Theorem uhgrstrrepe 25745

Description: Replacing (or adding) the edges (between elements of the base set) of an extensible structure results in a hypergraph. Instead of requiring (𝜑 → 𝐺 Struct ⟨(Base‘ndx), 𝐼⟩), it would be sufficient to require (𝜑 → Fun (𝐺 ∖ {∅})) or only (𝜑 → Fun 𝐺). (Contributed by AV, 18-Jan-2020.) (Revised by AV, 7-Jun-2021.)

Hypotheses

Ref	Expression
uhgrstrrepe.v	⊢ 𝑉 = (Base‘𝐺)
uhgrstrrepe.i	⊢ 𝐼 = (.ef‘ndx)
uhgrstrrepe.s	⊢ (𝜑 → 𝐺 Struct ⟨(Base‘ndx), 𝐼⟩)
uhgrstrrepe.b	⊢ (𝜑 → (Base‘ndx) ∈ dom 𝐺)
uhgrstrrepe.g	⊢ (𝜑 → 𝐺 ∈ 𝑈)
uhgrstrrepe.e	⊢ (𝜑 → 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}))
uhgrstrrepe.w	⊢ (𝜑 → 𝐸 ∈ 𝑊)

Assertion

Ref	Expression
uhgrstrrepe	⊢ (𝜑 → (𝐺 sSet ⟨𝐼, 𝐸⟩) ∈ UHGraph )

Proof of Theorem uhgrstrrepe

Step	Hyp	Ref	Expression
1		uhgrstrrepe.e	. . . 4 ⊢ (𝜑 → 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}))
2		uhgrstrrepe.v	. . . . . . . 8 ⊢ 𝑉 = (Base‘𝐺)
3		baseid 15747	. . . . . . . . . 10 ⊢ Base = Slot (Base‘ndx)
4		slotsbaseefdif 25672	. . . . . . . . . . 11 ⊢ (Base‘ndx) ≠ (.ef‘ndx)
5		uhgrstrrepe.i	. . . . . . . . . . 11 ⊢ 𝐼 = (.ef‘ndx)
6	4, 5	neeqtrri 2855	. . . . . . . . . 10 ⊢ (Base‘ndx) ≠ 𝐼
7	3, 6	setsnid 15743	. . . . . . . . 9 ⊢ (Base‘𝐺) = (Base‘(𝐺 sSet ⟨𝐼, 𝐸⟩))
8		uhgrstrrepe.s	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 Struct ⟨(Base‘ndx), 𝐼⟩)
9		uhgrstrrepe.b	. . . . . . . . . . 11 ⊢ (𝜑 → (Base‘ndx) ∈ dom 𝐺)
10		uhgrstrrepe.g	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 ∈ 𝑈)
11		uhgrstrrepe.w	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∈ 𝑊)
12	2, 5, 8, 9, 10, 1, 11	uhgrstrrepelem 25744	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐺 sSet ⟨𝐼, 𝐸⟩) ∈ V ∧ Fun ((𝐺 sSet ⟨𝐼, 𝐸⟩) ∖ {∅}) ∧ {(Base‘ndx), (.ef‘ndx)} ⊆ dom (𝐺 sSet ⟨𝐼, 𝐸⟩)))
13		funvtxval 25695	. . . . . . . . . . 11 ⊢ (((𝐺 sSet ⟨𝐼, 𝐸⟩) ∈ V ∧ Fun ((𝐺 sSet ⟨𝐼, 𝐸⟩) ∖ {∅}) ∧ {(Base‘ndx), (.ef‘ndx)} ⊆ dom (𝐺 sSet ⟨𝐼, 𝐸⟩)) → (Vtx‘(𝐺 sSet ⟨𝐼, 𝐸⟩)) = (Base‘(𝐺 sSet ⟨𝐼, 𝐸⟩)))
14	13	eqcomd 2616	. . . . . . . . . 10 ⊢ (((𝐺 sSet ⟨𝐼, 𝐸⟩) ∈ V ∧ Fun ((𝐺 sSet ⟨𝐼, 𝐸⟩) ∖ {∅}) ∧ {(Base‘ndx), (.ef‘ndx)} ⊆ dom (𝐺 sSet ⟨𝐼, 𝐸⟩)) → (Base‘(𝐺 sSet ⟨𝐼, 𝐸⟩)) = (Vtx‘(𝐺 sSet ⟨𝐼, 𝐸⟩)))
15	12, 14	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (Base‘(𝐺 sSet ⟨𝐼, 𝐸⟩)) = (Vtx‘(𝐺 sSet ⟨𝐼, 𝐸⟩)))
16	7, 15	syl5eq 2656	. . . . . . . 8 ⊢ (𝜑 → (Base‘𝐺) = (Vtx‘(𝐺 sSet ⟨𝐼, 𝐸⟩)))
17	2, 16	syl5req 2657	. . . . . . 7 ⊢ (𝜑 → (Vtx‘(𝐺 sSet ⟨𝐼, 𝐸⟩)) = 𝑉)
18	17	pweqd 4113	. . . . . 6 ⊢ (𝜑 → 𝒫 (Vtx‘(𝐺 sSet ⟨𝐼, 𝐸⟩)) = 𝒫 𝑉)
19	18	difeq1d 3689	. . . . 5 ⊢ (𝜑 → (𝒫 (Vtx‘(𝐺 sSet ⟨𝐼, 𝐸⟩)) ∖ {∅}) = (𝒫 𝑉 ∖ {∅}))
20	19	feq3d 5945	. . . 4 ⊢ (𝜑 → (𝐸:dom 𝐸⟶(𝒫 (Vtx‘(𝐺 sSet ⟨𝐼, 𝐸⟩)) ∖ {∅}) ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
21	1, 20	mpbird 246	. . 3 ⊢ (𝜑 → 𝐸:dom 𝐸⟶(𝒫 (Vtx‘(𝐺 sSet ⟨𝐼, 𝐸⟩)) ∖ {∅}))
22	5	opeq1i 4343	. . . . . . 7 ⊢ ⟨𝐼, 𝐸⟩ = ⟨(.ef‘ndx), 𝐸⟩
23	22	oveq2i 6560	. . . . . 6 ⊢ (𝐺 sSet ⟨𝐼, 𝐸⟩) = (𝐺 sSet ⟨(.ef‘ndx), 𝐸⟩)
24	23	fveq2i 6106	. . . . 5 ⊢ (.ef‘(𝐺 sSet ⟨𝐼, 𝐸⟩)) = (.ef‘(𝐺 sSet ⟨(.ef‘ndx), 𝐸⟩))
25		funiedgval 25696	. . . . . 6 ⊢ (((𝐺 sSet ⟨𝐼, 𝐸⟩) ∈ V ∧ Fun ((𝐺 sSet ⟨𝐼, 𝐸⟩) ∖ {∅}) ∧ {(Base‘ndx), (.ef‘ndx)} ⊆ dom (𝐺 sSet ⟨𝐼, 𝐸⟩)) → (iEdg‘(𝐺 sSet ⟨𝐼, 𝐸⟩)) = (.ef‘(𝐺 sSet ⟨𝐼, 𝐸⟩)))
26	12, 25	syl 17	. . . . 5 ⊢ (𝜑 → (iEdg‘(𝐺 sSet ⟨𝐼, 𝐸⟩)) = (.ef‘(𝐺 sSet ⟨𝐼, 𝐸⟩)))
27		df-edgf 25668	. . . . . . . 8 ⊢ .ef = Slot ;18
28		1nn0 11185	. . . . . . . . 9 ⊢ 1 ∈ ℕ0
29		8nn 11068	. . . . . . . . 9 ⊢ 8 ∈ ℕ
30	28, 29	decnncl 11394	. . . . . . . 8 ⊢ ;18 ∈ ℕ
31	27, 30	ndxid 15716	. . . . . . 7 ⊢ .ef = Slot (.ef‘ndx)
32	31	setsid 15742	. . . . . 6 ⊢ ((𝐺 ∈ 𝑈 ∧ 𝐸 ∈ 𝑊) → 𝐸 = (.ef‘(𝐺 sSet ⟨(.ef‘ndx), 𝐸⟩)))
33	10, 11, 32	syl2anc 691	. . . . 5 ⊢ (𝜑 → 𝐸 = (.ef‘(𝐺 sSet ⟨(.ef‘ndx), 𝐸⟩)))
34	24, 26, 33	3eqtr4a 2670	. . . 4 ⊢ (𝜑 → (iEdg‘(𝐺 sSet ⟨𝐼, 𝐸⟩)) = 𝐸)
35	34	dmeqd 5248	. . . 4 ⊢ (𝜑 → dom (iEdg‘(𝐺 sSet ⟨𝐼, 𝐸⟩)) = dom 𝐸)
36	34, 35	feq12d 5946	. . 3 ⊢ (𝜑 → ((iEdg‘(𝐺 sSet ⟨𝐼, 𝐸⟩)):dom (iEdg‘(𝐺 sSet ⟨𝐼, 𝐸⟩))⟶(𝒫 (Vtx‘(𝐺 sSet ⟨𝐼, 𝐸⟩)) ∖ {∅}) ↔ 𝐸:dom 𝐸⟶(𝒫 (Vtx‘(𝐺 sSet ⟨𝐼, 𝐸⟩)) ∖ {∅})))
37	21, 36	mpbird 246	. 2 ⊢ (𝜑 → (iEdg‘(𝐺 sSet ⟨𝐼, 𝐸⟩)):dom (iEdg‘(𝐺 sSet ⟨𝐼, 𝐸⟩))⟶(𝒫 (Vtx‘(𝐺 sSet ⟨𝐼, 𝐸⟩)) ∖ {∅}))
38		ovex 6577	. . 3 ⊢ (𝐺 sSet ⟨𝐼, 𝐸⟩) ∈ V
39		eqid 2610	. . . 4 ⊢ (Vtx‘(𝐺 sSet ⟨𝐼, 𝐸⟩)) = (Vtx‘(𝐺 sSet ⟨𝐼, 𝐸⟩))
40		eqid 2610	. . . 4 ⊢ (iEdg‘(𝐺 sSet ⟨𝐼, 𝐸⟩)) = (iEdg‘(𝐺 sSet ⟨𝐼, 𝐸⟩))
41	39, 40	isuhgr 25726	. . 3 ⊢ ((𝐺 sSet ⟨𝐼, 𝐸⟩) ∈ V → ((𝐺 sSet ⟨𝐼, 𝐸⟩) ∈ UHGraph ↔ (iEdg‘(𝐺 sSet ⟨𝐼, 𝐸⟩)):dom (iEdg‘(𝐺 sSet ⟨𝐼, 𝐸⟩))⟶(𝒫 (Vtx‘(𝐺 sSet ⟨𝐼, 𝐸⟩)) ∖ {∅})))
42	38, 41	mp1i 13	. 2 ⊢ (𝜑 → ((𝐺 sSet ⟨𝐼, 𝐸⟩) ∈ UHGraph ↔ (iEdg‘(𝐺 sSet ⟨𝐼, 𝐸⟩)):dom (iEdg‘(𝐺 sSet ⟨𝐼, 𝐸⟩))⟶(𝒫 (Vtx‘(𝐺 sSet ⟨𝐼, 𝐸⟩)) ∖ {∅})))
43	37, 42	mpbird 246	1 ⊢ (𝜑 → (𝐺 sSet ⟨𝐼, 𝐸⟩) ∈ UHGraph )

Syntax hints:   → wi 4   ↔ wb 195   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  {csn 4125  {cpr 4127  ⟨cop 4131   class class class wbr 4583  dom cdm 5038  Fun wfun 5798  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  1c1 9816  8c8 10953  ;cdc 11369   Struct cstr 15691  ndxcnx 15692   sSet csts 15693  Basecbs 15695  .efcedgf 25667  Vtxcvtx 25673  iEdgciedg 25674   UHGraph cuhgr 25722

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-xnn0 11241  df-z 11255  df-dec 11370  df-uz 11564  df-fz 12198  df-hash 12980  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-edgf 25668  df-vtx 25675  df-iedg 25676  df-uhgr 25724

This theorem is referenced by: (None)