Theorem uhgrm 16202

Description: An edge is an inhabited subset of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 15-Dec-2020.)

Hypothesis

Ref	Expression
uhgrfun.e	|- E = (iEdg ` G )

Assertion

Ref	Expression
uhgrm	|- ( ( G e. UHGraph /\ 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)

Proof of Theorem uhgrm

Dummy variable s is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2234	. . . . . . . 8 |- (Vtx ` G ) = (Vtx ` G )
2		uhgrfun.e	. . . . . . . 8 |- E = (iEdg ` G )
3	1, 2	uhgrfm 16197	. . . . . . 7 |- ( G e. UHGraph -> E : dom E --> { s e. ~P (Vtx ` G ) | E. j j e. s } )
4		fndm 5460	. . . . . . . 8 |- ( E Fn A -> dom E = A )
5	4	feq2d 5501	. . . . . . 7 |- ( E Fn A -> ( E : dom E --> { s e. ~P (Vtx ` G ) | E. j j e. s } <-> E : A --> { s e. ~P (Vtx ` G ) | E. j j e. s } ) )
6	3, 5	syl5ibcom 155	. . . . . 6 |- ( G e. UHGraph -> ( E Fn A -> E : A --> { s e. ~P (Vtx ` G ) | E. j j e. s } ) )
7	6	imp 124	. . . . 5 |- ( ( G e. UHGraph /\ E Fn A ) -> E : A --> { s e. ~P (Vtx ` G ) | E. j j e. s } )
8	7	ffvelcdmda 5817	. . . 4 |- ( ( ( G e. UHGraph /\ E Fn A ) /\ F e. A ) -> ( E ` F ) e. { s e. ~P (Vtx ` G ) | E. j j e. s } )
9	8	3impa 1221	. . 3 |- ( ( G e. UHGraph /\ E Fn A /\ F e. A ) -> ( E ` F ) e. { s e. ~P (Vtx ` G ) | E. j j e. s } )
10		eleq2 2298	. . . . 5 |- ( s = ( E ` F ) -> ( j e. s <-> j e. ( E ` F ) ) )
11	10	exbidv 1874	. . . 4 |- ( s = ( E ` F ) -> ( E. j j e. s <-> E. j j e. ( E ` F ) ) )
12	11	elrab 2976	. . 3 |- ( ( E ` F ) e. { s e. ~P (Vtx ` G ) | E. j j e. s } <-> ( ( E ` F ) e. ~P (Vtx ` G ) /\ E. j j e. ( E ` F ) ) )
13	9, 12	sylib 122	. 2 |- ( ( G e. UHGraph /\ E Fn A /\ F e. A ) -> ( ( E ` F ) e. ~P (Vtx ` G ) /\ E. j j e. ( E ` F ) ) )
14	13	simprd 114	1 |- ( ( G e. UHGraph /\ E Fn A /\ F e. A ) -> E. j j e. ( E ` F ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   E.wex 1541   e. wcel 2205   {crab 2526   ~Pcpw 3674   dom cdm 4754   Fn wfn 5352   -->wf 5353   ` cfv 5357  Vtxcvtx 16136  iEdgciedg 16137  UHGraphcuhgr 16191

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-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-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-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-fo 5363  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-sub 8463  df-inn 9258  df-2 9316  df-3 9317  df-4 9318  df-5 9319  df-6 9320  df-7 9321  df-8 9322  df-9 9323  df-n0 9517  df-dec 9731  df-ndx 13302  df-slot 13303  df-base 13305  df-edgf 16129  df-vtx 16138  df-iedg 16139  df-uhgrm 16193

This theorem is referenced by:  lpvtx  16203  subgruhgredgdm  16394