Theorem uhgrspansubgrlem 16126

Description: Lemma for uhgrspansubgr 16127: 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 16127. (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 15910	. . . 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 15927	. . . . . . 7 |- ( G e. UHGraph -> Fun E )
6		funres 5367	. . . . . . 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 5348	. . . . . 6 |- ( ph -> ( Fun (iEdg ` S ) <-> Fun ( E |` A ) ) )
10	7, 9	mpbird 167	. . . . 5 |- ( ph -> Fun (iEdg ` S ) )
11		elrnrexdmb 5787	. . . . 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 5641	. . . . . . . 8 |- ( ( ph /\ i e. dom (iEdg ` S ) ) -> ( (iEdg ` S ) ` i ) = ( ( E |` A ) ` i ) )
15	8	dmeqd 4933	. . . . . . . . . . . . 13 |- ( ph -> dom (iEdg ` S ) = dom ( E |` A ) )
16		dmres 5034	. . . . . . . . . . . . 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 3393	. . . . . . . . . . 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 5664	. . . . . . . 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 3394	. . . . . . . . . . 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 15925	. . . . . . . . 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 3657	. . . . . . . . . . 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 15869	. . . . . . . . . . . . . 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 2805	. . . . . . . . . . . 12 |- i e. _V
38		fvexg 5658	. . . . . . . . . . . 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 3660	. . . . . . . . . . 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 2650	. . . 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 3233	1 |- ( ph -> (Edg ` S ) C_ ~P (Vtx ` S ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1397   e. wcel 2202   E.wrex 2511   _Vcvv 2802   i^i cin 3199   C_ wss 3200   ~Pcpw 3652   dom cdm 4725   ran crn 4726   |` cres 4727   Fun wfun 5320   ` cfv 5326  Vtxcvtx 15862  iEdgciedg 15863  Edgcedg 15907  UHGraphcuhgr 15917

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-cnre 8142

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fo 5332  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-sub 8351  df-inn 9143  df-2 9201  df-3 9202  df-4 9203  df-5 9204  df-6 9205  df-7 9206  df-8 9207  df-9 9208  df-n0 9402  df-dec 9611  df-ndx 13084  df-slot 13085  df-base 13087  df-edgf 15855  df-vtx 15864  df-iedg 15865  df-edg 15908  df-uhgrm 15919

This theorem is referenced by:  uhgrspansubgr  16127