Theorem uhgrspansubgrlem 16200

Description: Lemma for uhgrspansubgr 16201: 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 16201. (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 15984	. . . 4 |- (Edg ` S ) = ran (iEdg ` S )
2	1	eleq2i 2298	. . 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 16001	. . . . . . 7 |- ( G e. UHGraph -> Fun E )
6		funres 5374	. . . . . . 7 |- ( Fun E -> Fun ( E |` A ) )
7	3, 5, 6	3syl 17	. . . . . 6 |- ( ph -> Fun ( E |` A ) )
8		uhgrspan.r	. . . . . . 7 |- ( ph -> (iEdg ` S ) = ( E |` A ) )
9	8	funeqd 5355	. . . . . 6 |- ( ph -> ( Fun (iEdg ` S ) <-> Fun ( E |` A ) ) )
10	7, 9	mpbird 167	. . . . 5 |- ( ph -> Fun (iEdg ` S ) )
11		elrnrexdmb 5795	. . . . 5 |- ( Fun (iEdg ` S ) -> ( e e. ran (iEdg ` S ) <-> E. i e. dom (iEdg ` S ) e = ( (iEdg ` S ) ` i ) ) )
12	10, 11	syl 14	. . . 4 |- ( ph -> ( e e. ran (iEdg ` S ) <-> E. i e. dom (iEdg ` S ) e = ( (iEdg ` S ) ` i ) ) )
13	8	adantr 276	. . . . . . . . 9 |- ( ( ph /\ i e. dom (iEdg ` S ) ) -> (iEdg ` S ) = ( E |` A ) )
14	13	fveq1d 5650	. . . . . . . 8 |- ( ( ph /\ i e. dom (iEdg ` S ) ) -> ( (iEdg ` S ) ` i ) = ( ( E |` A ) ` i ) )
15	8	dmeqd 4939	. . . . . . . . . . . . 13 |- ( ph -> dom (iEdg ` S ) = dom ( E |` A ) )
16		dmres 5040	. . . . . . . . . . . . 13 |- dom ( E |` A ) = ( A i^i dom E )
17	15, 16	eqtrdi 2280	. . . . . . . . . . . 12 |- ( ph -> dom (iEdg ` S ) = ( A i^i dom E ) )
18	17	eleq2d 2301	. . . . . . . . . . 11 |- ( ph -> ( i e. dom (iEdg ` S ) <-> i e. ( A i^i dom E ) ) )
19		elinel1 3395	. . . . . . . . . . 11 |- ( i e. ( A i^i dom E ) -> i e. A )
20	18, 19	biimtrdi 163	. . . . . . . . . 10 |- ( ph -> ( i e. dom (iEdg ` S ) -> i e. A ) )
21	20	imp 124	. . . . . . . . 9 |- ( ( ph /\ i e. dom (iEdg ` S ) ) -> i e. A )
22	21	fvresd 5673	. . . . . . . 8 |- ( ( ph /\ i e. dom (iEdg ` S ) ) -> ( ( E |` A ) ` i ) = ( E ` i ) )
23	14, 22	eqtrd 2264	. . . . . . 7 |- ( ( ph /\ i e. dom (iEdg ` S ) ) -> ( (iEdg ` S ) ` i ) = ( E ` i ) )
24		elinel2 3396	. . . . . . . . . . 11 |- ( i e. ( A i^i dom E ) -> i e. dom E )
25	18, 24	biimtrdi 163	. . . . . . . . . 10 |- ( ph -> ( i e. dom (iEdg ` S ) -> i e. dom E ) )
26	25	imp 124	. . . . . . . . 9 |- ( ( ph /\ i e. dom (iEdg ` S ) ) -> i e. dom E )
27		uhgrspan.v	. . . . . . . . . 10 |- V = (Vtx ` G )
28	27, 4	uhgrss 15999	. . . . . . . . 9 |- ( ( G e. UHGraph /\ i e. dom E ) -> ( E ` i ) C_ V )
29	3, 26, 28	syl2an2r 599	. . . . . . . 8 |- ( ( ph /\ i e. dom (iEdg ` S ) ) -> ( E ` i ) C_ V )
30		uhgrspan.q	. . . . . . . . . . . 12 |- ( ph -> (Vtx ` S ) = V )
31	30	pweqd 3661	. . . . . . . . . . 11 |- ( ph -> ~P (Vtx ` S ) = ~P V )
32	31	eleq2d 2301	. . . . . . . . . 10 |- ( ph -> ( ( E ` i ) e. ~P (Vtx ` S ) <-> ( E ` i ) e. ~P V ) )
33	32	adantr 276	. . . . . . . . 9 |- ( ( ph /\ i e. dom (iEdg ` S ) ) -> ( ( E ` i ) e. ~P (Vtx ` S ) <-> ( E ` i ) e. ~P V ) )
34		iedgex 15943	. . . . . . . . . . . . . 14 |- ( G e. UHGraph -> (iEdg ` G ) e. _V )
35	3, 34	syl 14	. . . . . . . . . . . . 13 |- ( ph -> (iEdg ` G ) e. _V )
36	4, 35	eqeltrid 2318	. . . . . . . . . . . 12 |- ( ph -> E e. _V )
37		vex 2806	. . . . . . . . . . . 12 |- i e. _V
38		fvexg 5667	. . . . . . . . . . . 12 |- ( ( E e. _V /\ i e. _V ) -> ( E ` i ) e. _V )
39	36, 37, 38	sylancl 413	. . . . . . . . . . 11 |- ( ph -> ( E ` i ) e. _V )
40		elpwg 3664	. . . . . . . . . . 11 |- ( ( E ` i ) e. _V -> ( ( E ` i ) e. ~P V <-> ( E ` i ) C_ V ) )
41	39, 40	syl 14	. . . . . . . . . 10 |- ( ph -> ( ( E ` i ) e. ~P V <-> ( E ` i ) C_ V ) )
42	41	adantr 276	. . . . . . . . 9 |- ( ( ph /\ i e. dom (iEdg ` S ) ) -> ( ( E ` i ) e. ~P V <-> ( E ` i ) C_ V ) )
43	33, 42	bitrd 188	. . . . . . . 8 |- ( ( ph /\ i e. dom (iEdg ` S ) ) -> ( ( E ` i ) e. ~P (Vtx ` S ) <-> ( E ` i ) C_ V ) )
44	29, 43	mpbird 167	. . . . . . 7 |- ( ( ph /\ i e. dom (iEdg ` S ) ) -> ( E ` i ) e. ~P (Vtx ` S ) )
45	23, 44	eqeltrd 2308	. . . . . 6 |- ( ( ph /\ i e. dom (iEdg ` S ) ) -> ( (iEdg ` S ) ` i ) e. ~P (Vtx ` S ) )
46		eleq1 2294	. . . . . 6 |- ( e = ( (iEdg ` S ) ` i ) -> ( e e. ~P (Vtx ` S ) <-> ( (iEdg ` S ) ` i ) e. ~P (Vtx ` S ) ) )
47	45, 46	syl5ibrcom 157	. . . . 5 |- ( ( ph /\ i e. dom (iEdg ` S ) ) -> ( e = ( (iEdg ` S ) ` i ) -> e e. ~P (Vtx ` S ) ) )
48	47	rexlimdva 2651	. . . 4 |- ( ph -> ( E. i e. dom (iEdg ` S ) e = ( (iEdg ` S ) ` i ) -> e e. ~P (Vtx ` S ) ) )
49	12, 48	sylbid 150	. . 3 |- ( ph -> ( e e. ran (iEdg ` S ) -> e e. ~P (Vtx ` S ) ) )
50	2, 49	biimtrid 152	. 2 |- ( ph -> ( e e. (Edg ` S ) -> e e. ~P (Vtx ` S ) ) )
51	50	ssrdv 3234	1 |- ( ph -> (Edg ` S ) C_ ~P (Vtx ` S ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2202   E.wrex 2512   _Vcvv 2803   i^i cin 3200   C_ wss 3201   ~Pcpw 3656   dom cdm 4731   ran crn 4732   |` cres 4733   Fun wfun 5327   ` cfv 5333  Vtxcvtx 15936  iEdgciedg 15937  Edgcedg 15981  UHGraphcuhgr 15991

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 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-cnre 8186

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fo 5339  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-sub 8394  df-inn 9186  df-2 9244  df-3 9245  df-4 9246  df-5 9247  df-6 9248  df-7 9249  df-8 9250  df-9 9251  df-n0 9445  df-dec 9656  df-ndx 13148  df-slot 13149  df-base 13151  df-edgf 15929  df-vtx 15938  df-iedg 15939  df-edg 15982  df-uhgrm 15993

This theorem is referenced by:  uhgrspansubgr  16201