Theorem usgrres1 28569

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 28522 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 6871	. . . . 5 ⊢ ( I ↾ 𝐹):𝐹–1-1-onto→𝐹
2		f1of1 6832	. . . . 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 2733	. . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → ( I ↾ 𝐹) = ( I ↾ 𝐹))
5		dmresi 6051	. . . . . 6 ⊢ dom ( I ↾ 𝐹) = 𝐹
6	5	a1i 11	. . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → dom ( I ↾ 𝐹) = 𝐹)
7		eqidd 2733	. . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → 𝐹 = 𝐹)
8	4, 6, 7	f1eq123d 6825	. . . 4 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → (( I ↾ 𝐹):dom ( I ↾ 𝐹)–1-1→𝐹 ↔ ( I ↾ 𝐹):𝐹–1-1→𝐹))
9	3, 8	mpbird 256	. . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → ( I ↾ 𝐹):dom ( I ↾ 𝐹)–1-1→𝐹)
10		usgrumgr 28436	. . . 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 28564	. . . 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 6794	. . 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 5464	. . . 4 ⊢ ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩ ∈ V
20	18, 19	eqeltri 2829	. . 3 ⊢ 𝑆 ∈ V
21	11, 12, 13, 18	upgrres1lem2 28565	. . . . 5 ⊢ (Vtx‘𝑆) = (𝑉 ∖ {𝑁})
22	21	eqcomi 2741	. . . 4 ⊢ (𝑉 ∖ {𝑁}) = (Vtx‘𝑆)
23	11, 12, 13, 18	upgrres1lem3 28566	. . . . 5 ⊢ (iEdg‘𝑆) = ( I ↾ 𝐹)
24	23	eqcomi 2741	. . . 4 ⊢ ( I ↾ 𝐹) = (iEdg‘𝑆)
25	22, 24	isusgrs 28413	. . 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 256	1 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → 𝑆 ∈ USGraph)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106   ∉ wnel 3046  {crab 3432  Vcvv 3474   ∖ cdif 3945   ⊆ wss 3948  𝒫 cpw 4602  {csn 4628  ⟨cop 4634   I cid 5573  dom cdm 5676  ran crn 5677   ↾ cres 5678  –1-1→wf1 6540  –1-1-onto→wf1o 6542  ‘cfv 6543  2c2 12266  ♯chash 14289  Vtxcvtx 28253  iEdgciedg 28254  Edgcedg 28304  UMGraphcumgr 28338  USGraphcusgr 28406

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-1st 7974  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-1o 8465  df-er 8702  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942  df-card 9933  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11445  df-neg 11446  df-nn 12212  df-2 12274  df-n0 12472  df-z 12558  df-uz 12822  df-fz 13484  df-hash 14290  df-vtx 28255  df-iedg 28256  df-edg 28305  df-uhgr 28315  df-upgr 28339  df-umgr 28340  df-usgr 28408

This theorem is referenced by:  fusgrfis  28584  cusgrres  28702