Theorem uhgrspansubgrlem 16271

Description: Lemma for uhgrspansubgr 16272: 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 16272. (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 16055	. . . 4 |- (Edg ` S ) = ran (iEdg ` S )
2	1	eleq2i 2299	. . 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 16072	. . . . . . 7 |- ( G e. UHGraph -> Fun E )
6		funres 5393	. . . . . . 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 5374	. . . . . 6 |- ( ph -> ( Fun (iEdg ` S ) <-> Fun ( E |` A ) ) )
10	7, 9	mpbird 167	. . . . 5 |- ( ph -> Fun (iEdg ` S ) )
11		elrnrexdmb 5817	. . . . 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 5672	. . . . . . . 8 |- ( ( ph /\ i e. dom (iEdg ` S ) ) -> ( (iEdg ` S ) ` i ) = ( ( E |` A ) ` i ) )
15	8	dmeqd 4958	. . . . . . . . . . . . 13 |- ( ph -> dom (iEdg ` S ) = dom ( E |` A ) )
16		dmres 5059	. . . . . . . . . . . . 13 |- dom ( E |` A ) = ( A i^i dom E )
17	15, 16	eqtrdi 2281	. . . . . . . . . . . 12 |- ( ph -> dom (iEdg ` S ) = ( A i^i dom E ) )
18	17	eleq2d 2302	. . . . . . . . . . 11 |- ( ph -> ( i e. dom (iEdg ` S ) <-> i e. ( A i^i dom E ) ) )
19		elinel1 3405	. . . . . . . . . . 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 5695	. . . . . . . 8 |- ( ( ph /\ i e. dom (iEdg ` S ) ) -> ( ( E |` A ) ` i ) = ( E ` i ) )
23	14, 22	eqtrd 2265	. . . . . . 7 |- ( ( ph /\ i e. dom (iEdg ` S ) ) -> ( (iEdg ` S ) ` i ) = ( E ` i ) )
24		elinel2 3406	. . . . . . . . . . 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 16070	. . . . . . . . 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 3674	. . . . . . . . . . 11 |- ( ph -> ~P (Vtx ` S ) = ~P V )
32	31	eleq2d 2302	. . . . . . . . . 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 16014	. . . . . . . . . . . . . 14 |- ( G e. UHGraph -> (iEdg ` G ) e. _V )
35	3, 34	syl 14	. . . . . . . . . . . . 13 |- ( ph -> (iEdg ` G ) e. _V )
36	4, 35	eqeltrid 2319	. . . . . . . . . . . 12 |- ( ph -> E e. _V )
37		vex 2816	. . . . . . . . . . . 12 |- i e. _V
38		fvexg 5689	. . . . . . . . . . . 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 3677	. . . . . . . . . . 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 2309	. . . . . 6 |- ( ( ph /\ i e. dom (iEdg ` S ) ) -> ( (iEdg ` S ) ` i ) e. ~P (Vtx ` S ) )
46		eleq1 2295	. . . . . 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 2660	. . . 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 3244	1 |- ( ph -> (Edg ` S ) C_ ~P (Vtx ` S ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2203   E.wrex 2521   _Vcvv 2813   i^i cin 3210   C_ wss 3211   ~Pcpw 3669   dom cdm 4749   ran crn 4750   |` cres 4751   Fun wfun 5346   ` cfv 5352  Vtxcvtx 16007  iEdgciedg 16008  Edgcedg 16052  UHGraphcuhgr 16062

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 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-cnre 8238

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 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-fo 5358  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-sub 8446  df-inn 9238  df-2 9296  df-3 9297  df-4 9298  df-5 9299  df-6 9300  df-7 9301  df-8 9302  df-9 9303  df-n0 9497  df-dec 9710  df-ndx 13215  df-slot 13216  df-base 13218  df-edgf 16000  df-vtx 16009  df-iedg 16010  df-edg 16053  df-uhgrm 16064

This theorem is referenced by:  uhgrspansubgr  16272