![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > upgredg2vtx | Structured version Visualization version GIF version |
Description: For a vertex incident to an edge there is another vertex incident to the edge in a pseudograph. (Contributed by AV, 18-Oct-2020.) (Revised by AV, 5-Dec-2020.) |
Ref | Expression |
---|---|
upgredg.v | ⊢ 𝑉 = (Vtx‘𝐺) |
upgredg.e | ⊢ 𝐸 = (Edg‘𝐺) |
Ref | Expression |
---|---|
upgredg2vtx | ⊢ ((𝐺 ∈ UPGraph ∧ 𝐶 ∈ 𝐸 ∧ 𝐴 ∈ 𝐶) → ∃𝑏 ∈ 𝑉 𝐶 = {𝐴, 𝑏}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | upgredg.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | upgredg.e | . . . 4 ⊢ 𝐸 = (Edg‘𝐺) | |
3 | 1, 2 | upgredg 28924 | . . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐶 ∈ 𝐸) → ∃𝑎 ∈ 𝑉 ∃𝑐 ∈ 𝑉 𝐶 = {𝑎, 𝑐}) |
4 | 3 | 3adant3 1130 | . 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐶 ∈ 𝐸 ∧ 𝐴 ∈ 𝐶) → ∃𝑎 ∈ 𝑉 ∃𝑐 ∈ 𝑉 𝐶 = {𝑎, 𝑐}) |
5 | elpr2elpr 4865 | . . . . . . 7 ⊢ ((𝑎 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉 ∧ 𝐴 ∈ {𝑎, 𝑐}) → ∃𝑏 ∈ 𝑉 {𝑎, 𝑐} = {𝐴, 𝑏}) | |
6 | 5 | 3expia 1119 | . . . . . 6 ⊢ ((𝑎 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉) → (𝐴 ∈ {𝑎, 𝑐} → ∃𝑏 ∈ 𝑉 {𝑎, 𝑐} = {𝐴, 𝑏})) |
7 | eleq2 2817 | . . . . . . 7 ⊢ (𝐶 = {𝑎, 𝑐} → (𝐴 ∈ 𝐶 ↔ 𝐴 ∈ {𝑎, 𝑐})) | |
8 | eqeq1 2731 | . . . . . . . 8 ⊢ (𝐶 = {𝑎, 𝑐} → (𝐶 = {𝐴, 𝑏} ↔ {𝑎, 𝑐} = {𝐴, 𝑏})) | |
9 | 8 | rexbidv 3173 | . . . . . . 7 ⊢ (𝐶 = {𝑎, 𝑐} → (∃𝑏 ∈ 𝑉 𝐶 = {𝐴, 𝑏} ↔ ∃𝑏 ∈ 𝑉 {𝑎, 𝑐} = {𝐴, 𝑏})) |
10 | 7, 9 | imbi12d 344 | . . . . . 6 ⊢ (𝐶 = {𝑎, 𝑐} → ((𝐴 ∈ 𝐶 → ∃𝑏 ∈ 𝑉 𝐶 = {𝐴, 𝑏}) ↔ (𝐴 ∈ {𝑎, 𝑐} → ∃𝑏 ∈ 𝑉 {𝑎, 𝑐} = {𝐴, 𝑏}))) |
11 | 6, 10 | imbitrrid 245 | . . . . 5 ⊢ (𝐶 = {𝑎, 𝑐} → ((𝑎 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉) → (𝐴 ∈ 𝐶 → ∃𝑏 ∈ 𝑉 𝐶 = {𝐴, 𝑏}))) |
12 | 11 | com13 88 | . . . 4 ⊢ (𝐴 ∈ 𝐶 → ((𝑎 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉) → (𝐶 = {𝑎, 𝑐} → ∃𝑏 ∈ 𝑉 𝐶 = {𝐴, 𝑏}))) |
13 | 12 | 3ad2ant3 1133 | . . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐶 ∈ 𝐸 ∧ 𝐴 ∈ 𝐶) → ((𝑎 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉) → (𝐶 = {𝑎, 𝑐} → ∃𝑏 ∈ 𝑉 𝐶 = {𝐴, 𝑏}))) |
14 | 13 | rexlimdvv 3205 | . 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐶 ∈ 𝐸 ∧ 𝐴 ∈ 𝐶) → (∃𝑎 ∈ 𝑉 ∃𝑐 ∈ 𝑉 𝐶 = {𝑎, 𝑐} → ∃𝑏 ∈ 𝑉 𝐶 = {𝐴, 𝑏})) |
15 | 4, 14 | mpd 15 | 1 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐶 ∈ 𝐸 ∧ 𝐴 ∈ 𝐶) → ∃𝑏 ∈ 𝑉 𝐶 = {𝐴, 𝑏}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ∃wrex 3065 {cpr 4626 ‘cfv 6542 Vtxcvtx 28783 Edgcedg 28834 UPGraphcupgr 28867 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7863 df-1st 7985 df-2nd 7986 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-2o 8479 df-oadd 8482 df-er 8716 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-dju 9910 df-card 9948 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-nn 12229 df-2 12291 df-n0 12489 df-xnn0 12561 df-z 12575 df-uz 12839 df-fz 13503 df-hash 14308 df-edg 28835 df-upgr 28869 |
This theorem is referenced by: usgredg2vtx 29006 uspgredg2vtxeu 29007 |
Copyright terms: Public domain | W3C validator |