Theorem edgstruct 15858

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 15853	. . 3 ⊢ Edg = (𝑔 ∈ V ↦ ran (iEdg‘𝑔))
2		fveq2 5626	. . . 4 ⊢ (𝑔 = 𝐺 → (iEdg‘𝑔) = (iEdg‘𝐺))
3	2	rneqd 4952	. . 3 ⊢ (𝑔 = 𝐺 → ran (iEdg‘𝑔) = ran (iEdg‘𝐺))
4		edgstruct.s	. . . 4 ⊢ 𝐺 = {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), 𝐸⟩}
5		basendxnn 13083	. . . . . 6 ⊢ (Base‘ndx) ∈ ℕ
6		simpl 109	. . . . . 6 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → 𝑉 ∈ 𝑊)
7		opexg 4313	. . . . . 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 15803	. . . . . 6 ⊢ (.ef‘ndx) ∈ ℕ
10		simpr 110	. . . . . 6 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → 𝐸 ∈ 𝑋)
11		opexg 4313	. . . . . 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 4294	. . . . 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 2316	. . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → 𝐺 ∈ V)
16	5	elexi 2812	. . . . . 6 ⊢ (Base‘ndx) ∈ V
17	9	elexi 2812	. . . . . 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 15806	. . . . . . . . . 10 ⊢ (Base‘ndx) ≠ (.ef‘ndx)
21	20	a1i 9	. . . . . . . . 9 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (Base‘ndx) ≠ (.ef‘ndx))
22		fnprg 5375	. . . . . . . . 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 1282	. . . . . . . 8 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), 𝐸⟩} Fn {(Base‘ndx), (.ef‘ndx)})
24	4	fneq1i 5414	. . . . . . . 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 5417	. . . . . . 7 ⊢ (𝐺 Fn {(Base‘ndx), (.ef‘ndx)} → Fun 𝐺)
27		fundif 5364	. . . . . . 7 ⊢ (Fun 𝐺 → Fun (𝐺 ∖ {∅}))
28	25, 26, 27	3syl 17	. . . . . 6 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → Fun (𝐺 ∖ {∅}))
29	25	fndmd 5421	. . . . . . 7 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → dom 𝐺 = {(Base‘ndx), (.ef‘ndx)})
30		eqimss2 3279	. . . . . . 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 15827	. . . . 5 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (iEdg‘𝐺) = (.ef‘𝐺))
33		edgfid 15801	. . . . . . . 8 ⊢ .ef = Slot (.ef‘ndx)
34	33, 9	ndxslid 13052	. . . . . . 7 ⊢ (.ef = Slot (.ef‘ndx) ∧ (.ef‘ndx) ∈ ℕ)
35	34	slotex 13054	. . . . . 6 ⊢ (𝐺 ∈ V → (.ef‘𝐺) ∈ V)
36	15, 35	syl 14	. . . . 5 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (.ef‘𝐺) ∈ V)
37	32, 36	eqeltrd 2306	. . . 4 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (iEdg‘𝐺) ∈ V)
38		rnexg 4988	. . . 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 5727	. 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (Edg‘𝐺) = ran (iEdg‘𝐺))
41	4	struct2griedg 15841	. . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (iEdg‘𝐺) = 𝐸)
42	41	rneqd 4952	. 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → ran (iEdg‘𝐺) = ran 𝐸)
43	40, 42	eqtrd 2262	1 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (Edg‘𝐺) = ran 𝐸)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1395   ∈ wcel 2200   ≠ wne 2400  Vcvv 2799   ∖ cdif 3194   ⊆ wss 3197  ∅c0 3491  {csn 3666  {cpr 3667  ⟨cop 3669  dom cdm 4718  ran crn 4719  Fun wfun 5311   Fn wfn 5312  ‘cfv 5317  ℕcn 9106  ndxcnx 13024  Basecbs 13027  .efcedgf 15799  iEdgciedg 15808  Edgcedg 15852

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-mulrcl 8094  ax-addcom 8095  ax-mulcom 8096  ax-addass 8097  ax-mulass 8098  ax-distr 8099  ax-i2m1 8100  ax-0lt1 8101  ax-1rid 8102  ax-0id 8103  ax-rnegex 8104  ax-precex 8105  ax-cnre 8106  ax-pre-ltirr 8107  ax-pre-ltwlin 8108  ax-pre-lttrn 8109  ax-pre-apti 8110  ax-pre-ltadd 8111  ax-pre-mulgt0 8112

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-iord 4456  df-on 4458  df-suc 4461  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-2nd 6285  df-1o 6560  df-2o 6561  df-en 6886  df-dom 6887  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182  df-le 8183  df-sub 8315  df-neg 8316  df-inn 9107  df-2 9165  df-3 9166  df-4 9167  df-5 9168  df-6 9169  df-7 9170  df-8 9171  df-9 9172  df-n0 9366  df-z 9443  df-dec 9575  df-uz 9719  df-fz 10201  df-struct 13029  df-ndx 13030  df-slot 13031  df-base 13033  df-edgf 15800  df-iedg 15810  df-edg 15853

This theorem is referenced by: (None)