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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 (𝜑𝐺 Struct ⟨(Base‘ndx), 𝐼⟩)
uhgrstrrepe.b (𝜑 → (Base‘ndx) ∈ dom 𝐺)
uhgrstrrepe.g (𝜑𝐺𝑈)
uhgrstrrepe.e (𝜑𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}))
uhgrstrrepe.w (𝜑𝐸𝑊)
Assertion
Ref Expression
uhgrstrrepe (𝜑 → (𝐺 sSet ⟨𝐼, 𝐸⟩) ∈ UHGraph )

Proof of Theorem uhgrstrrepe
StepHypRef Expression
1 uhgrstrrepe.e . . . 4 (𝜑𝐸: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)
64, 5neeqtrri 2855 . . . . . . . . . 10 (Base‘ndx) ≠ 𝐼
73, 6setsnid 15743 . . . . . . . . 9 (Base‘𝐺) = (Base‘(𝐺 sSet ⟨𝐼, 𝐸⟩))
8 uhgrstrrepe.s . . . . . . . . . . 11 (𝜑𝐺 Struct ⟨(Base‘ndx), 𝐼⟩)
9 uhgrstrrepe.b . . . . . . . . . . 11 (𝜑 → (Base‘ndx) ∈ dom 𝐺)
10 uhgrstrrepe.g . . . . . . . . . . 11 (𝜑𝐺𝑈)
11 uhgrstrrepe.w . . . . . . . . . . 11 (𝜑𝐸𝑊)
122, 5, 8, 9, 10, 1, 11uhgrstrrepelem 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⟩)))
1413eqcomd 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⟩)))
1512, 14syl 17 . . . . . . . . 9 (𝜑 → (Base‘(𝐺 sSet ⟨𝐼, 𝐸⟩)) = (Vtx‘(𝐺 sSet ⟨𝐼, 𝐸⟩)))
167, 15syl5eq 2656 . . . . . . . 8 (𝜑 → (Base‘𝐺) = (Vtx‘(𝐺 sSet ⟨𝐼, 𝐸⟩)))
172, 16syl5req 2657 . . . . . . 7 (𝜑 → (Vtx‘(𝐺 sSet ⟨𝐼, 𝐸⟩)) = 𝑉)
1817pweqd 4113 . . . . . 6 (𝜑 → 𝒫 (Vtx‘(𝐺 sSet ⟨𝐼, 𝐸⟩)) = 𝒫 𝑉)
1918difeq1d 3689 . . . . 5 (𝜑 → (𝒫 (Vtx‘(𝐺 sSet ⟨𝐼, 𝐸⟩)) ∖ {∅}) = (𝒫 𝑉 ∖ {∅}))
2019feq3d 5945 . . . 4 (𝜑 → (𝐸:dom 𝐸⟶(𝒫 (Vtx‘(𝐺 sSet ⟨𝐼, 𝐸⟩)) ∖ {∅}) ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
211, 20mpbird 246 . . 3 (𝜑𝐸:dom 𝐸⟶(𝒫 (Vtx‘(𝐺 sSet ⟨𝐼, 𝐸⟩)) ∖ {∅}))
225opeq1i 4343 . . . . . . 7 𝐼, 𝐸⟩ = ⟨(.ef‘ndx), 𝐸
2322oveq2i 6560 . . . . . 6 (𝐺 sSet ⟨𝐼, 𝐸⟩) = (𝐺 sSet ⟨(.ef‘ndx), 𝐸⟩)
2423fveq2i 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⟩)))
2612, 25syl 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
3028, 29decnncl 11394 . . . . . . . 8 18 ∈ ℕ
3127, 30ndxid 15716 . . . . . . 7 .ef = Slot (.ef‘ndx)
3231setsid 15742 . . . . . 6 ((𝐺𝑈𝐸𝑊) → 𝐸 = (.ef‘(𝐺 sSet ⟨(.ef‘ndx), 𝐸⟩)))
3310, 11, 32syl2anc 691 . . . . 5 (𝜑𝐸 = (.ef‘(𝐺 sSet ⟨(.ef‘ndx), 𝐸⟩)))
3424, 26, 333eqtr4a 2670 . . . 4 (𝜑 → (iEdg‘(𝐺 sSet ⟨𝐼, 𝐸⟩)) = 𝐸)
3534dmeqd 5248 . . . 4 (𝜑 → dom (iEdg‘(𝐺 sSet ⟨𝐼, 𝐸⟩)) = dom 𝐸)
3634, 35feq12d 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⟩)) ∖ {∅})))
3721, 36mpbird 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⟩))
4139, 40isuhgr 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⟩)) ∖ {∅})))
4238, 41mp1i 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⟩)) ∖ {∅})))
4337, 42mpbird 246 1 (𝜑 → (𝐺 sSet ⟨𝐼, 𝐸⟩) ∈ UHGraph )
 Colors of variables: wff setvar class 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)
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