Theorem edgstruct 16185

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 16179	. . 3 ⊢ Edg = (𝑔 ∈ V ↦ ran (iEdg‘𝑔))
2		fveq2 5675	. . . 4 ⊢ (𝑔 = 𝐺 → (iEdg‘𝑔) = (iEdg‘𝐺))
3	2	rneqd 4991	. . 3 ⊢ (𝑔 = 𝐺 → ran (iEdg‘𝑔) = ran (iEdg‘𝐺))
4		edgstruct.s	. . . 4 ⊢ 𝐺 = {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), 𝐸⟩}
5		basendxnn 13352	. . . . . 6 ⊢ (Base‘ndx) ∈ ℕ
6		simpl 109	. . . . . 6 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → 𝑉 ∈ 𝑊)
7		opexg 4349	. . . . . 6 ⊢ (((Base‘ndx) ∈ ℕ ∧ 𝑉 ∈ 𝑊) → ⟨(Base‘ndx), 𝑉⟩ ∈ V)
8	5, 6, 7	sylancr 414	. . . . 5 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → ⟨(Base‘ndx), 𝑉⟩ ∈ V)
9		edgfndxnn 16129	. . . . . 6 ⊢ (.ef‘ndx) ∈ ℕ
10		simpr 110	. . . . . 6 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → 𝐸 ∈ 𝑋)
11		opexg 4349	. . . . . 6 ⊢ (((.ef‘ndx) ∈ ℕ ∧ 𝐸 ∈ 𝑋) → ⟨(.ef‘ndx), 𝐸⟩ ∈ V)
12	9, 10, 11	sylancr 414	. . . . 5 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → ⟨(.ef‘ndx), 𝐸⟩ ∈ V)
13		prexg 4330	. . . . 5 ⊢ ((⟨(Base‘ndx), 𝑉⟩ ∈ V ∧ ⟨(.ef‘ndx), 𝐸⟩ ∈ V) → {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), 𝐸⟩} ∈ V)
14	8, 12, 13	syl2anc 411	. . . 4 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), 𝐸⟩} ∈ V)
15	4, 14	eqeltrid 2321	. . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → 𝐺 ∈ V)
16	5	elexi 2828	. . . . . 6 ⊢ (Base‘ndx) ∈ V
17	9	elexi 2828	. . . . . 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 16132	. . . . . . . . . 10 ⊢ (Base‘ndx) ≠ (.ef‘ndx)
21	20	a1i 9	. . . . . . . . 9 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (Base‘ndx) ≠ (.ef‘ndx))
22		fnprg 5416	. . . . . . . . 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 1285	. . . . . . . 8 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), 𝐸⟩} Fn {(Base‘ndx), (.ef‘ndx)})
24	4	fneq1i 5455	. . . . . . . 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 5458	. . . . . . 7 ⊢ (𝐺 Fn {(Base‘ndx), (.ef‘ndx)} → Fun 𝐺)
27		fundif 5405	. . . . . . 7 ⊢ (Fun 𝐺 → Fun (𝐺 ∖ {∅}))
28	25, 26, 27	3syl 17	. . . . . 6 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → Fun (𝐺 ∖ {∅}))
29	25	fndmd 5462	. . . . . . 7 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → dom 𝐺 = {(Base‘ndx), (.ef‘ndx)})
30		eqimss2 3297	. . . . . . 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 16153	. . . . 5 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (iEdg‘𝐺) = (.ef‘𝐺))
33		edgfid 16127	. . . . . . . 8 ⊢ .ef = Slot (.ef‘ndx)
34	33, 9	ndxslid 13321	. . . . . . 7 ⊢ (.ef = Slot (.ef‘ndx) ∧ (.ef‘ndx) ∈ ℕ)
35	34	slotex 13323	. . . . . 6 ⊢ (𝐺 ∈ V → (.ef‘𝐺) ∈ V)
36	15, 35	syl 14	. . . . 5 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (.ef‘𝐺) ∈ V)
37	32, 36	eqeltrd 2311	. . . 4 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (iEdg‘𝐺) ∈ V)
38		rnexg 5027	. . . 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 5776	. 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (Edg‘𝐺) = ran (iEdg‘𝐺))
41	4	struct2griedg 16167	. . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (iEdg‘𝐺) = 𝐸)
42	41	rneqd 4991	. 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → ran (iEdg‘𝐺) = ran 𝐸)
43	40, 42	eqtrd 2267	1 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (Edg‘𝐺) = ran 𝐸)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2205   ≠ wne 2414  Vcvv 2815   ∖ cdif 3211   ⊆ wss 3214  ∅c0 3512  {csn 3694  {cpr 3695  ⟨cop 3697  dom cdm 4754  ran crn 4755  Fun wfun 5351   Fn wfn 5352  ‘cfv 5357  ℕcn 9254  ndxcnx 13293  Basecbs 13296  .efcedgf 16125  iEdgciedg 16134  Edgcedg 16178

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-nul 4241  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-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260

This theorem depends on definitions:  df-bi 117  df-3or 1006  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-nel 2510  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-nul 3513  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-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-suc 4497  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-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-2nd 6348  df-1o 6660  df-2o 6661  df-en 6989  df-dom 6990  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  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-z 9595  df-dec 9728  df-uz 9872  df-fz 10362  df-struct 13298  df-ndx 13299  df-slot 13300  df-base 13302  df-edgf 16126  df-iedg 16136  df-edg 16179

This theorem is referenced by: (None)