Theorem subgruhgredgd 26176

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

Step	Hyp	Ref	Expression
1		subgruhgredgd.s	. . 3 |- ( ph -> S SubGraph G )
2		subgruhgredgd.v	. . . 4 |- V = (Vtx ` S )
3		eqid 2622	. . . 4 |- (Vtx ` G ) = (Vtx ` G )
4		subgruhgredgd.i	. . . 4 |- I = (iEdg ` S )
5		eqid 2622	. . . 4 |- (iEdg ` G ) = (iEdg ` G )
6		eqid 2622	. . . 4 |- (Edg ` S ) = (Edg ` S )
7	2, 3, 4, 5, 6	subgrprop2 26166	. . 3 |- ( S SubGraph G -> ( V C_ (Vtx ` G ) /\ I C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P V ) )
8	1, 7	syl 17	. 2 |- ( ph -> ( V C_ (Vtx ` G ) /\ I C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P V ) )
9		simpr3 1069	. . . 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 26174	. . . . . . . . 9 |- ( ( G e. UHGraph /\ S SubGraph G ) -> Fun (iEdg ` S ) )
12	10, 1, 11	syl2anc 693	. . . . . . . 8 |- ( ph -> Fun (iEdg ` S ) )
13		subgruhgredgd.x	. . . . . . . . 9 |- ( ph -> X e. dom I )
14	4	dmeqi 5325	. . . . . . . . 9 |- dom I = dom (iEdg ` S )
15	13, 14	syl6eleq 2711	. . . . . . . 8 |- ( ph -> X e. dom (iEdg ` S ) )
16	12, 15	jca 554	. . . . . . 7 |- ( ph -> ( Fun (iEdg ` S ) /\ X e. dom (iEdg ` S ) ) )
17	16	adantr 481	. . . . . 6 |- ( ( ph /\ ( V C_ (Vtx ` G ) /\ I C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P V ) ) -> ( Fun (iEdg ` S ) /\ X e. dom (iEdg ` S ) ) )
18	4	fveq1i 6192	. . . . . . 7 |- ( I ` X ) = ( (iEdg ` S ) ` X )
19		fvelrn 6352	. . . . . . 7 |- ( ( Fun (iEdg ` S ) /\ X e. dom (iEdg ` S ) ) -> ( (iEdg ` S ) ` X ) e. ran (iEdg ` S ) )
20	18, 19	syl5eqel 2705	. . . . . 6 |- ( ( Fun (iEdg ` S ) /\ X e. dom (iEdg ` S ) ) -> ( I ` X ) e. ran (iEdg ` S ) )
21	17, 20	syl 17	. . . . 5 |- ( ( ph /\ ( V C_ (Vtx ` G ) /\ I C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P V ) ) -> ( I ` X ) e. ran (iEdg ` S ) )
22		edgval 25941	. . . . 5 |- (Edg ` S ) = ran (iEdg ` S )
23	21, 22	syl6eleqr 2712	. . . 4 |- ( ( ph /\ ( V C_ (Vtx ` G ) /\ I C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P V ) ) -> ( I ` X ) e. (Edg ` S ) )
24	9, 23	sseldd 3604	. . 3 |- ( ( ph /\ ( V C_ (Vtx ` G ) /\ I C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P V ) ) -> ( I ` X ) e. ~P V )
25	5	uhgrfun 25961	. . . . . . 7 |- ( G e. UHGraph -> Fun (iEdg ` G ) )
26	10, 25	syl 17	. . . . . 6 |- ( ph -> Fun (iEdg ` G ) )
27	26	adantr 481	. . . . 5 |- ( ( ph /\ ( V C_ (Vtx ` G ) /\ I C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P V ) ) -> Fun (iEdg ` G ) )
28		simpr2 1068	. . . . 5 |- ( ( ph /\ ( V C_ (Vtx ` G ) /\ I C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P V ) ) -> I C_ (iEdg ` G ) )
29	13	adantr 481	. . . . 5 |- ( ( ph /\ ( V C_ (Vtx ` G ) /\ I C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P V ) ) -> X e. dom I )
30		funssfv 6209	. . . . . 6 |- ( ( Fun (iEdg ` G ) /\ I C_ (iEdg ` G ) /\ X e. dom I ) -> ( (iEdg ` G ) ` X ) = ( I ` X ) )
31	30	eqcomd 2628	. . . . 5 |- ( ( Fun (iEdg ` G ) /\ I C_ (iEdg ` G ) /\ X e. dom I ) -> ( I ` X ) = ( (iEdg ` G ) ` X ) )
32	27, 28, 29, 31	syl3anc 1326	. . . 4 |- ( ( ph /\ ( V C_ (Vtx ` G ) /\ I C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P V ) ) -> ( I ` X ) = ( (iEdg ` G ) ` X ) )
33	10	adantr 481	. . . . 5 |- ( ( ph /\ ( V C_ (Vtx ` G ) /\ I C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P V ) ) -> G e. UHGraph )
34		funfn 5918	. . . . . . 7 |- ( Fun (iEdg ` G ) <-> (iEdg ` G ) Fn dom (iEdg ` G ) )
35	26, 34	sylib 208	. . . . . 6 |- ( ph -> (iEdg ` G ) Fn dom (iEdg ` G ) )
36	35	adantr 481	. . . . 5 |- ( ( ph /\ ( V C_ (Vtx ` G ) /\ I C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P V ) ) -> (iEdg ` G ) Fn dom (iEdg ` G ) )
37		subgreldmiedg 26175	. . . . . . 7 |- ( ( S SubGraph G /\ X e. dom (iEdg ` S ) ) -> X e. dom (iEdg ` G ) )
38	1, 15, 37	syl2anc 693	. . . . . 6 |- ( ph -> X e. dom (iEdg ` G ) )
39	38	adantr 481	. . . . 5 |- ( ( ph /\ ( V C_ (Vtx ` G ) /\ I C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P V ) ) -> X e. dom (iEdg ` G ) )
40	5	uhgrn0 25962	. . . . 5 |- ( ( G e. UHGraph /\ (iEdg ` G ) Fn dom (iEdg ` G ) /\ X e. dom (iEdg ` G ) ) -> ( (iEdg ` G ) ` X ) =/= (/) )
41	33, 36, 39, 40	syl3anc 1326	. . . 4 |- ( ( ph /\ ( V C_ (Vtx ` G ) /\ I C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P V ) ) -> ( (iEdg ` G ) ` X ) =/= (/) )
42	32, 41	eqnetrd 2861	. . 3 |- ( ( ph /\ ( V C_ (Vtx ` G ) /\ I C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P V ) ) -> ( I ` X ) =/= (/) )
43		eldifsn 4317	. . 3 |- ( ( I ` X ) e. ( ~P V \ { (/) } ) <-> ( ( I ` X ) e. ~P V /\ ( I ` X ) =/= (/) ) )
44	24, 42, 43	sylanbrc 698	. 2 |- ( ( ph /\ ( V C_ (Vtx ` G ) /\ I C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P V ) ) -> ( I ` X ) e. ( ~P V \ { (/) } ) )
45	8, 44	mpdan 702	1 |- ( ph -> ( I ` X ) e. ( ~P V \ { (/) } ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   \ cdif 3571   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   {csn 4177   class class class wbr 4653   dom cdm 5114   ran crn 5115   Fun wfun 5882   Fn wfn 5883   ` cfv 5888  Vtxcvtx 25874  iEdgciedg 25875  Edgcedg 25939   UHGraph cuhgr 25951   SubGraph csubgr 26159

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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-ral 2917  df-rex 2918  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-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-edg 25940  df-uhgr 25953  df-subgr 26160

This theorem is referenced by:  subumgredg2  26177  subuhgr  26178  subupgr  26179