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| Mirrors > Home > MPE Home > Th. List > usgrres1 | Structured version Visualization version GIF version | ||
| 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 29285 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.) |
| 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)〉 |
| Ref | Expression |
|---|---|
| usgrres1 | ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → 𝑆 ∈ USGraph) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | f1oi 6886 | . . . . 5 ⊢ ( I ↾ 𝐹):𝐹–1-1-onto→𝐹 | |
| 2 | f1of1 6847 | . . . . 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 2738 | . . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → ( I ↾ 𝐹) = ( I ↾ 𝐹)) | |
| 5 | dmresi 6070 | . . . . . 6 ⊢ dom ( I ↾ 𝐹) = 𝐹 | |
| 6 | 5 | a1i 11 | . . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → dom ( I ↾ 𝐹) = 𝐹) |
| 7 | eqidd 2738 | . . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → 𝐹 = 𝐹) | |
| 8 | 4, 6, 7 | f1eq123d 6840 | . . . 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 29198 | . . . 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 29327 | . . . 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 6810 | . . 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 5469 | . . . 4 ⊢ 〈(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)〉 ∈ V | |
| 20 | 18, 19 | eqeltri 2837 | . . 3 ⊢ 𝑆 ∈ V |
| 21 | 11, 12, 13, 18 | upgrres1lem2 29328 | . . . . 5 ⊢ (Vtx‘𝑆) = (𝑉 ∖ {𝑁}) |
| 22 | 21 | eqcomi 2746 | . . . 4 ⊢ (𝑉 ∖ {𝑁}) = (Vtx‘𝑆) |
| 23 | 11, 12, 13, 18 | upgrres1lem3 29329 | . . . . 5 ⊢ (iEdg‘𝑆) = ( I ↾ 𝐹) |
| 24 | 23 | eqcomi 2746 | . . . 4 ⊢ ( I ↾ 𝐹) = (iEdg‘𝑆) |
| 25 | 22, 24 | isusgrs 29173 | . . 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) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∉ wnel 3046 {crab 3436 Vcvv 3480 ∖ cdif 3948 ⊆ wss 3951 𝒫 cpw 4600 {csn 4626 〈cop 4632 I cid 5577 dom cdm 5685 ran crn 5686 ↾ cres 5687 –1-1→wf1 6558 –1-1-onto→wf1o 6560 ‘cfv 6561 2c2 12321 ♯chash 14369 Vtxcvtx 29013 iEdgciedg 29014 Edgcedg 29064 UMGraphcumgr 29098 USGraphcusgr 29166 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-hash 14370 df-vtx 29015 df-iedg 29016 df-edg 29065 df-uhgr 29075 df-upgr 29099 df-umgr 29100 df-usgr 29168 |
| This theorem is referenced by: fusgrfis 29347 cusgrres 29466 |
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