Theorem uhgrspansubgrlem 26182

Description: Lemma for uhgrspansubgr 26183: The edges of the graph S obtained by removing some edges of a hypergraph G are subsets of its vertices (a spanning subgraph, see comment for uhgrspansubgr 26183. (Contributed by AV, 18-Nov-2020.)

Hypotheses

Ref	Expression
uhgrspan.v	|- V = (Vtx ` G )
uhgrspan.e	|- E = (iEdg ` G )
uhgrspan.s	|- ( ph -> S e. W )
uhgrspan.q	|- ( ph -> (Vtx ` S ) = V )
uhgrspan.r	|- ( ph -> (iEdg ` S ) = ( E |` A ) )
uhgrspan.g	|- ( ph -> G e. UHGraph )

Assertion

Ref	Expression
uhgrspansubgrlem	|- ( ph -> (Edg ` S ) C_ ~P (Vtx ` S ) )

Proof of Theorem uhgrspansubgrlem

Dummy variables e i are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		edgval 25941	. . . 4 |- (Edg ` S ) = ran (iEdg ` S )
2	1	eleq2i 2693	. . 3 |- ( e e. (Edg ` S ) <-> e e. ran (iEdg ` S ) )
3		uhgrspan.g	. . . . . . 7 |- ( ph -> G e. UHGraph )
4		uhgrspan.e	. . . . . . . 8 |- E = (iEdg ` G )
5	4	uhgrfun 25961	. . . . . . 7 |- ( G e. UHGraph -> Fun E )
6		funres 5929	. . . . . . 7 |- ( Fun E -> Fun ( E |` A ) )
7	3, 5, 6	3syl 18	. . . . . 6 |- ( ph -> Fun ( E |` A ) )
8		uhgrspan.r	. . . . . . 7 |- ( ph -> (iEdg ` S ) = ( E |` A ) )
9	8	funeqd 5910	. . . . . 6 |- ( ph -> ( Fun (iEdg ` S ) <-> Fun ( E |` A ) ) )
10	7, 9	mpbird 247	. . . . 5 |- ( ph -> Fun (iEdg ` S ) )
11		elrnrexdmb 6364	. . . . 5 |- ( Fun (iEdg ` S ) -> ( e e. ran (iEdg ` S ) <-> E. i e. dom (iEdg ` S ) e = ( (iEdg ` S ) ` i ) ) )
12	10, 11	syl 17	. . . 4 |- ( ph -> ( e e. ran (iEdg ` S ) <-> E. i e. dom (iEdg ` S ) e = ( (iEdg ` S ) ` i ) ) )
13	8	adantr 481	. . . . . . . . 9 |- ( ( ph /\ i e. dom (iEdg ` S ) ) -> (iEdg ` S ) = ( E |` A ) )
14	13	fveq1d 6193	. . . . . . . 8 |- ( ( ph /\ i e. dom (iEdg ` S ) ) -> ( (iEdg ` S ) ` i ) = ( ( E |` A ) ` i ) )
15	8	dmeqd 5326	. . . . . . . . . . . . 13 |- ( ph -> dom (iEdg ` S ) = dom ( E |` A ) )
16		dmres 5419	. . . . . . . . . . . . 13 |- dom ( E |` A ) = ( A i^i dom E )
17	15, 16	syl6eq 2672	. . . . . . . . . . . 12 |- ( ph -> dom (iEdg ` S ) = ( A i^i dom E ) )
18	17	eleq2d 2687	. . . . . . . . . . 11 |- ( ph -> ( i e. dom (iEdg ` S ) <-> i e. ( A i^i dom E ) ) )
19		elinel1 3799	. . . . . . . . . . 11 |- ( i e. ( A i^i dom E ) -> i e. A )
20	18, 19	syl6bi 243	. . . . . . . . . 10 |- ( ph -> ( i e. dom (iEdg ` S ) -> i e. A ) )
21	20	imp 445	. . . . . . . . 9 |- ( ( ph /\ i e. dom (iEdg ` S ) ) -> i e. A )
22	21	fvresd 6208	. . . . . . . 8 |- ( ( ph /\ i e. dom (iEdg ` S ) ) -> ( ( E |` A ) ` i ) = ( E ` i ) )
23	14, 22	eqtrd 2656	. . . . . . 7 |- ( ( ph /\ i e. dom (iEdg ` S ) ) -> ( (iEdg ` S ) ` i ) = ( E ` i ) )
24		elinel2 3800	. . . . . . . . . . 11 |- ( i e. ( A i^i dom E ) -> i e. dom E )
25	18, 24	syl6bi 243	. . . . . . . . . 10 |- ( ph -> ( i e. dom (iEdg ` S ) -> i e. dom E ) )
26	25	imp 445	. . . . . . . . 9 |- ( ( ph /\ i e. dom (iEdg ` S ) ) -> i e. dom E )
27		uhgrspan.v	. . . . . . . . . 10 |- V = (Vtx ` G )
28	27, 4	uhgrss 25959	. . . . . . . . 9 |- ( ( G e. UHGraph /\ i e. dom E ) -> ( E ` i ) C_ V )
29	3, 26, 28	syl2an2r 876	. . . . . . . 8 |- ( ( ph /\ i e. dom (iEdg ` S ) ) -> ( E ` i ) C_ V )
30		uhgrspan.q	. . . . . . . . . . . 12 |- ( ph -> (Vtx ` S ) = V )
31	30	pweqd 4163	. . . . . . . . . . 11 |- ( ph -> ~P (Vtx ` S ) = ~P V )
32	31	eleq2d 2687	. . . . . . . . . 10 |- ( ph -> ( ( E ` i ) e. ~P (Vtx ` S ) <-> ( E ` i ) e. ~P V ) )
33	32	adantr 481	. . . . . . . . 9 |- ( ( ph /\ i e. dom (iEdg ` S ) ) -> ( ( E ` i ) e. ~P (Vtx ` S ) <-> ( E ` i ) e. ~P V ) )
34		fvex 6201	. . . . . . . . . 10 |- ( E ` i ) e. _V
35	34	elpw 4164	. . . . . . . . 9 |- ( ( E ` i ) e. ~P V <-> ( E ` i ) C_ V )
36	33, 35	syl6bb 276	. . . . . . . 8 |- ( ( ph /\ i e. dom (iEdg ` S ) ) -> ( ( E ` i ) e. ~P (Vtx ` S ) <-> ( E ` i ) C_ V ) )
37	29, 36	mpbird 247	. . . . . . 7 |- ( ( ph /\ i e. dom (iEdg ` S ) ) -> ( E ` i ) e. ~P (Vtx ` S ) )
38	23, 37	eqeltrd 2701	. . . . . 6 |- ( ( ph /\ i e. dom (iEdg ` S ) ) -> ( (iEdg ` S ) ` i ) e. ~P (Vtx ` S ) )
39		eleq1 2689	. . . . . 6 |- ( e = ( (iEdg ` S ) ` i ) -> ( e e. ~P (Vtx ` S ) <-> ( (iEdg ` S ) ` i ) e. ~P (Vtx ` S ) ) )
40	38, 39	syl5ibrcom 237	. . . . 5 |- ( ( ph /\ i e. dom (iEdg ` S ) ) -> ( e = ( (iEdg ` S ) ` i ) -> e e. ~P (Vtx ` S ) ) )
41	40	rexlimdva 3031	. . . 4 |- ( ph -> ( E. i e. dom (iEdg ` S ) e = ( (iEdg ` S ) ` i ) -> e e. ~P (Vtx ` S ) ) )
42	12, 41	sylbid 230	. . 3 |- ( ph -> ( e e. ran (iEdg ` S ) -> e e. ~P (Vtx ` S ) ) )
43	2, 42	syl5bi 232	. 2 |- ( ph -> ( e e. (Edg ` S ) -> e e. ~P (Vtx ` S ) ) )
44	43	ssrdv 3609	1 |- ( ph -> (Edg ` S ) C_ ~P (Vtx ` S ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   dom cdm 5114   ran crn 5115   |` cres 5116   Fun wfun 5882   ` cfv 5888  Vtxcvtx 25874  iEdgciedg 25875  Edgcedg 25939   UHGraph cuhgr 25951

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-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

This theorem is referenced by:  uhgrspansubgr  26183