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Mirrors > Home > MPE Home > Th. List > uspgredg2vlem | Structured version Visualization version GIF version |
Description: Lemma for uspgredg2v 26933. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 6-Dec-2020.) |
Ref | Expression |
---|---|
uspgredg2v.v | ⊢ 𝑉 = (Vtx‘𝐺) |
uspgredg2v.e | ⊢ 𝐸 = (Edg‘𝐺) |
uspgredg2v.a | ⊢ 𝐴 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒} |
Ref | Expression |
---|---|
uspgredg2vlem | ⊢ ((𝐺 ∈ USPGraph ∧ 𝑌 ∈ 𝐴) → (℩𝑧 ∈ 𝑉 𝑌 = {𝑁, 𝑧}) ∈ 𝑉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq2 2898 | . . 3 ⊢ (𝑒 = 𝑌 → (𝑁 ∈ 𝑒 ↔ 𝑁 ∈ 𝑌)) | |
2 | uspgredg2v.a | . . 3 ⊢ 𝐴 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒} | |
3 | 1, 2 | elrab2 3680 | . 2 ⊢ (𝑌 ∈ 𝐴 ↔ (𝑌 ∈ 𝐸 ∧ 𝑁 ∈ 𝑌)) |
4 | simpl 483 | . . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑌 ∈ 𝐸 ∧ 𝑁 ∈ 𝑌)) → 𝐺 ∈ USPGraph) | |
5 | uspgredg2v.e | . . . . . . 7 ⊢ 𝐸 = (Edg‘𝐺) | |
6 | 5 | eleq2i 2901 | . . . . . 6 ⊢ (𝑌 ∈ 𝐸 ↔ 𝑌 ∈ (Edg‘𝐺)) |
7 | 6 | biimpi 217 | . . . . 5 ⊢ (𝑌 ∈ 𝐸 → 𝑌 ∈ (Edg‘𝐺)) |
8 | 7 | ad2antrl 724 | . . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑌 ∈ 𝐸 ∧ 𝑁 ∈ 𝑌)) → 𝑌 ∈ (Edg‘𝐺)) |
9 | simprr 769 | . . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑌 ∈ 𝐸 ∧ 𝑁 ∈ 𝑌)) → 𝑁 ∈ 𝑌) | |
10 | 4, 8, 9 | 3jca 1120 | . . 3 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑌 ∈ 𝐸 ∧ 𝑁 ∈ 𝑌)) → (𝐺 ∈ USPGraph ∧ 𝑌 ∈ (Edg‘𝐺) ∧ 𝑁 ∈ 𝑌)) |
11 | uspgredg2vtxeu 26929 | . . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑌 ∈ (Edg‘𝐺) ∧ 𝑁 ∈ 𝑌) → ∃!𝑧 ∈ (Vtx‘𝐺)𝑌 = {𝑁, 𝑧}) | |
12 | uspgredg2v.v | . . . . 5 ⊢ 𝑉 = (Vtx‘𝐺) | |
13 | reueq1 3405 | . . . . 5 ⊢ (𝑉 = (Vtx‘𝐺) → (∃!𝑧 ∈ 𝑉 𝑌 = {𝑁, 𝑧} ↔ ∃!𝑧 ∈ (Vtx‘𝐺)𝑌 = {𝑁, 𝑧})) | |
14 | 12, 13 | ax-mp 5 | . . . 4 ⊢ (∃!𝑧 ∈ 𝑉 𝑌 = {𝑁, 𝑧} ↔ ∃!𝑧 ∈ (Vtx‘𝐺)𝑌 = {𝑁, 𝑧}) |
15 | 11, 14 | sylibr 235 | . . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑌 ∈ (Edg‘𝐺) ∧ 𝑁 ∈ 𝑌) → ∃!𝑧 ∈ 𝑉 𝑌 = {𝑁, 𝑧}) |
16 | riotacl 7120 | . . 3 ⊢ (∃!𝑧 ∈ 𝑉 𝑌 = {𝑁, 𝑧} → (℩𝑧 ∈ 𝑉 𝑌 = {𝑁, 𝑧}) ∈ 𝑉) | |
17 | 10, 15, 16 | 3syl 18 | . 2 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑌 ∈ 𝐸 ∧ 𝑁 ∈ 𝑌)) → (℩𝑧 ∈ 𝑉 𝑌 = {𝑁, 𝑧}) ∈ 𝑉) |
18 | 3, 17 | sylan2b 593 | 1 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑌 ∈ 𝐴) → (℩𝑧 ∈ 𝑉 𝑌 = {𝑁, 𝑧}) ∈ 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ∃!wreu 3137 {crab 3139 {cpr 4559 ‘cfv 6348 ℩crio 7102 Vtxcvtx 26708 Edgcedg 26759 USPGraphcuspgr 26860 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-oadd 8095 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-dju 9318 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-n0 11886 df-xnn0 11956 df-z 11970 df-uz 12232 df-fz 12881 df-hash 13679 df-edg 26760 df-upgr 26794 df-uspgr 26862 |
This theorem is referenced by: uspgredg2v 26933 |
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