244  df-3 9245  df-4 9246  df-5 9247  df-6 9248  df-7 9249  df-8 9250  df-9 9251  df-n0 9445  df-dec 9656  df-ndx 13148  df-slot 13149  df-base 13151  df-edgf 15929  df-vtx 15938  df-iedg 15939  df-edg 15982  df-uhgrm 15993  df-upgren 16017  df-subgr 16178

This theorem is referenced by:  upgrspan  16203