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Theorem subgruhgredgd 39520
Description: An edge of a subgraph of a hypergraph is a nonempty subset of its vertices. (Contributed by AV, 17-Nov-2020.) (Revised by AV, 21-Nov-2020.)
Hypotheses
Ref Expression
subgruhgredgd.v  |-  V  =  (Vtx `  S )
subgruhgredgd.i  |-  I  =  (iEdg `  S )
subgruhgredgd.g  |-  ( ph  ->  G  e. UHGraph  )
subgruhgredgd.s  |-  ( ph  ->  S SubGraph  G )
subgruhgredgd.x  |-  ( ph  ->  X  e.  dom  I
)
Assertion
Ref Expression
subgruhgredgd  |-  ( ph  ->  ( I `  X
)  e.  ( ~P V  \  { (/) } ) )

Proof of Theorem subgruhgredgd
StepHypRef Expression
1 subgruhgredgd.s . . 3  |-  ( ph  ->  S SubGraph  G )
2 subgruhgredgd.v . . . 4  |-  V  =  (Vtx `  S )
3 eqid 2471 . . . 4  |-  (Vtx `  G )  =  (Vtx
`  G )
4 subgruhgredgd.i . . . 4  |-  I  =  (iEdg `  S )
5 eqid 2471 . . . 4  |-  (iEdg `  G )  =  (iEdg `  G )
6 eqid 2471 . . . 4  |-  (Edg `  S )  =  (Edg
`  S )
72, 3, 4, 5, 6subgrprop2 39510 . . 3  |-  ( S SubGraph  G  ->  ( V  C_  (Vtx `  G )  /\  I  C_  (iEdg `  G
)  /\  (Edg `  S
)  C_  ~P V
) )
81, 7syl 17 . 2  |-  ( ph  ->  ( V  C_  (Vtx `  G )  /\  I  C_  (iEdg `  G )  /\  (Edg `  S )  C_ 
~P V ) )
9 simpr3 1038 . . . 4  |-  ( (
ph  /\  ( V  C_  (Vtx `  G )  /\  I  C_  (iEdg `  G )  /\  (Edg `  S )  C_  ~P V ) )  -> 
(Edg `  S )  C_ 
~P V )
10 subgruhgredgd.g . . . . . . . . 9  |-  ( ph  ->  G  e. UHGraph  )
11 subgruhgrfun 39518 . . . . . . . . 9  |-  ( ( G  e. UHGraph  /\  S SubGraph  G )  ->  Fun  (iEdg `  S
) )
1210, 1, 11syl2anc 673 . . . . . . . 8  |-  ( ph  ->  Fun  (iEdg `  S
) )
13 subgruhgredgd.x . . . . . . . . 9  |-  ( ph  ->  X  e.  dom  I
)
144dmeqi 5041 . . . . . . . . 9  |-  dom  I  =  dom  (iEdg `  S
)
1513, 14syl6eleq 2559 . . . . . . . 8  |-  ( ph  ->  X  e.  dom  (iEdg `  S ) )
1612, 15jca 541 . . . . . . 7  |-  ( ph  ->  ( Fun  (iEdg `  S )  /\  X  e.  dom  (iEdg `  S
) ) )
1716adantr 472 . . . . . 6  |-  ( (
ph  /\  ( V  C_  (Vtx `  G )  /\  I  C_  (iEdg `  G )  /\  (Edg `  S )  C_  ~P V ) )  -> 
( Fun  (iEdg `  S
)  /\  X  e.  dom  (iEdg `  S )
) )
184fveq1i 5880 . . . . . . 7  |-  ( I `
 X )  =  ( (iEdg `  S
) `  X )
19 fvelrn 6030 . . . . . . 7  |-  ( ( Fun  (iEdg `  S
)  /\  X  e.  dom  (iEdg `  S )
)  ->  ( (iEdg `  S ) `  X
)  e.  ran  (iEdg `  S ) )
2018, 19syl5eqel 2553 . . . . . 6  |-  ( ( Fun  (iEdg `  S
)  /\  X  e.  dom  (iEdg `  S )
)  ->  ( I `  X )  e.  ran  (iEdg `  S ) )
2117, 20syl 17 . . . . 5  |-  ( (
ph  /\  ( V  C_  (Vtx `  G )  /\  I  C_  (iEdg `  G )  /\  (Edg `  S )  C_  ~P V ) )  -> 
( I `  X
)  e.  ran  (iEdg `  S ) )
22 subgrv 39506 . . . . . . . . 9  |-  ( S SubGraph  G  ->  ( S  e. 
_V  /\  G  e.  _V ) )
2322simpld 466 . . . . . . . 8  |-  ( S SubGraph  G  ->  S  e.  _V )
241, 23syl 17 . . . . . . 7  |-  ( ph  ->  S  e.  _V )
25 edgaval 39373 . . . . . . 7  |-  ( S  e.  _V  ->  (Edg `  S )  =  ran  (iEdg `  S ) )
2624, 25syl 17 . . . . . 6  |-  ( ph  ->  (Edg `  S )  =  ran  (iEdg `  S
) )
2726adantr 472 . . . . 5  |-  ( (
ph  /\  ( V  C_  (Vtx `  G )  /\  I  C_  (iEdg `  G )  /\  (Edg `  S )  C_  ~P V ) )  -> 
(Edg `  S )  =  ran  (iEdg `  S
) )
2821, 27eleqtrrd 2552 . . . 4  |-  ( (
ph  /\  ( V  C_  (Vtx `  G )  /\  I  C_  (iEdg `  G )  /\  (Edg `  S )  C_  ~P V ) )  -> 
( I `  X
)  e.  (Edg `  S ) )
299, 28sseldd 3419 . . 3  |-  ( (
ph  /\  ( V  C_  (Vtx `  G )  /\  I  C_  (iEdg `  G )  /\  (Edg `  S )  C_  ~P V ) )  -> 
( I `  X
)  e.  ~P V
)
305uhgrfun 39310 . . . . . . 7  |-  ( G  e. UHGraph  ->  Fun  (iEdg `  G
) )
3110, 30syl 17 . . . . . 6  |-  ( ph  ->  Fun  (iEdg `  G
) )
3231adantr 472 . . . . 5  |-  ( (
ph  /\  ( V  C_  (Vtx `  G )  /\  I  C_  (iEdg `  G )  /\  (Edg `  S )  C_  ~P V ) )  ->  Fun  (iEdg `  G )
)
33 simpr2 1037 . . . . 5  |-  ( (
ph  /\  ( V  C_  (Vtx `  G )  /\  I  C_  (iEdg `  G )  /\  (Edg `  S )  C_  ~P V ) )  ->  I  C_  (iEdg `  G
) )
3413adantr 472 . . . . 5  |-  ( (
ph  /\  ( V  C_  (Vtx `  G )  /\  I  C_  (iEdg `  G )  /\  (Edg `  S )  C_  ~P V ) )  ->  X  e.  dom  I )
35 funssfv 5894 . . . . . 6  |-  ( ( Fun  (iEdg `  G
)  /\  I  C_  (iEdg `  G )  /\  X  e.  dom  I )  -> 
( (iEdg `  G
) `  X )  =  ( I `  X ) )
3635eqcomd 2477 . . . . 5  |-  ( ( Fun  (iEdg `  G
)  /\  I  C_  (iEdg `  G )  /\  X  e.  dom  I )  -> 
( I `  X
)  =  ( (iEdg `  G ) `  X
) )
3732, 33, 34, 36syl3anc 1292 . . . 4  |-  ( (
ph  /\  ( V  C_  (Vtx `  G )  /\  I  C_  (iEdg `  G )  /\  (Edg `  S )  C_  ~P V ) )  -> 
( I `  X
)  =  ( (iEdg `  G ) `  X
) )
3810adantr 472 . . . . 5  |-  ( (
ph  /\  ( V  C_  (Vtx `  G )  /\  I  C_  (iEdg `  G )  /\  (Edg `  S )  C_  ~P V ) )  ->  G  e. UHGraph  )
39 funfn 5618 . . . . . . 7  |-  ( Fun  (iEdg `  G )  <->  (iEdg `  G )  Fn  dom  (iEdg `  G ) )
4031, 39sylib 201 . . . . . 6  |-  ( ph  ->  (iEdg `  G )  Fn  dom  (iEdg `  G
) )
4140adantr 472 . . . . 5  |-  ( (
ph  /\  ( V  C_  (Vtx `  G )  /\  I  C_  (iEdg `  G )  /\  (Edg `  S )  C_  ~P V ) )  -> 
(iEdg `  G )  Fn  dom  (iEdg `  G
) )
42 subgreldmiedg 39519 . . . . . . 7  |-  ( ( S SubGraph  G  /\  X  e. 
dom  (iEdg `  S )
)  ->  X  e.  dom  (iEdg `  G )
)
431, 15, 42syl2anc 673 . . . . . 6  |-  ( ph  ->  X  e.  dom  (iEdg `  G ) )
4443adantr 472 . . . . 5  |-  ( (
ph  /\  ( V  C_  (Vtx `  G )  /\  I  C_  (iEdg `  G )  /\  (Edg `  S )  C_  ~P V ) )  ->  X  e.  dom  (iEdg `  G ) )
455uhgrn0 39311 . . . . 5  |-  ( ( G  e. UHGraph  /\  (iEdg `  G )  Fn  dom  (iEdg `  G )  /\  X  e.  dom  (iEdg `  G ) )  -> 
( (iEdg `  G
) `  X )  =/=  (/) )
4638, 41, 44, 45syl3anc 1292 . . . 4  |-  ( (
ph  /\  ( V  C_  (Vtx `  G )  /\  I  C_  (iEdg `  G )  /\  (Edg `  S )  C_  ~P V ) )  -> 
( (iEdg `  G
) `  X )  =/=  (/) )
4737, 46eqnetrd 2710 . . 3  |-  ( (
ph  /\  ( V  C_  (Vtx `  G )  /\  I  C_  (iEdg `  G )  /\  (Edg `  S )  C_  ~P V ) )  -> 
( I `  X
)  =/=  (/) )
48 eldifsn 4088 . . 3  |-  ( ( I `  X )  e.  ( ~P V  \  { (/) } )  <->  ( (
I `  X )  e.  ~P V  /\  (
I `  X )  =/=  (/) ) )
4929, 47, 48sylanbrc 677 . 2  |-  ( (
ph  /\  ( V  C_  (Vtx `  G )  /\  I  C_  (iEdg `  G )  /\  (Edg `  S )  C_  ~P V ) )  -> 
( I `  X
)  e.  ( ~P V  \  { (/) } ) )
508, 49mpdan 681 1  |-  ( ph  ->  ( I `  X
)  e.  ( ~P V  \  { (/) } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   _Vcvv 3031    \ cdif 3387    C_ wss 3390   (/)c0 3722   ~Pcpw 3942   {csn 3959   class class class wbr 4395   dom cdm 4839   ran crn 4840   Fun wfun 5583    Fn wfn 5584   ` cfv 5589  Vtxcvtx 39251  iEdgciedg 39252   UHGraph cuhgr 39300  Edgcedga 39371   SubGraph csubgr 39503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-fv 5597  df-uhgr 39302  df-edga 39372  df-subgr 39504
This theorem is referenced by:  subumgredg2  39521  subuhgr  39522  subupgr  39523
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