Theorem edgstruct 15914

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 15908	. . 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 13137	. . . . . 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 15858	. . . . . 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 15861	. . . . . . . . . 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 15882	. . . . 5 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (iEdg‘𝐺) = (.ef‘𝐺))
33		edgfid 15856	. . . . . . . 8 ⊢ .ef = Slot (.ef‘ndx)
34	33, 9	ndxslid 13106	. . . . . . 7 ⊢ (.ef = Slot (.ef‘ndx) ∧ (.ef‘ndx) ∈ ℕ)
35	34	slotex 13108	. . . . . 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 15896	. . 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 9142  ndxcnx 13078  Basecbs 13081  .efcedgf 15854  iEdgciedg 15863  Edgcedg 15907

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 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148

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 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-2nd 6303  df-1o 6581  df-2o 6582  df-en 6909  df-dom 6910  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-inn 9143  df-2 9201  df-3 9202  df-4 9203  df-5 9204  df-6 9205  df-7 9206  df-8 9207  df-9 9208  df-n0 9402  df-z 9479  df-dec 9611  df-uz 9755  df-fz 10243  df-struct 13083  df-ndx 13084  df-slot 13085  df-base 13087  df-edgf 15855  df-iedg 15865  df-edg 15908

This theorem is referenced by: (None)