Theorem uhgrm 16060

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 2232	. . . . . . . 8 |- (Vtx ` G ) = (Vtx ` G )
2		uhgrfun.e	. . . . . . . 8 |- E = (iEdg ` G )
3	1, 2	uhgrfm 16055	. . . . . . 7 |- ( G e. UHGraph -> E : dom E --> { s e. ~P (Vtx ` G ) | E. j j e. s } )
4		fndm 5454	. . . . . . . 8 |- ( E Fn A -> dom E = A )
5	4	feq2d 5495	. . . . . . 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 5811	. . . 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 2296	. . . . 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 2972	. . 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 2203   {crab 2524   ~Pcpw 3668   dom cdm 4748   Fn wfn 5346   -->wf 5347   ` cfv 5351  Vtxcvtx 15994  iEdgciedg 15995  UHGraphcuhgr 16049

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 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-cnex 8214  ax-resscn 8215  ax-1cn 8216  ax-1re 8217  ax-icn 8218  ax-addcl 8219  ax-addrcl 8220  ax-mulcl 8221  ax-addcom 8223  ax-mulcom 8224  ax-addass 8225  ax-mulass 8226  ax-distr 8227  ax-i2m1 8228  ax-1rid 8230  ax-0id 8231  ax-rnegex 8232  ax-cnre 8234

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-fo 5357  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-sub 8442  df-inn 9234  df-2 9292  df-3 9293  df-4 9294  df-5 9295  df-6 9296  df-7 9297  df-8 9298  df-9 9299  df-n0 9493  df-dec 9706  df-ndx 13204  df-slot 13205  df-base 13207  df-edgf 15987  df-vtx 15996  df-iedg 15997  df-uhgrm 16051

This theorem is referenced by:  lpvtx  16061  subgruhgredgdm  16252