Theorem upgrex 25987

Description: An edge is an unordered pair of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 10-Oct-2020.)

Hypotheses

Ref	Expression
isupgr.v	|- V = (Vtx ` G )
isupgr.e	|- E = (iEdg ` G )

Assertion

Ref	Expression
upgrex	|- ( ( G e. UPGraph /\ E Fn A /\ F e. A ) -> E. x e. V E. y e. V ( E ` F ) = { x , y } )

Distinct variable groups:   x, G   x, V   x, E   x, F   x, A, y   y, E   y, F   y, G   y, V

Proof of Theorem upgrex

Step	Hyp	Ref	Expression
1		isupgr.v	. . . . 5 |- V = (Vtx ` G )
2		isupgr.e	. . . . 5 |- E = (iEdg ` G )
3	1, 2	upgrn0 25984	. . . 4 |- ( ( G e. UPGraph /\ E Fn A /\ F e. A ) -> ( E ` F ) =/= (/) )
4		n0 3931	. . . 4 |- ( ( E ` F ) =/= (/) <-> E. x x e. ( E ` F ) )
5	3, 4	sylib 208	. . 3 |- ( ( G e. UPGraph /\ E Fn A /\ F e. A ) -> E. x x e. ( E ` F ) )
6		simp1 1061	. . . . . . . 8 |- ( ( G e. UPGraph /\ E Fn A /\ F e. A ) -> G e. UPGraph )
7		fndm 5990	. . . . . . . . . . . . 13 |- ( E Fn A -> dom E = A )
8	7	eqcomd 2628	. . . . . . . . . . . 12 |- ( E Fn A -> A = dom E )
9	8	eleq2d 2687	. . . . . . . . . . 11 |- ( E Fn A -> ( F e. A <-> F e. dom E ) )
10	9	biimpd 219	. . . . . . . . . 10 |- ( E Fn A -> ( F e. A -> F e. dom E ) )
11	10	a1i 11	. . . . . . . . 9 |- ( G e. UPGraph -> ( E Fn A -> ( F e. A -> F e. dom E ) ) )
12	11	3imp 1256	. . . . . . . 8 |- ( ( G e. UPGraph /\ E Fn A /\ F e. A ) -> F e. dom E )
13	1, 2	upgrss 25983	. . . . . . . 8 |- ( ( G e. UPGraph /\ F e. dom E ) -> ( E ` F ) C_ V )
14	6, 12, 13	syl2anc 693	. . . . . . 7 |- ( ( G e. UPGraph /\ E Fn A /\ F e. A ) -> ( E ` F ) C_ V )
15	14	sselda 3603	. . . . . 6 |- ( ( ( G e. UPGraph /\ E Fn A /\ F e. A ) /\ x e. ( E ` F ) ) -> x e. V )
16	15	adantr 481	. . . . . . . 8 |- ( ( ( ( G e. UPGraph /\ E Fn A /\ F e. A ) /\ x e. ( E ` F ) ) /\ ( ( E ` F ) \ { x } ) = (/) ) -> x e. V )
17		simpr 477	. . . . . . . . . 10 |- ( ( ( ( G e. UPGraph /\ E Fn A /\ F e. A ) /\ x e. ( E ` F ) ) /\ ( ( E ` F ) \ { x } ) = (/) ) -> ( ( E ` F ) \ { x } ) = (/) )
18		ssdif0 3942	. . . . . . . . . 10 |- ( ( E ` F ) C_ { x } <-> ( ( E ` F ) \ { x } ) = (/) )
19	17, 18	sylibr 224	. . . . . . . . 9 |- ( ( ( ( G e. UPGraph /\ E Fn A /\ F e. A ) /\ x e. ( E ` F ) ) /\ ( ( E ` F ) \ { x } ) = (/) ) -> ( E ` F ) C_ { x } )
20		simpr 477	. . . . . . . . . . 11 |- ( ( ( G e. UPGraph /\ E Fn A /\ F e. A ) /\ x e. ( E ` F ) ) -> x e. ( E ` F ) )
21	20	snssd 4340	. . . . . . . . . 10 |- ( ( ( G e. UPGraph /\ E Fn A /\ F e. A ) /\ x e. ( E ` F ) ) -> { x } C_ ( E ` F ) )
22	21	adantr 481	. . . . . . . . 9 |- ( ( ( ( G e. UPGraph /\ E Fn A /\ F e. A ) /\ x e. ( E ` F ) ) /\ ( ( E ` F ) \ { x } ) = (/) ) -> { x } C_ ( E ` F ) )
23	19, 22	eqssd 3620	. . . . . . . 8 |- ( ( ( ( G e. UPGraph /\ E Fn A /\ F e. A ) /\ x e. ( E ` F ) ) /\ ( ( E ` F ) \ { x } ) = (/) ) -> ( E ` F ) = { x } )
24		preq2 4269	. . . . . . . . . . 11 |- ( y = x -> { x , y } = { x , x } )
25		dfsn2 4190	. . . . . . . . . . 11 |- { x } = { x , x }
26	24, 25	syl6eqr 2674	. . . . . . . . . 10 |- ( y = x -> { x , y } = { x } )
27	26	eqeq2d 2632	. . . . . . . . 9 |- ( y = x -> ( ( E ` F ) = { x , y } <-> ( E ` F ) = { x } ) )
28	27	rspcev 3309	. . . . . . . 8 |- ( ( x e. V /\ ( E ` F ) = { x } ) -> E. y e. V ( E ` F ) = { x , y } )
29	16, 23, 28	syl2anc 693	. . . . . . 7 |- ( ( ( ( G e. UPGraph /\ E Fn A /\ F e. A ) /\ x e. ( E ` F ) ) /\ ( ( E ` F ) \ { x } ) = (/) ) -> E. y e. V ( E ` F ) = { x , y } )
30		n0 3931	. . . . . . . 8 |- ( ( ( E ` F ) \ { x } ) =/= (/) <-> E. y y e. ( ( E ` F ) \ { x } ) )
31	14	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ( G e. UPGraph /\ E Fn A /\ F e. A ) /\ ( x e. ( E ` F ) /\ y e. ( ( E ` F ) \ { x } ) ) ) -> ( E ` F ) C_ V )
32		simprr 796	. . . . . . . . . . . . . . 15 |- ( ( ( G e. UPGraph /\ E Fn A /\ F e. A ) /\ ( x e. ( E ` F ) /\ y e. ( ( E ` F ) \ { x } ) ) ) -> y e. ( ( E ` F ) \ { x } ) )
33	32	eldifad 3586	. . . . . . . . . . . . . 14 |- ( ( ( G e. UPGraph /\ E Fn A /\ F e. A ) /\ ( x e. ( E ` F ) /\ y e. ( ( E ` F ) \ { x } ) ) ) -> y e. ( E ` F ) )
34	31, 33	sseldd 3604	. . . . . . . . . . . . 13 |- ( ( ( G e. UPGraph /\ E Fn A /\ F e. A ) /\ ( x e. ( E ` F ) /\ y e. ( ( E ` F ) \ { x } ) ) ) -> y e. V )
35	1, 2	upgrfi 25986	. . . . . . . . . . . . . . . 16 |- ( ( G e. UPGraph /\ E Fn A /\ F e. A ) -> ( E ` F ) e. Fin )
36	35	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ( G e. UPGraph /\ E Fn A /\ F e. A ) /\ ( x e. ( E ` F ) /\ y e. ( ( E ` F ) \ { x } ) ) ) -> ( E ` F ) e. Fin )
37		simprl 794	. . . . . . . . . . . . . . . 16 |- ( ( ( G e. UPGraph /\ E Fn A /\ F e. A ) /\ ( x e. ( E ` F ) /\ y e. ( ( E ` F ) \ { x } ) ) ) -> x e. ( E ` F ) )
38	37, 33	prssd 4354	. . . . . . . . . . . . . . 15 |- ( ( ( G e. UPGraph /\ E Fn A /\ F e. A ) /\ ( x e. ( E ` F ) /\ y e. ( ( E ` F ) \ { x } ) ) ) -> { x , y } C_ ( E ` F ) )
39		fvex 6201	. . . . . . . . . . . . . . . . 17 |- ( E ` F ) e. _V
40		ssdomg 8001	. . . . . . . . . . . . . . . . 17 |- ( ( E ` F ) e. _V -> ( { x , y } C_ ( E ` F ) -> { x , y } ~<_ ( E ` F ) ) )
41	39, 38, 40	mpsyl 68	. . . . . . . . . . . . . . . 16 |- ( ( ( G e. UPGraph /\ E Fn A /\ F e. A ) /\ ( x e. ( E ` F ) /\ y e. ( ( E ` F ) \ { x } ) ) ) -> { x , y } ~<_ ( E ` F ) )
42	1, 2	upgrle 25985	. . . . . . . . . . . . . . . . . . 19 |- ( ( G e. UPGraph /\ E Fn A /\ F e. A ) -> ( # ` ( E ` F ) ) <_ 2 )
43	42	adantr 481	. . . . . . . . . . . . . . . . . 18 |- ( ( ( G e. UPGraph /\ E Fn A /\ F e. A ) /\ ( x e. ( E ` F ) /\ y e. ( ( E ` F ) \ { x } ) ) ) -> ( # ` ( E ` F ) ) <_ 2 )
44		eldifsni 4320	. . . . . . . . . . . . . . . . . . . . 21 |- ( y e. ( ( E ` F ) \ { x } ) -> y =/= x )
45	44	ad2antll 765	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( G e. UPGraph /\ E Fn A /\ F e. A ) /\ ( x e. ( E ` F ) /\ y e. ( ( E ` F ) \ { x } ) ) ) -> y =/= x )
46	45	necomd 2849	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( G e. UPGraph /\ E Fn A /\ F e. A ) /\ ( x e. ( E ` F ) /\ y e. ( ( E ` F ) \ { x } ) ) ) -> x =/= y )
47		vex 3203	. . . . . . . . . . . . . . . . . . . 20 |- x e. _V
48		vex 3203	. . . . . . . . . . . . . . . . . . . 20 |- y e. _V
49		hashprg 13182	. . . . . . . . . . . . . . . . . . . 20 |- ( ( x e. _V /\ y e. _V ) -> ( x =/= y <-> ( # ` { x , y } ) = 2 ) )
50	47, 48, 49	mp2an 708	. . . . . . . . . . . . . . . . . . 19 |- ( x =/= y <-> ( # ` { x , y } ) = 2 )
51	46, 50	sylib 208	. . . . . . . . . . . . . . . . . 18 |- ( ( ( G e. UPGraph /\ E Fn A /\ F e. A ) /\ ( x e. ( E ` F ) /\ y e. ( ( E ` F ) \ { x } ) ) ) -> ( # ` { x , y } ) = 2 )
52	43, 51	breqtrrd 4681	. . . . . . . . . . . . . . . . 17 |- ( ( ( G e. UPGraph /\ E Fn A /\ F e. A ) /\ ( x e. ( E ` F ) /\ y e. ( ( E ` F ) \ { x } ) ) ) -> ( # ` ( E ` F ) ) <_ ( # ` { x , y } ) )
53		prfi 8235	. . . . . . . . . . . . . . . . . 18 |- { x , y } e. Fin
54		hashdom 13168	. . . . . . . . . . . . . . . . . 18 |- ( ( ( E ` F ) e. Fin /\ { x , y } e. Fin ) -> ( ( # ` ( E ` F ) ) <_ ( # ` { x , y } ) <-> ( E ` F ) ~<_ { x , y } ) )
55	36, 53, 54	sylancl 694	. . . . . . . . . . . . . . . . 17 |- ( ( ( G e. UPGraph /\ E Fn A /\ F e. A ) /\ ( x e. ( E ` F ) /\ y e. ( ( E ` F ) \ { x } ) ) ) -> ( ( # ` ( E ` F ) ) <_ ( # ` { x , y } ) <-> ( E ` F ) ~<_ { x , y } ) )
56	52, 55	mpbid 222	. . . . . . . . . . . . . . . 16 |- ( ( ( G e. UPGraph /\ E Fn A /\ F e. A ) /\ ( x e. ( E ` F ) /\ y e. ( ( E ` F ) \ { x } ) ) ) -> ( E ` F ) ~<_ { x , y } )
57		sbth 8080	. . . . . . . . . . . . . . . 16 |- ( ( { x , y } ~<_ ( E ` F ) /\ ( E ` F ) ~<_ { x , y } ) -> { x , y } ~~ ( E ` F ) )
58	41, 56, 57	syl2anc 693	. . . . . . . . . . . . . . 15 |- ( ( ( G e. UPGraph /\ E Fn A /\ F e. A ) /\ ( x e. ( E ` F ) /\ y e. ( ( E ` F ) \ { x } ) ) ) -> { x , y } ~~ ( E ` F ) )
59		fisseneq 8171	. . . . . . . . . . . . . . 15 |- ( ( ( E ` F ) e. Fin /\ { x , y } C_ ( E ` F ) /\ { x , y } ~~ ( E ` F ) ) -> { x , y } = ( E ` F ) )
60	36, 38, 58, 59	syl3anc 1326	. . . . . . . . . . . . . 14 |- ( ( ( G e. UPGraph /\ E Fn A /\ F e. A ) /\ ( x e. ( E ` F ) /\ y e. ( ( E ` F ) \ { x } ) ) ) -> { x , y } = ( E ` F ) )
61	60	eqcomd 2628	. . . . . . . . . . . . 13 |- ( ( ( G e. UPGraph /\ E Fn A /\ F e. A ) /\ ( x e. ( E ` F ) /\ y e. ( ( E ` F ) \ { x } ) ) ) -> ( E ` F ) = { x , y } )
62	34, 61	jca 554	. . . . . . . . . . . 12 |- ( ( ( G e. UPGraph /\ E Fn A /\ F e. A ) /\ ( x e. ( E ` F ) /\ y e. ( ( E ` F ) \ { x } ) ) ) -> ( y e. V /\ ( E ` F ) = { x , y } ) )
63	62	expr 643	. . . . . . . . . . 11 |- ( ( ( G e. UPGraph /\ E Fn A /\ F e. A ) /\ x e. ( E ` F ) ) -> ( y e. ( ( E ` F ) \ { x } ) -> ( y e. V /\ ( E ` F ) = { x , y } ) ) )
64	63	eximdv 1846	. . . . . . . . . 10 |- ( ( ( G e. UPGraph /\ E Fn A /\ F e. A ) /\ x e. ( E ` F ) ) -> ( E. y y e. ( ( E ` F ) \ { x } ) -> E. y ( y e. V /\ ( E ` F ) = { x , y } ) ) )
65	64	imp 445	. . . . . . . . 9 |- ( ( ( ( G e. UPGraph /\ E Fn A /\ F e. A ) /\ x e. ( E ` F ) ) /\ E. y y e. ( ( E ` F ) \ { x } ) ) -> E. y ( y e. V /\ ( E ` F ) = { x , y } ) )
66		df-rex 2918	. . . . . . . . 9 |- ( E. y e. V ( E ` F ) = { x , y } <-> E. y ( y e. V /\ ( E ` F ) = { x , y } ) )
67	65, 66	sylibr 224	. . . . . . . 8 |- ( ( ( ( G e. UPGraph /\ E Fn A /\ F e. A ) /\ x e. ( E ` F ) ) /\ E. y y e. ( ( E ` F ) \ { x } ) ) -> E. y e. V ( E ` F ) = { x , y } )
68	30, 67	sylan2b 492	. . . . . . 7 |- ( ( ( ( G e. UPGraph /\ E Fn A /\ F e. A ) /\ x e. ( E ` F ) ) /\ ( ( E ` F ) \ { x } ) =/= (/) ) -> E. y e. V ( E ` F ) = { x , y } )
69	29, 68	pm2.61dane 2881	. . . . . 6 |- ( ( ( G e. UPGraph /\ E Fn A /\ F e. A ) /\ x e. ( E ` F ) ) -> E. y e. V ( E ` F ) = { x , y } )
70	15, 69	jca 554	. . . . 5 |- ( ( ( G e. UPGraph /\ E Fn A /\ F e. A ) /\ x e. ( E ` F ) ) -> ( x e. V /\ E. y e. V ( E ` F ) = { x , y } ) )
71	70	ex 450	. . . 4 |- ( ( G e. UPGraph /\ E Fn A /\ F e. A ) -> ( x e. ( E ` F ) -> ( x e. V /\ E. y e. V ( E ` F ) = { x , y } ) ) )
72	71	eximdv 1846	. . 3 |- ( ( G e. UPGraph /\ E Fn A /\ F e. A ) -> ( E. x x e. ( E ` F ) -> E. x ( x e. V /\ E. y e. V ( E ` F ) = { x , y } ) ) )
73	5, 72	mpd 15	. 2 |- ( ( G e. UPGraph /\ E Fn A /\ F e. A ) -> E. x ( x e. V /\ E. y e. V ( E ` F ) = { x , y } ) )
74		df-rex 2918	. 2 |- ( E. x e. V E. y e. V ( E ` F ) = { x , y } <-> E. x ( x e. V /\ E. y e. V ( E ` F ) = { x , y } ) )
75	73, 74	sylibr 224	1 |- ( ( G e. UPGraph /\ E Fn A /\ F e. A ) -> E. x e. V E. y e. V ( E ` F ) = { x , y } )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   E.wrex 2913   _Vcvv 3200   \ cdif 3571   C_ wss 3574   (/)c0 3915   {csn 4177   {cpr 4179   class class class wbr 4653   dom cdm 5114   Fn wfn 5883   ` cfv 5888   ~~ cen 7952   ~<_ cdom 7953   Fincfn 7955   <_ cle 10075   2c2 11070   #chash 13117  Vtxcvtx 25874  iEdgciedg 25875   UPGraph cupgr 25975

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-hash 13118  df-upgr 25977

This theorem is referenced by: (None)