Theorem upgrm 16207

Description: An edge is an inhabited subset 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
upgrm	|- ( ( G e. UPGraph /\ E Fn A /\ F e. A ) -> E. j j e. ( E ` F ) )

Distinct variable groups:   j, E   j, F

Allowed substitution hints:   A( j)   G( j)   V( j)

Proof of Theorem upgrm

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isupgr.v	. . . . . 6 |- V = (Vtx ` G )
2		isupgr.e	. . . . . 6 |- E = (iEdg ` G )
3	1, 2	upgrfnen 16205	. . . . 5 |- ( ( G e. UPGraph /\ E Fn A ) -> E : A --> { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } )
4	3	ffvelcdmda 5817	. . . 4 |- ( ( ( G e. UPGraph /\ E Fn A ) /\ F e. A ) -> ( E ` F ) e. { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } )
5	4	3impa 1221	. . 3 |- ( ( G e. UPGraph /\ E Fn A /\ F e. A ) -> ( E ` F ) e. { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } )
6		breq1 4117	. . . . 5 |- ( x = ( E ` F ) -> ( x ~~ 1o <-> ( E ` F ) ~~ 1o ) )
7		breq1 4117	. . . . 5 |- ( x = ( E ` F ) -> ( x ~~ 2o <-> ( E ` F ) ~~ 2o ) )
8	6, 7	orbi12d 801	. . . 4 |- ( x = ( E ` F ) -> ( ( x ~~ 1o \/ x ~~ 2o ) <-> ( ( E ` F ) ~~ 1o \/ ( E ` F ) ~~ 2o ) ) )
9	8	elrab 2976	. . 3 |- ( ( E ` F ) e. { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } <-> ( ( E ` F ) e. ~P V /\ ( ( E ` F ) ~~ 1o \/ ( E ` F ) ~~ 2o ) ) )
10	5, 9	sylib 122	. 2 |- ( ( G e. UPGraph /\ E Fn A /\ F e. A ) -> ( ( E ` F ) e. ~P V /\ ( ( E ` F ) ~~ 1o \/ ( E ` F ) ~~ 2o ) ) )
11		en1m 7058	. . 3 |- ( ( E ` F ) ~~ 1o -> E. j j e. ( E ` F ) )
12		en2m 7079	. . 3 |- ( ( E ` F ) ~~ 2o -> E. j j e. ( E ` F ) )
13	11, 12	jaoi 724	. 2 |- ( ( ( E ` F ) ~~ 1o \/ ( E ` F ) ~~ 2o ) -> E. j j e. ( E ` F ) )
14	10, 13	simpl2im 386	1 |- ( ( G e. UPGraph /\ E Fn A /\ F e. A ) -> E. j j e. ( E ` F ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 716   /\ w3a 1005   = wceq 1398   E.wex 1541   e. wcel 2205   {crab 2526   ~Pcpw 3674   class class class wbr 4114   Fn wfn 5352   ` cfv 5357   1oc1o 6653   2oc2o 6654   ~~ cen 6986  Vtxcvtx 16119  iEdgciedg 16120  UPGraphcupgr 16198

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 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-suc 4497  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-1o 6660  df-2o 6661  df-en 6989  df-sub 8462  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-5 9316  df-6 9317  df-7 9318  df-8 9319  df-9 9320  df-n0 9514  df-dec 9728  df-ndx 13299  df-slot 13300  df-base 13302  df-edgf 16112  df-vtx 16121  df-iedg 16122  df-upgren 16200

This theorem is referenced by: (None)