Theorem usgrres1 29300

Description: Restricting a simple graph by removing one vertex results in a simple graph. Remark: This restricted graph is not a subgraph of the original graph in the sense of df-subgr 29253 since the domains of the edge functions may not be compatible. (Contributed by Alexander van der Vekens, 2-Jan-2018.) (Revised by AV, 10-Jan-2020.) (Revised by AV, 23-Oct-2020.) (Proof shortened by AV, 27-Nov-2020.)

Hypotheses

Ref	Expression
upgrres1.v	⊢ 𝑉 = (Vtx‘𝐺)
upgrres1.e	⊢ 𝐸 = (Edg‘𝐺)
upgrres1.f	⊢ 𝐹 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒}
upgrres1.s	⊢ 𝑆 = ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩

Assertion

Ref	Expression
usgrres1	⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → 𝑆 ∈ USGraph)

Distinct variable groups:   𝑒,𝐸   𝑒,𝐺   𝑒,𝑁   𝑒,𝑉

Allowed substitution hints:   𝑆(𝑒)   𝐹(𝑒)

Proof of Theorem usgrres1

Dummy variable 𝑝 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		f1oi 6807	. . . . 5 ⊢ ( I ↾ 𝐹):𝐹–1-1-onto→𝐹
2		f1of1 6768	. . . . 5 ⊢ (( I ↾ 𝐹):𝐹–1-1-onto→𝐹 → ( I ↾ 𝐹):𝐹–1-1→𝐹)
3	1, 2	mp1i 13	. . . 4 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → ( I ↾ 𝐹):𝐹–1-1→𝐹)
4		eqidd 2732	. . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → ( I ↾ 𝐹) = ( I ↾ 𝐹))
5		dmresi 6006	. . . . . 6 ⊢ dom ( I ↾ 𝐹) = 𝐹
6	5	a1i 11	. . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → dom ( I ↾ 𝐹) = 𝐹)
7		eqidd 2732	. . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → 𝐹 = 𝐹)
8	4, 6, 7	f1eq123d 6761	. . . 4 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → (( I ↾ 𝐹):dom ( I ↾ 𝐹)–1-1→𝐹 ↔ ( I ↾ 𝐹):𝐹–1-1→𝐹))
9	3, 8	mpbird 257	. . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → ( I ↾ 𝐹):dom ( I ↾ 𝐹)–1-1→𝐹)
10		usgrumgr 29166	. . . 4 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ UMGraph)
11		upgrres1.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
12		upgrres1.e	. . . . 5 ⊢ 𝐸 = (Edg‘𝐺)
13		upgrres1.f	. . . . 5 ⊢ 𝐹 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒}
14	11, 12, 13	umgrres1lem 29295	. . . 4 ⊢ ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ 𝑉) → ran ( I ↾ 𝐹) ⊆ {𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (♯‘𝑝) = 2})
15	10, 14	sylan 580	. . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → ran ( I ↾ 𝐹) ⊆ {𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (♯‘𝑝) = 2})
16		f1ssr 6731	. . 3 ⊢ ((( I ↾ 𝐹):dom ( I ↾ 𝐹)–1-1→𝐹 ∧ ran ( I ↾ 𝐹) ⊆ {𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (♯‘𝑝) = 2}) → ( I ↾ 𝐹):dom ( I ↾ 𝐹)–1-1→{𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (♯‘𝑝) = 2})
17	9, 15, 16	syl2anc 584	. 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → ( I ↾ 𝐹):dom ( I ↾ 𝐹)–1-1→{𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (♯‘𝑝) = 2})
18		upgrres1.s	. . . 4 ⊢ 𝑆 = ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩
19		opex 5407	. . . 4 ⊢ ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩ ∈ V
20	18, 19	eqeltri 2827	. . 3 ⊢ 𝑆 ∈ V
21	11, 12, 13, 18	upgrres1lem2 29296	. . . . 5 ⊢ (Vtx‘𝑆) = (𝑉 ∖ {𝑁})
22	21	eqcomi 2740	. . . 4 ⊢ (𝑉 ∖ {𝑁}) = (Vtx‘𝑆)
23	11, 12, 13, 18	upgrres1lem3 29297	. . . . 5 ⊢ (iEdg‘𝑆) = ( I ↾ 𝐹)
24	23	eqcomi 2740	. . . 4 ⊢ ( I ↾ 𝐹) = (iEdg‘𝑆)
25	22, 24	isusgrs 29141	. . 3 ⊢ (𝑆 ∈ V → (𝑆 ∈ USGraph ↔ ( I ↾ 𝐹):dom ( I ↾ 𝐹)–1-1→{𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (♯‘𝑝) = 2}))
26	20, 25	mp1i 13	. 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → (𝑆 ∈ USGraph ↔ ( I ↾ 𝐹):dom ( I ↾ 𝐹)–1-1→{𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (♯‘𝑝) = 2}))
27	17, 26	mpbird 257	1 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → 𝑆 ∈ USGraph)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111   ∉ wnel 3032  {crab 3395  Vcvv 3436   ∖ cdif 3894   ⊆ wss 3897  𝒫 cpw 4549  {csn 4575  ⟨cop 4581   I cid 5513  dom cdm 5619  ran crn 5620   ↾ cres 5621  –1-1→wf1 6484  –1-1-onto→wf1o 6486  ‘cfv 6487  2c2 12186  ♯chash 14243  Vtxcvtx 28981  iEdgciedg 28982  Edgcedg 29032  UMGraphcumgr 29066  USGraphcusgr 29134

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-cnex 11068  ax-resscn 11069  ax-1cn 11070  ax-icn 11071  ax-addcl 11072  ax-addrcl 11073  ax-mulcl 11074  ax-mulrcl 11075  ax-mulcom 11076  ax-addass 11077  ax-mulass 11078  ax-distr 11079  ax-i2m1 11080  ax-1ne0 11081  ax-1rid 11082  ax-rnegex 11083  ax-rrecex 11084  ax-cnre 11085  ax-pre-lttri 11086  ax-pre-lttrn 11087  ax-pre-ltadd 11088  ax-pre-mulgt0 11089

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-int 4898  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6254  df-ord 6315  df-on 6316  df-lim 6317  df-suc 6318  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-riota 7309  df-ov 7355  df-oprab 7356  df-mpo 7357  df-om 7803  df-1st 7927  df-2nd 7928  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-1o 8391  df-er 8628  df-en 8876  df-dom 8877  df-sdom 8878  df-fin 8879  df-card 9838  df-pnf 11154  df-mnf 11155  df-xr 11156  df-ltxr 11157  df-le 11158  df-sub 11352  df-neg 11353  df-nn 12132  df-2 12194  df-n0 12388  df-z 12475  df-uz 12739  df-fz 13414  df-hash 14244  df-vtx 28983  df-iedg 28984  df-edg 29033  df-uhgr 29043  df-upgr 29067  df-umgr 29068  df-usgr 29136

This theorem is referenced by:  fusgrfis  29315  cusgrres  29434