Theorem edgstruct 15921

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 15915	. . 3 ⊢ Edg = (𝑔 ∈ V ↦ ran (iEdg‘𝑔))
2		fveq2 5639	. . . 4 ⊢ (𝑔 = 𝐺 → (iEdg‘𝑔) = (iEdg‘𝐺))
3	2	rneqd 4961	. . 3 ⊢ (𝑔 = 𝐺 → ran (iEdg‘𝑔) = ran (iEdg‘𝐺))
4		edgstruct.s	. . . 4 ⊢ 𝐺 = {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), 𝐸⟩}
5		basendxnn 13143	. . . . . 6 ⊢ (Base‘ndx) ∈ ℕ
6		simpl 109	. . . . . 6 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → 𝑉 ∈ 𝑊)
7		opexg 4320	. . . . . 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 15865	. . . . . 6 ⊢ (.ef‘ndx) ∈ ℕ
10		simpr 110	. . . . . 6 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → 𝐸 ∈ 𝑋)
11		opexg 4320	. . . . . 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 4301	. . . . 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 2318	. . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → 𝐺 ∈ V)
16	5	elexi 2815	. . . . . 6 ⊢ (Base‘ndx) ∈ V
17	9	elexi 2815	. . . . . 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 15868	. . . . . . . . . 10 ⊢ (Base‘ndx) ≠ (.ef‘ndx)
21	20	a1i 9	. . . . . . . . 9 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (Base‘ndx) ≠ (.ef‘ndx))
22		fnprg 5385	. . . . . . . . 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 1284	. . . . . . . 8 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), 𝐸⟩} Fn {(Base‘ndx), (.ef‘ndx)})
24	4	fneq1i 5424	. . . . . . . 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 5427	. . . . . . 7 ⊢ (𝐺 Fn {(Base‘ndx), (.ef‘ndx)} → Fun 𝐺)
27		fundif 5374	. . . . . . 7 ⊢ (Fun 𝐺 → Fun (𝐺 ∖ {∅}))
28	25, 26, 27	3syl 17	. . . . . 6 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → Fun (𝐺 ∖ {∅}))
29	25	fndmd 5431	. . . . . . 7 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → dom 𝐺 = {(Base‘ndx), (.ef‘ndx)})
30		eqimss2 3282	. . . . . . 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 15889	. . . . 5 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (iEdg‘𝐺) = (.ef‘𝐺))
33		edgfid 15863	. . . . . . . 8 ⊢ .ef = Slot (.ef‘ndx)
34	33, 9	ndxslid 13112	. . . . . . 7 ⊢ (.ef = Slot (.ef‘ndx) ∧ (.ef‘ndx) ∈ ℕ)
35	34	slotex 13114	. . . . . 6 ⊢ (𝐺 ∈ V → (.ef‘𝐺) ∈ V)
36	15, 35	syl 14	. . . . 5 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (.ef‘𝐺) ∈ V)
37	32, 36	eqeltrd 2308	. . . 4 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (iEdg‘𝐺) ∈ V)
38		rnexg 4997	. . . 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 5740	. 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (Edg‘𝐺) = ran (iEdg‘𝐺))
41	4	struct2griedg 15903	. . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (iEdg‘𝐺) = 𝐸)
42	41	rneqd 4961	. 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → ran (iEdg‘𝐺) = ran 𝐸)
43	40, 42	eqtrd 2264	1 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (Edg‘𝐺) = ran 𝐸)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1397   ∈ wcel 2202   ≠ wne 2402  Vcvv 2802   ∖ cdif 3197   ⊆ wss 3200  ∅c0 3494  {csn 3669  {cpr 3670  ⟨cop 3672  dom cdm 4725  ran crn 4726  Fun wfun 5320   Fn wfn 5321  ‘cfv 5326  ℕcn 9143  ndxcnx 13084  Basecbs 13087  .efcedgf 15861  iEdgciedg 15870  Edgcedg 15914

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-mulrcl 8131  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-precex 8142  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148  ax-pre-mulgt0 8149

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-2nd 6304  df-1o 6582  df-2o 6583  df-en 6910  df-dom 6911  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-5 9205  df-6 9206  df-7 9207  df-8 9208  df-9 9209  df-n0 9403  df-z 9480  df-dec 9612  df-uz 9756  df-fz 10244  df-struct 13089  df-ndx 13090  df-slot 13091  df-base 13093  df-edgf 15862  df-iedg 15872  df-edg 15915

This theorem is referenced by: (None)