Theorem edgstruct 16471

Description: The edges of a graph represented as an extensible structure with vertices as base set and indexed edges. (Contributed by AV, 13-Oct-2020.)

Hypothesis

Ref	Expression
edgstruct.s	⊢ 𝐺 = {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), 𝐸⟩}

Assertion

Ref	Expression
edgstruct	⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (Edg‘𝐺) = ran 𝐸)

Proof of Theorem edgstruct

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-edg 16465	. . 3 ⊢ Edg = (𝑔 ∈ V ↦ ran (iEdg‘𝑔))
2		fveq2 5695	. . . 4 ⊢ (𝑔 = 𝐺 → (iEdg‘𝑔) = (iEdg‘𝐺))
3	2	rneqd 5011	. . 3 ⊢ (𝑔 = 𝐺 → ran (iEdg‘𝑔) = ran (iEdg‘𝐺))
4		edgstruct.s	. . . 4 ⊢ 𝐺 = {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), 𝐸⟩}
5		basendxnn 13460	. . . . . 6 ⊢ (Base‘ndx) ∈ ℕ
6		simpl 109	. . . . . 6 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → 𝑉 ∈ 𝑊)
7		opexg 4368	. . . . . 6 ⊢ (((Base‘ndx) ∈ ℕ ∧ 𝑉 ∈ 𝑊) → ⟨(Base‘ndx), 𝑉⟩ ∈ V)
8	5, 6, 7	sylancr 418	. . . . 5 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → ⟨(Base‘ndx), 𝑉⟩ ∈ V)
9		edgfndxnn 16415	. . . . . 6 ⊢ (.ef‘ndx) ∈ ℕ
10		simpr 110	. . . . . 6 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → 𝐸 ∈ 𝑋)
11		opexg 4368	. . . . . 6 ⊢ (((.ef‘ndx) ∈ ℕ ∧ 𝐸 ∈ 𝑋) → ⟨(.ef‘ndx), 𝐸⟩ ∈ V)
12	9, 10, 11	sylancr 418	. . . . 5 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → ⟨(.ef‘ndx), 𝐸⟩ ∈ V)
13		prexg 4349	. . . . 5 ⊢ ((⟨(Base‘ndx), 𝑉⟩ ∈ V ∧ ⟨(.ef‘ndx), 𝐸⟩ ∈ V) → {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), 𝐸⟩} ∈ V)
14	8, 12, 13	syl2anc 415	. . . 4 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), 𝐸⟩} ∈ V)
15	4, 14	eqeltrid 2325	. . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → 𝐺 ∈ V)
16	5	elexi 2834	. . . . . 6 ⊢ (Base‘ndx) ∈ V
17	9	elexi 2834	. . . . . 6 ⊢ (.ef‘ndx) ∈ V
18	5	a1i 9	. . . . . . . . 9 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (Base‘ndx) ∈ ℕ)
19	9	a1i 9	. . . . . . . . 9 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (.ef‘ndx) ∈ ℕ)
20		basendxnedgfndx 16418	. . . . . . . . . 10 ⊢ (Base‘ndx) ≠ (.ef‘ndx)
21	20	a1i 9	. . . . . . . . 9 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (Base‘ndx) ≠ (.ef‘ndx))
22		fnprg 5436	. . . . . . . . 9 ⊢ ((((Base‘ndx) ∈ ℕ ∧ (.ef‘ndx) ∈ ℕ) ∧ (𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) ∧ (Base‘ndx) ≠ (.ef‘ndx)) → {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), 𝐸⟩} Fn {(Base‘ndx), (.ef‘ndx)})
23	18, 19, 6, 10, 21, 22	syl221anc 1289	. . . . . . . 8 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), 𝐸⟩} Fn {(Base‘ndx), (.ef‘ndx)})
24	4	fneq1i 5475	. . . . . . . 8 ⊢ (𝐺 Fn {(Base‘ndx), (.ef‘ndx)} ↔ {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), 𝐸⟩} Fn {(Base‘ndx), (.ef‘ndx)})
25	23, 24	sylibr 134	. . . . . . 7 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → 𝐺 Fn {(Base‘ndx), (.ef‘ndx)})
26		fnfun 5478	. . . . . . 7 ⊢ (𝐺 Fn {(Base‘ndx), (.ef‘ndx)} → Fun 𝐺)
27		fundif 5425	. . . . . . 7 ⊢ (Fun 𝐺 → Fun (𝐺 ∖ {∅}))
28	25, 26, 27	3syl 17	. . . . . 6 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → Fun (𝐺 ∖ {∅}))
29	25	fndmd 5482	. . . . . . 7 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → dom 𝐺 = {(Base‘ndx), (.ef‘ndx)})
30		eqimss2 3303	. . . . . . 7 ⊢ (dom 𝐺 = {(Base‘ndx), (.ef‘ndx)} → {(Base‘ndx), (.ef‘ndx)} ⊆ dom 𝐺)
31	29, 30	syl 14	. . . . . 6 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → {(Base‘ndx), (.ef‘ndx)} ⊆ dom 𝐺)
32	16, 17, 15, 28, 21, 31	funiedgdm2vald 16439	. . . . 5 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (iEdg‘𝐺) = (.ef‘𝐺))
33		edgfid 16413	. . . . . . . 8 ⊢ .ef = Slot (.ef‘ndx)
34	33, 9	ndxslid 13429	. . . . . . 7 ⊢ (.ef = Slot (.ef‘ndx) ∧ (.ef‘ndx) ∈ ℕ)
35	34	slotex 13431	. . . . . 6 ⊢ (𝐺 ∈ V → (.ef‘𝐺) ∈ V)
36	15, 35	syl 14	. . . . 5 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (.ef‘𝐺) ∈ V)
37	32, 36	eqeltrd 2315	. . . 4 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (iEdg‘𝐺) ∈ V)
38		rnexg 5047	. . . 4 ⊢ ((iEdg‘𝐺) ∈ V → ran (iEdg‘𝐺) ∈ V)
39	37, 38	syl 14	. . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → ran (iEdg‘𝐺) ∈ V)
40	1, 3, 15, 39	fvmptd3 5799	. 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (Edg‘𝐺) = ran (iEdg‘𝐺))
41	4	struct2griedg 16453	. . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (iEdg‘𝐺) = 𝐸)
42	41	rneqd 5011	. 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → ran (iEdg‘𝐺) = ran 𝐸)
43	40, 42	eqtrd 2271	1 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (Edg‘𝐺) = ran 𝐸)

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209   ≠ wne 2420  Vcvv 2821   ∖ cdif 3217   ⊆ wss 3220  ∅c0 3520  {csn 3709  {cpr 3710  ⟨cop 3712  dom cdm 4774  ran crn 4775  Fun wfun 5371   Fn wfn 5372  ‘cfv 5377  ℕcn 9307  ndxcnx 13401  Basecbs 13404  .efcedgf 16411  iEdgciedg 16420  Edgcedg 16464

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8271  ax-resscn 8272  ax-1cn 8273  ax-1re 8274  ax-icn 8275  ax-addcl 8276  ax-addrcl 8277  ax-mulcl 8278  ax-mulrcl 8279  ax-addcom 8280  ax-mulcom 8281  ax-addass 8282  ax-mulass 8283  ax-distr 8284  ax-i2m1 8285  ax-0lt1 8286  ax-1rid 8287  ax-0id 8288  ax-rnegex 8289  ax-precex 8290  ax-cnre 8291  ax-pre-ltirr 8292  ax-pre-ltwlin 8293  ax-pre-lttrn 8294  ax-pre-apti 8295  ax-pre-ltadd 8296  ax-pre-mulgt0 8297

This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-suc 4516  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-2nd 6375  df-1o 6687  df-2o 6688  df-en 7023  df-dom 7024  df-pnf 8363  df-mnf 8364  df-xr 8365  df-ltxr 8366  df-le 8367  df-sub 8501  df-neg 8502  df-inn 9308  df-2 9366  df-3 9367  df-4 9368  df-5 9369  df-6 9370  df-7 9371  df-8 9372  df-9 9373  df-n0 9569  df-z 9650  df-dec 9783  df-uz 9932  df-fz 10423  df-struct 13406  df-ndx 13407  df-slot 13408  df-base 13410  df-edgf 16412  df-iedg 16422  df-edg 16465

This theorem is used by: (None)