Theorem uhgrspansubgrlem 16397

Description: Lemma for uhgrspansubgr 16398: 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 16398. (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 16181	. . . 4 |- (Edg ` S ) = ran (iEdg ` S )
2	1	eleq2i 2301	. . 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 16198	. . . . . . 7 |- ( G e. UHGraph -> Fun E )
6		funres 5398	. . . . . . 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 5379	. . . . . 6 |- ( ph -> ( Fun (iEdg ` S ) <-> Fun ( E |` A ) ) )
10	7, 9	mpbird 167	. . . . 5 |- ( ph -> Fun (iEdg ` S ) )
11		elrnrexdmb 5822	. . . . 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 5677	. . . . . . . 8 |- ( ( ph /\ i e. dom (iEdg ` S ) ) -> ( (iEdg ` S ) ` i ) = ( ( E |` A ) ` i ) )
15	8	dmeqd 4963	. . . . . . . . . . . . 13 |- ( ph -> dom (iEdg ` S ) = dom ( E |` A ) )
16		dmres 5064	. . . . . . . . . . . . 13 |- dom ( E |` A ) = ( A i^i dom E )
17	15, 16	eqtrdi 2283	. . . . . . . . . . . 12 |- ( ph -> dom (iEdg ` S ) = ( A i^i dom E ) )
18	17	eleq2d 2304	. . . . . . . . . . 11 |- ( ph -> ( i e. dom (iEdg ` S ) <-> i e. ( A i^i dom E ) ) )
19		elinel1 3409	. . . . . . . . . . 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 5700	. . . . . . . 8 |- ( ( ph /\ i e. dom (iEdg ` S ) ) -> ( ( E |` A ) ` i ) = ( E ` i ) )
23	14, 22	eqtrd 2267	. . . . . . 7 |- ( ( ph /\ i e. dom (iEdg ` S ) ) -> ( (iEdg ` S ) ` i ) = ( E ` i ) )
24		elinel2 3410	. . . . . . . . . . 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 16196	. . . . . . . . 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 3679	. . . . . . . . . . 11 |- ( ph -> ~P (Vtx ` S ) = ~P V )
32	31	eleq2d 2304	. . . . . . . . . 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 16140	. . . . . . . . . . . . . 14 |- ( G e. UHGraph -> (iEdg ` G ) e. _V )
35	3, 34	syl 14	. . . . . . . . . . . . 13 |- ( ph -> (iEdg ` G ) e. _V )
36	4, 35	eqeltrid 2321	. . . . . . . . . . . 12 |- ( ph -> E e. _V )
37		vex 2818	. . . . . . . . . . . 12 |- i e. _V
38		fvexg 5694	. . . . . . . . . . . 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 3682	. . . . . . . . . . 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 2311	. . . . . 6 |- ( ( ph /\ i e. dom (iEdg ` S ) ) -> ( (iEdg ` S ) ` i ) e. ~P (Vtx ` S ) )
46		eleq1 2297	. . . . . 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 2662	. . . 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 3248	1 |- ( ph -> (Edg ` S ) C_ ~P (Vtx ` S ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2205   E.wrex 2523   _Vcvv 2815   i^i cin 3213   C_ wss 3214   ~Pcpw 3674   dom cdm 4754   ran crn 4755   |` cres 4756   Fun wfun 5351   ` cfv 5357  Vtxcvtx 16133  iEdgciedg 16134  Edgcedg 16178  UHGraphcuhgr 16188

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 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254

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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-fo 5363  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-sub 8462  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-5 9316  df-6 9317  df-7 9318  df-8 9319  df-9 9320  df-n0 9514  df-dec 9728  df-ndx 13299  df-slot 13300  df-base 13302  df-edgf 16126  df-vtx 16135  df-iedg 16136  df-edg 16179  df-uhgrm 16190

This theorem is referenced by:  uhgrspansubgr  16398